Mathematics
· Sew the seeds of mathematical logical thinking.
· Greek word that means things that is learned. Studies shapes occurring in space, which may be thought of as a world of points, surfaces, and solids. Studies the properties of different shapes and the relations between them and how to measure them. Clearly defines the ideas that are to be discussed and clearly states the assumptions that can be made. Then, on the basis of both the definitions and assumptions, it forges a chain of proofs, each link in the chain being as strong as any other.
· The secret to life can be found in mathematical thought.
· This is not the old doctrine of art for art sake; it is art for humanity’s sake.
· All that is needed is interest and an un-distracted head.
· “They say, what they say, let them say,” motto of Marischal College, Aberdeen.
· “The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit,” A.N. Whitehead (1925).
· “It is easier to square the circle than to get round a mathematician,” Auguste de Moran.
· Over the centuries, mathematics has become one of the most useful and fascinating divisions of human knowledge.
· The human race’s view of the world has at its foundations the principles of mathematics. From city planning to developments in outer space, to measurements of space and time as pertaining to the relationships of objects within a given space at any given time, it has the ability to be perceived from many simultaneous points of view.
· Through the use of all different types of mathematics, especially the field of geometrics, there have been many applications in the realm of art. Artists have attempted quite successfully to reveal to the world the beautiful experience of mathematics and the underlying basic human need of mathematical form and structure in the world.
· It helps us in many important areas of study and has the power to solve some of the deepest puzzles man must face. It is the study of quantities and relations through the use of numbers and symbols. Arithmetic deals with quantities expressed by numbers. Algebra uses quantities and relations expressed by symbols. Geometry involves quantities associated with figures in space. Trigonometry is concerned with the measurement of angles and the relationships of angles. Analytic geometry applies algebra to geometric studies. And calculus works with pairs of associated quantities and the way one quantity changes in relation to the other.
· In the study of geometry one can follow the gradual unfolding of mathematical thought from its earliest beginnings to present time. The periods of important geometrical interest that have played such a large part in the development of not only mathematics but in the evolution of the human mind started to sprout up during the Greek period (300 BC-300 AD), Cartesian geometry and the Calculus period (1620-1720 AD), non-Euclidean, Projective and Algebraic geometric period (1800 AD onward), and the Foundations period (1880 AD onward). The history of mathematics has a direct bearing upon the concepts of measure and control, which are an integral part in the organization of society from its city planning to the proportions of its doorways and the movement of its people. This is an attempt to analyze, from all angles, a gestalt, if you will, and compare geometrises and other theories as relevant to not only the human experience and perception of space and time in mathematics but also as to its significance and applications in the realm of art.
· Geometry is an important branch of mathematics and consists of finding many interesting and unusual facts about points, lines, and planes. It also includes finding lengths, areas, and volumes of geometric figures. The name geometry comes from Greek words that mean earth and measure. Excavations of ancient cities show that land and buildings were carefully laid out. Egyptian architects and engineers designed and built huge temples and pyramids. They also used geometry to determine the size of their fields, and to find the boundaries of their farms after the yearly floods of the Nile River washed away old landmarks.
Pi
· In what would appear to be at first a simple mathematical function, the division of a circle’s circumference by its diameter, has fascinated scientists, mathematicians and laymen for thousands of years. Derived from the ratio of a circles diameter to its circumference (d divided by c = pi).
· This number, 3.1415 and of to infinity, has served as something of a Mount Everest for computer programmers and mathematicians.
· What is the attraction to this number? Perhaps the fact that a circle is probably the most perfect and simple form known to man. And lying at the heart of it is a specific, unchanging number that also manages to appear in functions of geometry, statistics, and biology, everywhere. It keeps popping its head up, reminding us that it is there and defying us to understand why.
· Very much like the universe itself, the more technologically advanced we become and as our picture of pi grows larger, the more its mysteries grow.
Phi
· Phi = 1.61803399887…
· The Golden Ratio is a geometric proportion discovered in antiquity that turns up in sculpture, botany/leaves/seashells, and galaxies/planetary orbits, and in the thermodynamics of black holes.
· Nature appears to have chosen the logarithmic spiral as one of its favourite shapes.
· The golden ratio is embedded in such geometric forms as the cube, dodecahedron (twelve faces), icosahedrons (twenty faces), octahedron (eight faces), and tetrahedron (four faces).
· Euclid stumbled upon its significance in antiquity.
· Make a square inside a rectangle and the draw a square outside it. The new rectangles are also golden rectangles, and the ratio between the squares is the golden ratio.
· Leonardo Da Vinci, an Italian 15th century artist, inventor, and sculptor, rediscovered the balanced perfection of the golden rectangle and pencilled it into his masterpieces. Connecting a curve through the concentric golden rectangles, you generate the mythical golden spiral.
Full Version: Mathematics and Art
Mathematical American
The ultimate convergence of truth, beauty and science
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." So wrote British philosopher and logician Bertrand Russell nearly 100 years ago. He was not alone in this sentiment. French mathematician Henri Poincaré declared that "the mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." Likewise, Einstein described pure mathematics as "the poetry of logical ideas." Indeed, many a scholar has remarked on the elegance of the science."
"Geometry" means "measure of the earth". In ancient Egypt, from which Greece inherited this study, the Nile would flood its banks each year and when the waters receded the work of re-defining and re-establishing the boundaries was called geometry. The Rhind papyrus,named after the Scottish Egyptologist A Henry Rhind, who purchased it in 1858, was written around 1650 BC by the scribe Ahmes who is copying a document which is 200 years older. It shows a number of practical mathematical problems, several of which are concerned with geometrical shapes.
Thales The history of straight-edge and compass constructions has its roots in Greek mathematics. Thales was born about 624 BC in Miletus, Asia Minor and died about 547 BC in Miletus. Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is supposed to have visited Egypt and brought back the study of geometry.
In many textbooks on the history of mathematics he is credited with five theorems of elementary geometry:-
A circle is bisected by any diameter.
The base angles of an isosceles triangle are equal.
The angles between two intersecting straight lines are equal.
Two triangles are congruent if they have two angles and one side equal.
An angle in a semicircle is a right angle.
Euclid Not a great deal is known about Euclid. It is thought that he was born about 325 BC and died about 265 BC in Alexandria, Egypt. The Thirteen Books of Euclid's Elements was one of the first great works of Mathematics, setting out definitions, postulates and "common notions" of the most basic terms in geometry and then describing in precise order the mathematics that can henceforth be deduced. Not all the book was original, he used the first great Greek mathematician Thales as a major source, but nevertheless the book as served as a model for mathematics writing and research ever since.
Here are the first four definitions from The Elements:
A point is that which has no part.
A line is breadthless length.
The extremities of a line are points.
A straight line is a line which lies evenly with the points on itself
The Elements Is Sir Thomas Heath's translation of The Thirteen Books of Euclid's Elements, published in 1925, a translation of the words which Euclid wrote in 300 BC? In fact it is likely to be a translation of a copy of a copy of a copy of.... traced back to a version written (with alterations and additions) in AD 888 for Arethas, bishop of Caesarea Cappadociae (now in central Turkey). 888 is nearer to now than it is to Euclid's time!
There are three classical problems in Greek geometry that have fascinated mathematicans for centuries:
squaring the circle,
doubling the cube
trisecting an angle.
The first problem was quite popular in 414 BC, and appears in Aristophanes' Birds 1001-1005. Plutarch writes that Anaxagorus worked on the problem while in prison . Many ancient Greek philosophers including Aristotle, Themistius, Philoponus and Simplicius also worked or commented on the problem. Its history has become linked with that of p, the ratio of the circumference of a circle to its diameter. In fact the final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Carl Louis Ferdinand von Lindemann proved that p was transcendental, that is it is not the root of any polynomial equation with rational coefficients.
Trisecting an angle: Archimedes' method
Archimedes Archimedes of Syracuse (born 287 BC, died 212 BC) is crediting with one solution of the third problem of trisecting an angle. Given an angle CAB draw a circle with centre A so that AC and AB are radii of the circle. From C draw a line to cut BA produced at E. Have this line cut the circle at F and have the property that EF is equal to the radius of the circle. Finally draw from A the radius AX of the circle with AX parallel to EC. Then AX trisects angle CAB.
The legacy of Greek mathematics, particularly in the fields of geometry and geometric science, was enormous. From an early period the Greeks formulated the objectives of mathematics not in terms of practical procedures but as a theoretical discipline committed to the development of general propositions and formal demonstrations. The range and diversity of their findings, especially those of the masters of the 3rd century BC, supplied geometers with subject matter for centuries thereafter, even though the tradition that was transmitted into the Middle Ages and Renaissance was incomplete and defective. Art, religion, mysticism, architecture have all been heavily influenced by constructions based on lines and circles - see sacred geometry.
Geometry flourished in many countries over the centuries. The artist and mathematician Albrecht Durer (who studied mathematics and architecture from ancient classics by himself) wrote a Treatise on measurement with compasses and straight edge in 1525, in which he said...
"And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art".
The Measurers
Margarita philosophica, 1583 Plato believed that geometry was an incredible mode of immersing oneself into philosophical contemplation. In fact, the notice above his porch read, "Let no one unversed in geometry enter my doors."
The connection between the practical and philosophical aspects of geometry are illustrated in Margarita philosophica (Basle 1583).
In 1672 Georg Mohr, a Danish geometer, published Euclides Danicus, in which he proved that every compass and straight edge construction can be done with compasses alone. This amazing fact is usually attributed to Lorenzo Mascheroni, an Italian mathematician in the eighteen century and consequently, such constructions are often refered to as Mascheroni constructions.
Descartes The French mathematician René Descartes (1596 - 1650) signalled a break from a purely geometric approach with his publication La Géométrie. His creation of coordinate geometry (also called analytic geometry, or Cartesian geometry), for which Pierre de Fermat must also be credited, has been called the first really great advance in mathematical technique since the Greeks. It laid the foundations not only for modern mathematics, but for modern science as well. It led directly to the creation of the calculus by Newton and Leibniz.
The Thinker, Rodin 1880 "I have come to know that Geometry is at the very heart of feeling, and that each expression of feeling is made by a movement governed by Geometry. Geometry is everywhere in Nature. This is the Concert of Nature."
Auguste Rodin (1840-1917)
Thales The history of straight-edge and compass constructions has its roots in Greek mathematics. Thales was born about 624 BC in Miletus, Asia Minor and died about 547 BC in Miletus. Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is supposed to have visited Egypt and brought back the study of geometry.
In many textbooks on the history of mathematics he is credited with five theorems of elementary geometry:-
A circle is bisected by any diameter.
The base angles of an isosceles triangle are equal.
The angles between two intersecting straight lines are equal.
Two triangles are congruent if they have two angles and one side equal.
An angle in a semicircle is a right angle.
Euclid Not a great deal is known about Euclid. It is thought that he was born about 325 BC and died about 265 BC in Alexandria, Egypt. The Thirteen Books of Euclid's Elements was one of the first great works of Mathematics, setting out definitions, postulates and "common notions" of the most basic terms in geometry and then describing in precise order the mathematics that can henceforth be deduced. Not all the book was original, he used the first great Greek mathematician Thales as a major source, but nevertheless the book as served as a model for mathematics writing and research ever since.
Here are the first four definitions from The Elements:
A point is that which has no part.
A line is breadthless length.
The extremities of a line are points.
A straight line is a line which lies evenly with the points on itself
The Elements Is Sir Thomas Heath's translation of The Thirteen Books of Euclid's Elements, published in 1925, a translation of the words which Euclid wrote in 300 BC? In fact it is likely to be a translation of a copy of a copy of a copy of.... traced back to a version written (with alterations and additions) in AD 888 for Arethas, bishop of Caesarea Cappadociae (now in central Turkey). 888 is nearer to now than it is to Euclid's time!
There are three classical problems in Greek geometry that have fascinated mathematicans for centuries:
squaring the circle,
doubling the cube
trisecting an angle.
The first problem was quite popular in 414 BC, and appears in Aristophanes' Birds 1001-1005. Plutarch writes that Anaxagorus worked on the problem while in prison . Many ancient Greek philosophers including Aristotle, Themistius, Philoponus and Simplicius also worked or commented on the problem. Its history has become linked with that of p, the ratio of the circumference of a circle to its diameter. In fact the final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Carl Louis Ferdinand von Lindemann proved that p was transcendental, that is it is not the root of any polynomial equation with rational coefficients.
Trisecting an angle: Archimedes' method
Archimedes Archimedes of Syracuse (born 287 BC, died 212 BC) is crediting with one solution of the third problem of trisecting an angle. Given an angle CAB draw a circle with centre A so that AC and AB are radii of the circle. From C draw a line to cut BA produced at E. Have this line cut the circle at F and have the property that EF is equal to the radius of the circle. Finally draw from A the radius AX of the circle with AX parallel to EC. Then AX trisects angle CAB.
The legacy of Greek mathematics, particularly in the fields of geometry and geometric science, was enormous. From an early period the Greeks formulated the objectives of mathematics not in terms of practical procedures but as a theoretical discipline committed to the development of general propositions and formal demonstrations. The range and diversity of their findings, especially those of the masters of the 3rd century BC, supplied geometers with subject matter for centuries thereafter, even though the tradition that was transmitted into the Middle Ages and Renaissance was incomplete and defective. Art, religion, mysticism, architecture have all been heavily influenced by constructions based on lines and circles - see sacred geometry.
Geometry flourished in many countries over the centuries. The artist and mathematician Albrecht Durer (who studied mathematics and architecture from ancient classics by himself) wrote a Treatise on measurement with compasses and straight edge in 1525, in which he said...
"And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art".
The Measurers
Margarita philosophica, 1583 Plato believed that geometry was an incredible mode of immersing oneself into philosophical contemplation. In fact, the notice above his porch read, "Let no one unversed in geometry enter my doors."
The connection between the practical and philosophical aspects of geometry are illustrated in Margarita philosophica (Basle 1583).
In 1672 Georg Mohr, a Danish geometer, published Euclides Danicus, in which he proved that every compass and straight edge construction can be done with compasses alone. This amazing fact is usually attributed to Lorenzo Mascheroni, an Italian mathematician in the eighteen century and consequently, such constructions are often refered to as Mascheroni constructions.
Descartes The French mathematician René Descartes (1596 - 1650) signalled a break from a purely geometric approach with his publication La Géométrie. His creation of coordinate geometry (also called analytic geometry, or Cartesian geometry), for which Pierre de Fermat must also be credited, has been called the first really great advance in mathematical technique since the Greeks. It laid the foundations not only for modern mathematics, but for modern science as well. It led directly to the creation of the calculus by Newton and Leibniz.
The Thinker, Rodin 1880 "I have come to know that Geometry is at the very heart of feeling, and that each expression of feeling is made by a movement governed by Geometry. Geometry is everywhere in Nature. This is the Concert of Nature."
Auguste Rodin (1840-1917)
In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the sacred foundation of all things created.
The Golden Ratio
The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental measure that seems to crop up almost everywhere, including crops.
(The golden ratio is about 1.618033988749894848204586834365638117720309180...)
The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage.
The golden ratio has some unique properties and makes some interesting appearances:
phi = phi^2 - 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.
phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2
phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...)))))...
phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry enthusiasts
ratio of segments in 5-pointed star (pentagram) considered sacred to Plato & Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4, phi^5, etc.
phi = apothem to bisected base ratio in the Great Pyramid of Giza
phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144...
(interesting that the 12th term is 12 "raised to a higher power", which appears prominently in a vast collection of metaphysical literature)
The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly
Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect.
The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio. This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!) the construction of the Great Pyramid.
As scholars and artists of eras gone by discovered (such as Leonardo da Vinci, Plato , & Pythagoras), the intentional use of these natural proportions in art of various forms expands our sense of beauty, balance & harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus. The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.
The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental measure that seems to crop up almost everywhere, including crops.
(The golden ratio is about 1.618033988749894848204586834365638117720309180...)
The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage.
The golden ratio has some unique properties and makes some interesting appearances:
phi = phi^2 - 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.
phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2
phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...)))))...
phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry enthusiasts
ratio of segments in 5-pointed star (pentagram) considered sacred to Plato & Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4, phi^5, etc.
phi = apothem to bisected base ratio in the Great Pyramid of Giza
phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144...
(interesting that the 12th term is 12 "raised to a higher power", which appears prominently in a vast collection of metaphysical literature)
The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly
Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect.
The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio. This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!) the construction of the Great Pyramid.
As scholars and artists of eras gone by discovered (such as Leonardo da Vinci, Plato , & Pythagoras), the intentional use of these natural proportions in art of various forms expands our sense of beauty, balance & harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus. The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.
Art and the Golden Ratio
The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms. For years, experts have been finding examples in everything from the facades of ancient Greece to art masterpieces. In recent times, the validity of its link with beauty has been widely debated.
Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity. They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty, proportions, and continuous rhythm.
The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms. For years, experts have been finding examples in everything from the facades of ancient Greece to art masterpieces. In recent times, the validity of its link with beauty has been widely debated.
Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity. They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty, proportions, and continuous rhythm.
Phi and the Egyptian Pyramids?
The Rhind Papyrus of about 1650 BC is one of the oldest mathematical works in existence, giving methods and problems used by the ancient Babylonians and Egyptians. It includes the solution to some problems about pyramids but it does not mention anything about the golden ratio Phi.
The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1·6. It is a matter of debate whether this was "intended" to be the golden section number or not.
According to Elmer Robinson, using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi.
The golden section in The Kings Tomb in Egypt.
How to Find the "Golden Number" without really trying Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pp 406 - 410
Case studies include the Great Pyramid of Cheops and the various theories propounded to explain its dimensions, the golden section in architecture, its use by Le Corbusier and Seurat and in the visual arts. He concludes that several of the works that purport to show Phi was used are, in fact, fallacious and "without any foundation whatever".
The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in Fibonacci Quarterly vol 6, 1968, pages 85 - 87;
has three separate theories (only one of which involves the golden section) which agree quite well with the dimensions as measured in 1880.
A Note on the Geometry of the Great Pyramid Elmer D Robinson in The Fibonacci Quarterly vol 20 (1982) page 343
shows that the theory involving pi fits much better than the one regarding Phi.
George Markowsky's Misconceptions about the Golden ratio in The College Mathematics Journal Vol 23, January 1992, pages 2-19.
This is readable and well presented. You may or may not agree with all that Markowsky says, but this is a very good article that tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not! He has some convincing arguments that Phi does not occur in the measurements of the Egyptian pyramids.
Other names for Phi
Euclid (about 300BC) in his "Elements" calls dividing a line at the 0.6180399.. point dividing a line in the extreme and mean ratio. This later gave rise to the name golden mean.
There are no extant records of the Greek architects' plans for their most famous temples and buildings (such as the Parthenon). So we do not know if they deliberately used the golden section in their architectural plans. The American mathematician Mark Barr used the Greek letter phi to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (also written as Paccioli) wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).
The Rhind Papyrus of about 1650 BC is one of the oldest mathematical works in existence, giving methods and problems used by the ancient Babylonians and Egyptians. It includes the solution to some problems about pyramids but it does not mention anything about the golden ratio Phi.
The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1·6. It is a matter of debate whether this was "intended" to be the golden section number or not.
According to Elmer Robinson, using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi.
The golden section in The Kings Tomb in Egypt.
How to Find the "Golden Number" without really trying Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pp 406 - 410
Case studies include the Great Pyramid of Cheops and the various theories propounded to explain its dimensions, the golden section in architecture, its use by Le Corbusier and Seurat and in the visual arts. He concludes that several of the works that purport to show Phi was used are, in fact, fallacious and "without any foundation whatever".
The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in Fibonacci Quarterly vol 6, 1968, pages 85 - 87;
has three separate theories (only one of which involves the golden section) which agree quite well with the dimensions as measured in 1880.
A Note on the Geometry of the Great Pyramid Elmer D Robinson in The Fibonacci Quarterly vol 20 (1982) page 343
shows that the theory involving pi fits much better than the one regarding Phi.
George Markowsky's Misconceptions about the Golden ratio in The College Mathematics Journal Vol 23, January 1992, pages 2-19.
This is readable and well presented. You may or may not agree with all that Markowsky says, but this is a very good article that tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not! He has some convincing arguments that Phi does not occur in the measurements of the Egyptian pyramids.
Other names for Phi
Euclid (about 300BC) in his "Elements" calls dividing a line at the 0.6180399.. point dividing a line in the extreme and mean ratio. This later gave rise to the name golden mean.
There are no extant records of the Greek architects' plans for their most famous temples and buildings (such as the Parthenon). So we do not know if they deliberately used the golden section in their architectural plans. The American mathematician Mark Barr used the Greek letter phi to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (also written as Paccioli) wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).
The Golden Mean
The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.
You can find it in a number of places:
Number Series
If you start with the numbers 0 and 1, and make a list in which each new number is the sum of the previous two, you get a list like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... to infinity-->
This is called a 'Fibonacci series'.
If you then take the ratio of any two sequential numbers in this series, you'll find that it falls into an increasingly narrow range:
1/0 = Whoa! That one doesn't count.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...
and so on, with each addition coming ever closer to multiplying by some as-yet-undetermined number.
The number that this ratio is oscillating around is phi (1.6180339887499...). It's interesting to note that the ratio 21/13 differs from phi by less than .003, and 34/21 by only about .001 (less than 1/10 of one percent!), thus providing our less technically-advanced ancestors an easy way to derive phi on a large scale in the real world with a high degree of precision.
Geometry
If you have a rectangle whose sides are related by phi (say, for instance, 13 x 8), that rectangle is said to be a Golden Rectangle. It has the interesting property that, if you create a new rectangle by 'swinging' the long side around one of its ends to create a new long side, then that new rectangle is also Golden.
In the case of our 13 x 8 rectangle, the new rectangle will be (13 + 8 =) 21 x 13. You can see this is the same thing that's going on in our number list, but when you discover it geometrically it looks downright magical. If you start with a square (1 x 1) and start swinging sides to make rectangles, you wind up with Golden rectangles without even trying. Here's the list, in case it isn't obvious:
1 x 1
2 x 1
3 x 2
5 x 3
8 x 5
13 x 8
21 x 13
34 x 21
and so on, with, again, each addition coming ever closer to multiplying by phi.
Ancient architecture is filled with Golden rectangles.
When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively. This division of a straight line into a phi proportion is what is actually meant by the term 'Golden Section'.
Pure Math
Proportion is the relationship of the size of two things.
Arithmetic proportion exists when a quantity is changed by adding some amount.
Geometric proportion exists when a quantity is changed by multiplying by some amount.
Phi possesses both qualities.
If you study the Fibonacci series and the Golden Rectangle carefully, you will eventually realize that
phi + 1 = phi * phi.
Consider: suppose that you start with a Golden rectangle having a short side one unit long. Since the long side of a Golden rectangle equals the short side multiplied by phi, the long side of our rectangle is one times phi. or simply phi.
Now suppose that you swing the long side to make a new Golden rectangle. The short side of the new rectangle is, of course, phi units long, and the long side is that times phi, or phi * phi. This describes a Geometric proportion.
But we also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or phi + 1. This describes an Arithmetic proportion.
Since these two expressions describe the same thing, they are equivalent, and so
phi + 1 = phi * phi.
(Now that we know this, we can discover the exact value of phi. )
The resulting proportion is both arithmetic and geometric. It is thus perfect proportion; you could think of it as the place on some imaginary graph where the curved line of multiplication crosses the straight line of addition.
Nature
In pure mathematics, an increase in size can be any imaginable number, even one like e or pi. But in the world of nature, things always grow by adding some unit, even if the unit is as small as a molecule. So it's not surprising that phi turns out to be an ideal rate of growth for things which grow by adding some quantity.
Some examples:
The Nautilus shell (Nautilus pompilius) grows larger on each spiral by phi.
The sunflower has 55 clockwise spirals overlaid on either 34 or 89 counterclockwise spirals, a phi proportion.
The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.
You can find it in a number of places:
Number Series
If you start with the numbers 0 and 1, and make a list in which each new number is the sum of the previous two, you get a list like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... to infinity-->
This is called a 'Fibonacci series'.
If you then take the ratio of any two sequential numbers in this series, you'll find that it falls into an increasingly narrow range:
1/0 = Whoa! That one doesn't count.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...
and so on, with each addition coming ever closer to multiplying by some as-yet-undetermined number.
The number that this ratio is oscillating around is phi (1.6180339887499...). It's interesting to note that the ratio 21/13 differs from phi by less than .003, and 34/21 by only about .001 (less than 1/10 of one percent!), thus providing our less technically-advanced ancestors an easy way to derive phi on a large scale in the real world with a high degree of precision.
Geometry
If you have a rectangle whose sides are related by phi (say, for instance, 13 x 8), that rectangle is said to be a Golden Rectangle. It has the interesting property that, if you create a new rectangle by 'swinging' the long side around one of its ends to create a new long side, then that new rectangle is also Golden.
In the case of our 13 x 8 rectangle, the new rectangle will be (13 + 8 =) 21 x 13. You can see this is the same thing that's going on in our number list, but when you discover it geometrically it looks downright magical. If you start with a square (1 x 1) and start swinging sides to make rectangles, you wind up with Golden rectangles without even trying. Here's the list, in case it isn't obvious:
1 x 1
2 x 1
3 x 2
5 x 3
8 x 5
13 x 8
21 x 13
34 x 21
and so on, with, again, each addition coming ever closer to multiplying by phi.
Ancient architecture is filled with Golden rectangles.
When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively. This division of a straight line into a phi proportion is what is actually meant by the term 'Golden Section'.
Pure Math
Proportion is the relationship of the size of two things.
Arithmetic proportion exists when a quantity is changed by adding some amount.
Geometric proportion exists when a quantity is changed by multiplying by some amount.
Phi possesses both qualities.
If you study the Fibonacci series and the Golden Rectangle carefully, you will eventually realize that
phi + 1 = phi * phi.
Consider: suppose that you start with a Golden rectangle having a short side one unit long. Since the long side of a Golden rectangle equals the short side multiplied by phi, the long side of our rectangle is one times phi. or simply phi.
Now suppose that you swing the long side to make a new Golden rectangle. The short side of the new rectangle is, of course, phi units long, and the long side is that times phi, or phi * phi. This describes a Geometric proportion.
But we also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or phi + 1. This describes an Arithmetic proportion.
Since these two expressions describe the same thing, they are equivalent, and so
phi + 1 = phi * phi.
(Now that we know this, we can discover the exact value of phi. )
The resulting proportion is both arithmetic and geometric. It is thus perfect proportion; you could think of it as the place on some imaginary graph where the curved line of multiplication crosses the straight line of addition.
Nature
In pure mathematics, an increase in size can be any imaginable number, even one like e or pi. But in the world of nature, things always grow by adding some unit, even if the unit is as small as a molecule. So it's not surprising that phi turns out to be an ideal rate of growth for things which grow by adding some quantity.
Some examples:
The Nautilus shell (Nautilus pompilius) grows larger on each spiral by phi.
The sunflower has 55 clockwise spirals overlaid on either 34 or 89 counterclockwise spirals, a phi proportion.
“The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit,” A.N. Whitehead (1925).
Shouldn't "creation" have been "discovery"? What do you think?
I like this topic, can we have more please Trip.
Shouldn't "creation" have been "discovery"? What do you think?
I like this topic, can we have more please Trip.
discovery before creation?
or
creation before discover?.
I suppose its all in how you look at it and, Hey Hey, that is definately one way to look at it.
or
creation before discover?.
I suppose its all in how you look at it and, Hey Hey, that is definately one way to look at it.
Kazemir Malevich – Russian Suprematism
·Was one of the pioneers of geometrical abstraction.
·Was recognized as an artist who works with numbers.
·He imbued geometry with meaning.
·The long history of mathematics in art is a tradition and a convention.
·He had contemporary theories of the 4 dimensions of time and space and their eternal relationship that Euclid laid out in his ‘three-plus-one’ dimensionality theory: forward-back, left-right, and before-after.
·He thought that the proportions of canvasses were important.
·He was concerned with geometry since the earliest days in his career.
·The square was the Suprematist building block.
·He adopted the square, a basic element of constructivist art, as the format for paintings, where neither the vertical nor the horizontal dominated the painting.
·The square expanded equally in all directions.
·Reason in art could be found in a box of square dimensions with the secret of beauty being in the harmony of those dimensions.
·As humanity developed, human perception will develop into a four dimensional framework.
·Time is only an image of fourth dimensional space and each step into a new dimension appeared at first to be movement in time but became a dimension in space. All perceptions are filtered through the body and regarded from a plastic point of view, the fourth dimension appears to spring from the three known dimensions that represent the immensity of space externalizes itself in all directions at any given moment. It is space itself, the dimension of the infinite.
·The fourth dimension endows objects with plasticity and gives objects their right proportions on the whole. Dynamic form is a species of the fourth dimension, both in painting and in sculpture, which cannot exist perfectly without the complete concurrence of those three dimensions that determine volume: height, width, and depth. Art considered under these terms function as contemplative images that help in attaining an elevated level of consciousness with an evolutionary state of mind.
·Was one of the pioneers of geometrical abstraction.
·Was recognized as an artist who works with numbers.
·He imbued geometry with meaning.
·The long history of mathematics in art is a tradition and a convention.
·He had contemporary theories of the 4 dimensions of time and space and their eternal relationship that Euclid laid out in his ‘three-plus-one’ dimensionality theory: forward-back, left-right, and before-after.
·He thought that the proportions of canvasses were important.
·He was concerned with geometry since the earliest days in his career.
·The square was the Suprematist building block.
·He adopted the square, a basic element of constructivist art, as the format for paintings, where neither the vertical nor the horizontal dominated the painting.
·The square expanded equally in all directions.
·Reason in art could be found in a box of square dimensions with the secret of beauty being in the harmony of those dimensions.
·As humanity developed, human perception will develop into a four dimensional framework.
·Time is only an image of fourth dimensional space and each step into a new dimension appeared at first to be movement in time but became a dimension in space. All perceptions are filtered through the body and regarded from a plastic point of view, the fourth dimension appears to spring from the three known dimensions that represent the immensity of space externalizes itself in all directions at any given moment. It is space itself, the dimension of the infinite.
·The fourth dimension endows objects with plasticity and gives objects their right proportions on the whole. Dynamic form is a species of the fourth dimension, both in painting and in sculpture, which cannot exist perfectly without the complete concurrence of those three dimensions that determine volume: height, width, and depth. Art considered under these terms function as contemplative images that help in attaining an elevated level of consciousness with an evolutionary state of mind.
Al Held – Formalism/Contemporary Geometrics, Post-painterly Abstraction
·Wants to revive non-objective paintings.
·Makes wholly abstract images that look somehow representational.
·He crafts a pictorial domain in which he is architect, set designer, utopian visionary and ultimately, magician.
·Employed interlocking geometric forms.
·He was uncompromisingly abstract.
·Contained no trace of the artist’s own feelings of individuality.
·Valued social dialogue.
·Colour field painter.
·Valued the use of universal symbols.
·Revelations of ego in paintings (abstract expressionism).
·Childs developing spatial perception (Jean Piaget).
·A comprehension of space in perspective as the individual develops a point of view.
·Shallow layered space.
·An abstract understanding of space in three dimensions.
·Gradual evolution toward a more conspicuously geometric image.
·Employed both horizontal and vertical formats.
·Wanted to give gesture structure and logic.
·Implied elemental geometries – triangles, squares, circles, rectangles, etc.
·Which alluded to their-own manifestly physical constitution. depicting neither landscape nor figure, but depicting both.
·Self-awareness, impulse and order, fabricated space, bright, primary colours, diffused forms, implied depth, planes of colour, coloured shapes, stacked depth, secondary hints, coexist tentatively, forced colours together, biomorphic images, juxtaposition of colours, illusory space, visionary colour sense.
·Bringing order to things.
·Adjusting form, colour, and gesture toward a goal of balanced yet highly active iconography.
·Forms are compounded and superimposed on one another.
·Showing vestiges of previous shapes.
·Increasing idiosyncratic bend.
·Complicated almost to the point of contradiction.
·Symbols come from continuing fascination with psychology.
·Evince multiple under-layers.
·By name a piece of art becomes something.
·Frontal and lateral spatial effects.
·Employ illusions.
·Compress the paintings atmosphere.
·Static space.
·You might call this the geometric surreal, Candy Land for grown-ups (children permitted). This is retinal art to the max, curiously uplifting and enormously fun to look at, a trip without drugs.
·Wants to revive non-objective paintings.
·Makes wholly abstract images that look somehow representational.
·He crafts a pictorial domain in which he is architect, set designer, utopian visionary and ultimately, magician.
·Employed interlocking geometric forms.
·He was uncompromisingly abstract.
·Contained no trace of the artist’s own feelings of individuality.
·Valued social dialogue.
·Colour field painter.
·Valued the use of universal symbols.
·Revelations of ego in paintings (abstract expressionism).
·Childs developing spatial perception (Jean Piaget).
·A comprehension of space in perspective as the individual develops a point of view.
·Shallow layered space.
·An abstract understanding of space in three dimensions.
·Gradual evolution toward a more conspicuously geometric image.
·Employed both horizontal and vertical formats.
·Wanted to give gesture structure and logic.
·Implied elemental geometries – triangles, squares, circles, rectangles, etc.
·Which alluded to their-own manifestly physical constitution. depicting neither landscape nor figure, but depicting both.
·Self-awareness, impulse and order, fabricated space, bright, primary colours, diffused forms, implied depth, planes of colour, coloured shapes, stacked depth, secondary hints, coexist tentatively, forced colours together, biomorphic images, juxtaposition of colours, illusory space, visionary colour sense.
·Bringing order to things.
·Adjusting form, colour, and gesture toward a goal of balanced yet highly active iconography.
·Forms are compounded and superimposed on one another.
·Showing vestiges of previous shapes.
·Increasing idiosyncratic bend.
·Complicated almost to the point of contradiction.
·Symbols come from continuing fascination with psychology.
·Evince multiple under-layers.
·By name a piece of art becomes something.
·Frontal and lateral spatial effects.
·Employ illusions.
·Compress the paintings atmosphere.
·Static space.
·You might call this the geometric surreal, Candy Land for grown-ups (children permitted). This is retinal art to the max, curiously uplifting and enormously fun to look at, a trip without drugs.
John Ruskin – 1819-1900 AD
·His ideal was of an economy in which every industry would be an art, and every product would proudly express the personality and artistry of the artisan.
·Visual art is the quintessential manifestation of human endeavour.
·Visual culture is an enduring embodiment of public and private virtue.
·The teaching of art is the teaching of all things.
·Art is a product of human genius and divine inspiration, as objects of spiritual devotion and national pride, as instruments of emulation and instruction.
·“The moral and spiritual substance of human history, as the palpable texture of human culture.”
·In criticizing, reject nothing, select nothing, and scorn nothing, really care, really look, evolve into some sort of crazed monster.
·The rise of industrial modernity whose agency befouled the air, befouled the waters and regimented thought, thus making clear sight impossible.
·In criticizing, we should see what is before our eyes as clearly and as innocently as possible before passing judgement on it.
·Revere the intricate, irregular precision of tiny things, distant prospects and transient atmospheres, clearly seen.
·Loathe perfection and order.
·Never be satisfied that you’ve handled a subject properly until you’ve contradicted yourself at least three times.
·Carry words on the tip of your tongue in a condition of intricate verbal readiness, as a vehicle for his passion.
·Art is pure and flawless workmanship.
·Aesthetic denoted the consumers rather than the producer’s stand point.
·Skilful action is on the side of the artist who creates. Taste is on the side of the consumer.
·Perfection in execution cannot be measured or defined in terms of execution; it implies those who perceive and enjoy the product that is executed.
·Craftsmanship to be artistic in the final sense must love; it must care deeply for the subject matter upon which skill is exercised.
·Artistic perception of own works are colourless and cold recognition of what has been done.
·The aesthetic experience in its limited sense is thus seen to be inherently connected with the experience of making.
·His ideal was of an economy in which every industry would be an art, and every product would proudly express the personality and artistry of the artisan.
·Visual art is the quintessential manifestation of human endeavour.
·Visual culture is an enduring embodiment of public and private virtue.
·The teaching of art is the teaching of all things.
·Art is a product of human genius and divine inspiration, as objects of spiritual devotion and national pride, as instruments of emulation and instruction.
·“The moral and spiritual substance of human history, as the palpable texture of human culture.”
·In criticizing, reject nothing, select nothing, and scorn nothing, really care, really look, evolve into some sort of crazed monster.
·The rise of industrial modernity whose agency befouled the air, befouled the waters and regimented thought, thus making clear sight impossible.
·In criticizing, we should see what is before our eyes as clearly and as innocently as possible before passing judgement on it.
·Revere the intricate, irregular precision of tiny things, distant prospects and transient atmospheres, clearly seen.
·Loathe perfection and order.
·Never be satisfied that you’ve handled a subject properly until you’ve contradicted yourself at least three times.
·Carry words on the tip of your tongue in a condition of intricate verbal readiness, as a vehicle for his passion.
·Art is pure and flawless workmanship.
·Aesthetic denoted the consumers rather than the producer’s stand point.
·Skilful action is on the side of the artist who creates. Taste is on the side of the consumer.
·Perfection in execution cannot be measured or defined in terms of execution; it implies those who perceive and enjoy the product that is executed.
·Craftsmanship to be artistic in the final sense must love; it must care deeply for the subject matter upon which skill is exercised.
·Artistic perception of own works are colourless and cold recognition of what has been done.
·The aesthetic experience in its limited sense is thus seen to be inherently connected with the experience of making.
A New Era of Thought, Recognition of the 4th Dimension (1888) Charles Howard Hinton
· Test the limits of expressiveness in painting, as it is traditionally understood.
· Elevated level of consciousness with an evolutionary state of mind.
· Function as contemplative images, aid in attaining the zaum state.
· Teach about space and emptiness.
· All perceptions are filtered through our body.
· In art we already have the 1st experience of the language of the future. Art is the avant-garde of psychological evolution. It gives a glimpse, an impression of future mental capacities.
· Reason has now imprisoned art in a box of square dimensions.
· Regarded from a plastic point of view, the 4th dimension appears to spring forth from the three known dimensions and it represents the immensity of space externalizing itself in all directions at any given moment. It is space itself, the dimension of the infinite. The 4th dimension endows objects with plasticity.
· Dynamic form is a species of 4th dimension, both in painting and in sculpture, which cannot exist perfectly without the complete concurrence of the three known dimension that determine volume: height, width, and depth, Boccioni.
· Only motion crystallizes outward appearance into a single whole. Objectiveness is changed by motion. A speeding train fuses the separate-ness of its cars into a compact mass. Only in motion does vastness reside. The faster you move near a thick garden lattice the more clearly you see the general masses behind it. You can see in different ways simultaneously. When at last we shall rush past objectiveness we shall probably see the totality of the whole world, M. Matiushian.
· The future social order will take its shape from compatibility with the theoretical principles in art.
· The psychology of geometrical art.
· Mathematical truths are immortal, imperishable; the same yesterday, today, and forever.
· Reflect the psychical obsessions of the society, which produced it.
· It is said that analyzing pleasure or beauty destroys it.
· There is pleasure in looking and pleasure in being looked at.
· Recognition of yourself is joyous in that you imagine your mirror image to be more complete, more perfect than you experience in your own body.
· The first articulation of the I.
· Pursue aims in indifference to perceptual reality, and motivate eroticised phantasmagoria that affect the subjects perception of the world to make a mockery of empirical objectivity.
· Charismatic models command considerable attention, whereas persons low in interpersonal attractiveness tend to be ignored.
· People expect a great work of art to develop their minds by expressing and clarifying the best thoughts of great men and women of their own and earlier times.
· Provoke thought, I am diversity, I am multicultural, I will not stand for/in mediocrity.
· Manipulate the environment, attain an ability to predict and control behaviour.
· The value of a society is its ability to survive.
· Quest for essential reality, transport into endless emptiness, where you sense around you the creative points of the universe.
· Open eyes and teach the crowd to see the great beauty of the world concealed from them.
Hinton, Charles Howard (1853-1907)
An English-born mathematician best known for his writings and inventions aimed at helping to visualize the fourth dimension; he may also have coined the name tesseract for the four-dimensional analogue of a cube. Hinton matriculated at Oxford and continued to study there, earning a B.A. (1877) and an M.A. (1886), while he also taught, first at Cheltenham Ladies' School and then, from 1880 to 1886, at Uppingham School. At this time, another teacher at Uppingham was Howard Candler, who was a friend of Edwin Abbott and thus provides a possible link between these two explorers of other dimensions. In the early 1880s Hinton published a series of pamphlets starting with "What is the Fourth Dimension?" and "A Plane World" (a contemporary of Abbott's Flatland), which were reprinted in the two-volume Scientific Romances (1884). Hinton's descriptions owed much to the mathematical models of William Clifford, whose theories about 4-d spaces were then in vogue. But Hinton went much further in his attempts to break free of three-dimensional thought. He devised an elaborate set of small colored cubes to represent the various cross sections of a tesseract and then memorized the cubes and their many possible orientations in order to gain a window on the fourth dimension.
At the time he was teaching in England, Hinton married Mary Everest Boole, the eldest daughter of George Boole, the founder of mathematical logic. Regrettably, he also married a Maud Wheldon and was tried at the Old Bailey in London for bigamy. After serving a day in prison for the offence, he fled with his (first) family to Japan, where he taught for some years, before taking up a post at Princeton University. There, in 1897, he designed a species of baseball gun which, with the help of gunpowder charges, would shoot out balls at speeds of 40 to 70 miles per hour. It was used by the Princeton Nine for several seasons before being abandoned by the players in fear of their lives.
After a brief spell at the University of Minnesota, Hinton joined the Naval Observatory in Washington, D.C. At the same time, he developed more rigorously his ideas on the fourth dimension and presented his results before the Washington Philosophical Society in 1902. What would prove, Hinton asked, the existence of a real fourth spatial dimension? He offered three possibilities, two of which, involving a specific molecular structure and a particular case of electrical induction, have since been explained by science in more mundane ways. However, Hintons other case, to do with right and left handedness remains open because there are instances of right and left-handedness in nature, such as the spin of elementary particles, to which his example could be applied. In any event, Hintons final assessment that we can only regard a four-dimensional space as possible if three-dimensional mechanics fails to explain known physical phenomena still rings true today. See also Boole (Stott), Alicia.
· Test the limits of expressiveness in painting, as it is traditionally understood.
· Elevated level of consciousness with an evolutionary state of mind.
· Function as contemplative images, aid in attaining the zaum state.
· Teach about space and emptiness.
· All perceptions are filtered through our body.
· In art we already have the 1st experience of the language of the future. Art is the avant-garde of psychological evolution. It gives a glimpse, an impression of future mental capacities.
· Reason has now imprisoned art in a box of square dimensions.
· Regarded from a plastic point of view, the 4th dimension appears to spring forth from the three known dimensions and it represents the immensity of space externalizing itself in all directions at any given moment. It is space itself, the dimension of the infinite. The 4th dimension endows objects with plasticity.
· Dynamic form is a species of 4th dimension, both in painting and in sculpture, which cannot exist perfectly without the complete concurrence of the three known dimension that determine volume: height, width, and depth, Boccioni.
· Only motion crystallizes outward appearance into a single whole. Objectiveness is changed by motion. A speeding train fuses the separate-ness of its cars into a compact mass. Only in motion does vastness reside. The faster you move near a thick garden lattice the more clearly you see the general masses behind it. You can see in different ways simultaneously. When at last we shall rush past objectiveness we shall probably see the totality of the whole world, M. Matiushian.
· The future social order will take its shape from compatibility with the theoretical principles in art.
· The psychology of geometrical art.
· Mathematical truths are immortal, imperishable; the same yesterday, today, and forever.
· Reflect the psychical obsessions of the society, which produced it.
· It is said that analyzing pleasure or beauty destroys it.
· There is pleasure in looking and pleasure in being looked at.
· Recognition of yourself is joyous in that you imagine your mirror image to be more complete, more perfect than you experience in your own body.
· The first articulation of the I.
· Pursue aims in indifference to perceptual reality, and motivate eroticised phantasmagoria that affect the subjects perception of the world to make a mockery of empirical objectivity.
· Charismatic models command considerable attention, whereas persons low in interpersonal attractiveness tend to be ignored.
· People expect a great work of art to develop their minds by expressing and clarifying the best thoughts of great men and women of their own and earlier times.
· Provoke thought, I am diversity, I am multicultural, I will not stand for/in mediocrity.
· Manipulate the environment, attain an ability to predict and control behaviour.
· The value of a society is its ability to survive.
· Quest for essential reality, transport into endless emptiness, where you sense around you the creative points of the universe.
· Open eyes and teach the crowd to see the great beauty of the world concealed from them.
Hinton, Charles Howard (1853-1907)
An English-born mathematician best known for his writings and inventions aimed at helping to visualize the fourth dimension; he may also have coined the name tesseract for the four-dimensional analogue of a cube. Hinton matriculated at Oxford and continued to study there, earning a B.A. (1877) and an M.A. (1886), while he also taught, first at Cheltenham Ladies' School and then, from 1880 to 1886, at Uppingham School. At this time, another teacher at Uppingham was Howard Candler, who was a friend of Edwin Abbott and thus provides a possible link between these two explorers of other dimensions. In the early 1880s Hinton published a series of pamphlets starting with "What is the Fourth Dimension?" and "A Plane World" (a contemporary of Abbott's Flatland), which were reprinted in the two-volume Scientific Romances (1884). Hinton's descriptions owed much to the mathematical models of William Clifford, whose theories about 4-d spaces were then in vogue. But Hinton went much further in his attempts to break free of three-dimensional thought. He devised an elaborate set of small colored cubes to represent the various cross sections of a tesseract and then memorized the cubes and their many possible orientations in order to gain a window on the fourth dimension.
At the time he was teaching in England, Hinton married Mary Everest Boole, the eldest daughter of George Boole, the founder of mathematical logic. Regrettably, he also married a Maud Wheldon and was tried at the Old Bailey in London for bigamy. After serving a day in prison for the offence, he fled with his (first) family to Japan, where he taught for some years, before taking up a post at Princeton University. There, in 1897, he designed a species of baseball gun which, with the help of gunpowder charges, would shoot out balls at speeds of 40 to 70 miles per hour. It was used by the Princeton Nine for several seasons before being abandoned by the players in fear of their lives.
After a brief spell at the University of Minnesota, Hinton joined the Naval Observatory in Washington, D.C. At the same time, he developed more rigorously his ideas on the fourth dimension and presented his results before the Washington Philosophical Society in 1902. What would prove, Hinton asked, the existence of a real fourth spatial dimension? He offered three possibilities, two of which, involving a specific molecular structure and a particular case of electrical induction, have since been explained by science in more mundane ways. However, Hintons other case, to do with right and left handedness remains open because there are instances of right and left-handedness in nature, such as the spin of elementary particles, to which his example could be applied. In any event, Hintons final assessment that we can only regard a four-dimensional space as possible if three-dimensional mechanics fails to explain known physical phenomena still rings true today. See also Boole (Stott), Alicia.
Art & Physics – Parallel Visions in Space, Time, and Light (1991) – Leonard Shlain
· The artist employs illusion and metaphor.
· The physicist uses number and equation.
· Revolutionary art and visionary physics are both investigations into the nature of reality.
· Metaphor – is a mental innovation arising out of a unique combination of feeling states and images. It allows a leap across a chasm from one thought to the next. They have several different levels of meaning simultaneously perceived and supply a plasticity to language without which communication would be less interesting. They’re cousins are similes, analogies, allegories, proverbs, and parables and each in their own way allow multiple simultaneous means of interpreting one single set of words.
· The most compelling combination of metaphor and image is art. The artist frequently uses visual metaphor to transport a viewer from a neutral affective state to complex feeling state, for example, awe.
· When art is successful in metaphorically bearing us across and above, there are no transitions. It is an all at once quantum jump. When this happens, we somehow know we are in the presence of great art.
· The artist often juxtaposes different features of reality and synthesizes them.
· Art illuminates, imitates, and interprets reality.
· Some artists create a language of symbols for things, which there are yet to be words.
· Radical innovators in art embody the preverbal stages of new concepts that will eventually change a civilization.
· Begins with the assimilation of unfamiliar images.
· Abstract ideas give rise to a descriptive language.
· The brains highest function is thinking.
· To affix a name to something is the beginning of control over it.
· Revolutionary art is the preverbal stage of a civilization first contending with a major change in its perception of the world.
· Visionary art alerts the other members that a conceptual shift is about to occur in the thought system used to perceive the world.
· Volume, space, mass, force, light, colour, tension, relationship, and density.
· Artists often choose signs, symbols, and allegories.
· Art offers perceptions of reality that erases linguistic and national boundaries.
· The inner eye of imagination and the external world of things.
· Objective reality is seen through the filter of each person’s temperament.
· Opinion – inner spectral vision unique to each individual of how the world works.
· Consensus – when a critical mass of people agrees on one viewpoint.
· Paradigm – when an entire civilization reaches a consensus about how the world works (Priori Postulates). 2 + 2 will always equal 4, all right angles are equal.
· Euclidean space and Aristotelian time have formed a paradigm that has been remarkably enduring.
· When the time comes to change a paradigm the artist is most likely to be in the forefront.
· Subjectivity is the creative wellspring of all art.
· Begin with wonder and end in wisdom.
· Art generally anticipates scientific revisions of reality.
· The artist employs illusion and metaphor.
· The physicist uses number and equation.
· Revolutionary art and visionary physics are both investigations into the nature of reality.
· Metaphor – is a mental innovation arising out of a unique combination of feeling states and images. It allows a leap across a chasm from one thought to the next. They have several different levels of meaning simultaneously perceived and supply a plasticity to language without which communication would be less interesting. They’re cousins are similes, analogies, allegories, proverbs, and parables and each in their own way allow multiple simultaneous means of interpreting one single set of words.
· The most compelling combination of metaphor and image is art. The artist frequently uses visual metaphor to transport a viewer from a neutral affective state to complex feeling state, for example, awe.
· When art is successful in metaphorically bearing us across and above, there are no transitions. It is an all at once quantum jump. When this happens, we somehow know we are in the presence of great art.
· The artist often juxtaposes different features of reality and synthesizes them.
· Art illuminates, imitates, and interprets reality.
· Some artists create a language of symbols for things, which there are yet to be words.
· Radical innovators in art embody the preverbal stages of new concepts that will eventually change a civilization.
· Begins with the assimilation of unfamiliar images.
· Abstract ideas give rise to a descriptive language.
· The brains highest function is thinking.
· To affix a name to something is the beginning of control over it.
· Revolutionary art is the preverbal stage of a civilization first contending with a major change in its perception of the world.
· Visionary art alerts the other members that a conceptual shift is about to occur in the thought system used to perceive the world.
· Volume, space, mass, force, light, colour, tension, relationship, and density.
· Artists often choose signs, symbols, and allegories.
· Art offers perceptions of reality that erases linguistic and national boundaries.
· The inner eye of imagination and the external world of things.
· Objective reality is seen through the filter of each person’s temperament.
· Opinion – inner spectral vision unique to each individual of how the world works.
· Consensus – when a critical mass of people agrees on one viewpoint.
· Paradigm – when an entire civilization reaches a consensus about how the world works (Priori Postulates). 2 + 2 will always equal 4, all right angles are equal.
· Euclidean space and Aristotelian time have formed a paradigm that has been remarkably enduring.
· When the time comes to change a paradigm the artist is most likely to be in the forefront.
· Subjectivity is the creative wellspring of all art.
· Begin with wonder and end in wisdom.
· Art generally anticipates scientific revisions of reality.
The Universe and the Teacup: The Mathematics of Truth and Beauty (?) – K.C. Cole
· Probability – rich in philosophical interest and of the highest scientific importance. But it is also baffling.
· Chance – excuses us from all responsibility. What happens by chance is exempt from rhyme or reason.
· Phillip Morrison, physicist, “Chance and cause have been wonderfully married into a point of view in which precise pattern governs potential events, and yet in which the variety of potentialities allows the full growth of that novelty which we know to govern the world we live in.”
· Henri Ponticaire, “Fortuitous phenomena are, by definition, those whose laws we do not know.”
· James R. Newman, “The whole concept of chance is only a euphemism for ignorance.”
· Order within disorder.
· Disorder is far more probable than order, so disorder happens; disorder is an opportunistic disease, and it is highly predictable in the long run.
· All tend toward a state of inevitable disintegration.
· It is a probability that makes time a one-way street.
· To be a cause generally means to make a difference: in the hypothetical situation where the cause is absent, the effect would not have occurred.
· It takes energy to breath order into chaos.
· Cause and correlation, where there is smoke, there is fire.”
· We are primates evolved to gather fruit in the forest and when possible to reproduce, and I think it’s marvellous that we can do what we do. But we have to exercise almost intolerable discipline to not jump to conclusions. There might be a banana behind that leaf, or it might be a tiger’s tail. The one who makes the discrimination best and moves fastest gets away from the tiger. So this leaping to conclusions is a good strategy given that the choices are simple and nothing complicated is going on. But at the level of major social policy choices, jumping to conclusions is a serious concern.
· Measurement is the cornerstone of knowledge. It allows us to compare things with other things and to quantify relationships.
· Mine is bigger than yours amounts to a mathematical statement (M > Y).
· Everything boils down to quantity, if only because everything ultimately boils down to particles and forces.
· Answers depend on the questions.
· Probability – rich in philosophical interest and of the highest scientific importance. But it is also baffling.
· Chance – excuses us from all responsibility. What happens by chance is exempt from rhyme or reason.
· Phillip Morrison, physicist, “Chance and cause have been wonderfully married into a point of view in which precise pattern governs potential events, and yet in which the variety of potentialities allows the full growth of that novelty which we know to govern the world we live in.”
· Henri Ponticaire, “Fortuitous phenomena are, by definition, those whose laws we do not know.”
· James R. Newman, “The whole concept of chance is only a euphemism for ignorance.”
· Order within disorder.
· Disorder is far more probable than order, so disorder happens; disorder is an opportunistic disease, and it is highly predictable in the long run.
· All tend toward a state of inevitable disintegration.
· It is a probability that makes time a one-way street.
· To be a cause generally means to make a difference: in the hypothetical situation where the cause is absent, the effect would not have occurred.
· It takes energy to breath order into chaos.
· Cause and correlation, where there is smoke, there is fire.”
· We are primates evolved to gather fruit in the forest and when possible to reproduce, and I think it’s marvellous that we can do what we do. But we have to exercise almost intolerable discipline to not jump to conclusions. There might be a banana behind that leaf, or it might be a tiger’s tail. The one who makes the discrimination best and moves fastest gets away from the tiger. So this leaping to conclusions is a good strategy given that the choices are simple and nothing complicated is going on. But at the level of major social policy choices, jumping to conclusions is a serious concern.
· Measurement is the cornerstone of knowledge. It allows us to compare things with other things and to quantify relationships.
· Mine is bigger than yours amounts to a mathematical statement (M > Y).
· Everything boils down to quantity, if only because everything ultimately boils down to particles and forces.
· Answers depend on the questions.
Pi in the Sky: Counting, Thinking, and Being (1992) – John D. Barrow
· Mathematics has been founded upon the certainty that comes from speaking the language of science, a symbolic language that banishes ambiguity and doubt, the only language with a built-in logic, which enables us to intimate communion with the innermost working of nature.
· As science has progressed, it has become more mathematical in its expression and more unified in its structure.
· Why do we look to mathematics for answers to ultimate questions about the nature of physical reality?
· If we look at any technologically advanced society, we find them to be founded upon the language of mathematics.
· The development of our understanding of the natural world around us and within us.
· The work of scientific society is closely linked to number: to measuring, counting, surveying, dividing, and making patterns.
· Mathematics works as a description of the world and the things that occur within it.
· The world dances to a mathematical tune.
· What are numbers and how did we come to discover them? But, did we really discover them, or perhaps we merely invented them?
· Is it a natural activity for the human mind or just a curious skill possessed by few?
· Pythagoras and his followers saw the numerical as a symbolic picture of the true meaning of the universe.
· They believed all numbers to be of two sorts, either whole numbers (like 1,2,3, …) or fractions which were made by dividing any whole number by another. These were called rational numbers.
· Eventually it was proved that the square root of 2 could not be expressed as the ratio of any two whole numbers.
· These later became known as irrational numbers, (3,5,6,7, etc).
· The world is not static but Pythagorean’s numbers were.
· Euclid’s pristine geometry had subtle influences upon other areas of human thinking. All architecture and artistic composition, all navigation and astronomy.
· Geometry stood for absolute logical certainty.
· The discovery of non-Euclidean geometry was, for a long time regarded as a curiosity, probably flawed in some way. Although curved surfaces clearly did exist, almost everyone believed that the physical space we live and move in, was really Euclidean.
· Other geometries could exist.
· The term ‘non-Euclidean’ came to be employed as a label for unconventional, non-traditional, or radical thinking.
· Mathematics has been founded upon the certainty that comes from speaking the language of science, a symbolic language that banishes ambiguity and doubt, the only language with a built-in logic, which enables us to intimate communion with the innermost working of nature.
· As science has progressed, it has become more mathematical in its expression and more unified in its structure.
· Why do we look to mathematics for answers to ultimate questions about the nature of physical reality?
· If we look at any technologically advanced society, we find them to be founded upon the language of mathematics.
· The development of our understanding of the natural world around us and within us.
· The work of scientific society is closely linked to number: to measuring, counting, surveying, dividing, and making patterns.
· Mathematics works as a description of the world and the things that occur within it.
· The world dances to a mathematical tune.
· What are numbers and how did we come to discover them? But, did we really discover them, or perhaps we merely invented them?
· Is it a natural activity for the human mind or just a curious skill possessed by few?
· Pythagoras and his followers saw the numerical as a symbolic picture of the true meaning of the universe.
· They believed all numbers to be of two sorts, either whole numbers (like 1,2,3, …) or fractions which were made by dividing any whole number by another. These were called rational numbers.
· Eventually it was proved that the square root of 2 could not be expressed as the ratio of any two whole numbers.
· These later became known as irrational numbers, (3,5,6,7, etc).
· The world is not static but Pythagorean’s numbers were.
· Euclid’s pristine geometry had subtle influences upon other areas of human thinking. All architecture and artistic composition, all navigation and astronomy.
· Geometry stood for absolute logical certainty.
· The discovery of non-Euclidean geometry was, for a long time regarded as a curiosity, probably flawed in some way. Although curved surfaces clearly did exist, almost everyone believed that the physical space we live and move in, was really Euclidean.
· Other geometries could exist.
· The term ‘non-Euclidean’ came to be employed as a label for unconventional, non-traditional, or radical thinking.
Math and Gender - Fennema
Try to minimize the constraining effects of sex role on cognitive functioning.
There is a preference for males to be in active games and past-times concerned with skills and mastery of objects.
There is a preference for females for play in which they practice skills related to mastery over people and interpersonal relations.
While males favour achievement in the traditionally highly valued areas of intellectual expertise and leadership skills, females are considered more likely to aim for excellence in areas congruent with their traditional role, that is, areas that require social skills.
Thus mathematics stereotyping as a male domain has often been used to explain females lower performance and participation in post compulsory mathematics courses.
Components of general intelligence include numerical, verbal, and spatial skills.
Spatial skills are related to mathematical achievement.
Spatial skills are diverse such as mentally rotating a cube or reorienting oneself in relation to a fixed object.
Spatial visualization includes mentally manipulating all or part of an object.
Performance involving the reproduction and/or combination of spatial information at the internal level.
Involve complicated, multi-step manipulations of spatially presented information.
Space structuring, females left, males right.
Try to minimize the constraining effects of sex role on cognitive functioning.
There is a preference for males to be in active games and past-times concerned with skills and mastery of objects.
There is a preference for females for play in which they practice skills related to mastery over people and interpersonal relations.
While males favour achievement in the traditionally highly valued areas of intellectual expertise and leadership skills, females are considered more likely to aim for excellence in areas congruent with their traditional role, that is, areas that require social skills.
Thus mathematics stereotyping as a male domain has often been used to explain females lower performance and participation in post compulsory mathematics courses.
Components of general intelligence include numerical, verbal, and spatial skills.
Spatial skills are related to mathematical achievement.
Spatial skills are diverse such as mentally rotating a cube or reorienting oneself in relation to a fixed object.
Spatial visualization includes mentally manipulating all or part of an object.
Performance involving the reproduction and/or combination of spatial information at the internal level.
Involve complicated, multi-step manipulations of spatially presented information.
Space structuring, females left, males right.
Sorry unknown, I forgot that this was an art board.
Trip
What are numbers and how did we come to discover them? But, did we really discover them, or perhaps we merely invented them?
No, you can definitely count the stars and they were here before us. But to stop being serious, maybe we could explore the meaning of "number" a little more. Or perhaps, think of how we could describe the universe (everything) without numbers.
Unknown
Don't say paint it! Even I could have thought of that!
Trip
What are numbers and how did we come to discover them? But, did we really discover them, or perhaps we merely invented them?
No, you can definitely count the stars and they were here before us. But to stop being serious, maybe we could explore the meaning of "number" a little more. Or perhaps, think of how we could describe the universe (everything) without numbers.
Unknown
Don't say paint it! Even I could have thought of that!
One of the most important things humans ever did was invent (or was it discover?) numbers. It may seem odd to describe numbers as an important discovery or invention, but at some point someone must have realised that one sheep and another sheep made something more precise than 'some' sheep.
The invention of the counting numbers predates written history, but several numbers have since been added to the simple set of counting numbers: 1, 2, 3, 4...
For starters, it wasn't until 200 BCE that someone got an inkling that there ought to be a number to represent nothingness.
It was the Babylonians that first got this inkling, but it took another thousand years before the Hindus finally nailed down the idea, symbol and name for this new invention.
The word zero is ultimately derived from the Hindi for 'void' (sunya). Negative numbers had a similarly drawn-out birth. The slowness of the acceptance of negatives and zero is one of the reasons that the Western calendar lacks a year nought: 1 BCE changes to 1 CE, much to the suprise of today's more numerate historians.
Pythagoras, father of mathematics, ran a secret society of mathematicians. Women were welcomed into the 'brother'hood.
۰۱۲۳۴۵۶۷۸۹
०१२३४५६७८९
0123456789
The numerals we now use are ultimately derived from the Arabic numeral system (top row), via the Indian (Devanagari) numerals (middle row).
So, Indians invented zero, with a little help from the Babylonians, so creating the integers, ... -3, -2, -1, 0, 1, 2, 3, and from them the fractions, 1/2, 2/3, 345/2345.
However, being the endlessly inventive apes we are, there were plenty of other numbers yet to be discovered.
Pythagoras (born in 569 BCE in Samos, now Greece, died 475 BCE) was the leader of a highly secretive mathematical community, who was so horrified when it was proved that irrational numbers (like π, 3.142... infinite decimals that cannot be represented as simple fractions) existed that it is said that he had its discoverer Hippasus (c. 500 BCE) murdered.
It would appear that the Indian Aryabhatiya (475-550), who independently derived a value of π at 3.1416, seems to have been unworried that it might be irrational, which was very rational of him.
Similar problems arose when Georg Cantor (1845-1918) discovered that infinity ∞, hitherto thought of as a grubby concept to be swept under the table, came in several different sizes. Jealous rivals eventually sent Cantor mad, and he died in a lunatic asylum in 1918.
Much of the European Renaissance was spent relearning what the Greeks had already discovered in and around 400 BCE. The works of Socrates (469-399 BCE), Plato (427-347 BCE), Euclid (325-265 BCE) and Aristotle (384-322 BCE) covered fields of human endeavour as diverse as geometry, ethics, classification and logic. Their rediscovered works didn't just excite the fledgling scientist of southern Europe. Over four hundred years before, the Arab philosopher Abu Yusuf Al-Kindi (801-873), working in what is now Iraq, spent his life expounding Greek philosophy to an Islamic audience, and was the world's first cryptanalyst, inventing a way of cracking the letter-substitution cipher that until then had been assumed secure. Some hundred years later, the Iranian Al-Razi (864-930) worked on both medicine and alchemy: the word for one third of science, chemistry, is derived from the Arabic words al kimiya (alchemy).
Mikolaj Kopernik (Copernicus, 1473-1543), whom many don't realise was Polish, and the Italian Galileo Galilei (1564-1642) noticed that the Sun, rather than the Earth, was the centre of what we now call the solar system. Galileo was so forthright in his opinion that he was eventually declared a heretic by the Roman Inquisition and died under house arrest, an insult for which the Roman Catholic Church didn't quite apologise in 1992. Sometimes science isn't just about grant awarding councils and towing the line of dogma.
Revolutionary woman
The invention of the counting numbers predates written history, but several numbers have since been added to the simple set of counting numbers: 1, 2, 3, 4...
For starters, it wasn't until 200 BCE that someone got an inkling that there ought to be a number to represent nothingness.
It was the Babylonians that first got this inkling, but it took another thousand years before the Hindus finally nailed down the idea, symbol and name for this new invention.
The word zero is ultimately derived from the Hindi for 'void' (sunya). Negative numbers had a similarly drawn-out birth. The slowness of the acceptance of negatives and zero is one of the reasons that the Western calendar lacks a year nought: 1 BCE changes to 1 CE, much to the suprise of today's more numerate historians.
Pythagoras, father of mathematics, ran a secret society of mathematicians. Women were welcomed into the 'brother'hood.
۰۱۲۳۴۵۶۷۸۹
०१२३४५६७८९
0123456789
The numerals we now use are ultimately derived from the Arabic numeral system (top row), via the Indian (Devanagari) numerals (middle row).
So, Indians invented zero, with a little help from the Babylonians, so creating the integers, ... -3, -2, -1, 0, 1, 2, 3, and from them the fractions, 1/2, 2/3, 345/2345.
However, being the endlessly inventive apes we are, there were plenty of other numbers yet to be discovered.
Pythagoras (born in 569 BCE in Samos, now Greece, died 475 BCE) was the leader of a highly secretive mathematical community, who was so horrified when it was proved that irrational numbers (like π, 3.142... infinite decimals that cannot be represented as simple fractions) existed that it is said that he had its discoverer Hippasus (c. 500 BCE) murdered.
It would appear that the Indian Aryabhatiya (475-550), who independently derived a value of π at 3.1416, seems to have been unworried that it might be irrational, which was very rational of him.
Similar problems arose when Georg Cantor (1845-1918) discovered that infinity ∞, hitherto thought of as a grubby concept to be swept under the table, came in several different sizes. Jealous rivals eventually sent Cantor mad, and he died in a lunatic asylum in 1918.
Much of the European Renaissance was spent relearning what the Greeks had already discovered in and around 400 BCE. The works of Socrates (469-399 BCE), Plato (427-347 BCE), Euclid (325-265 BCE) and Aristotle (384-322 BCE) covered fields of human endeavour as diverse as geometry, ethics, classification and logic. Their rediscovered works didn't just excite the fledgling scientist of southern Europe. Over four hundred years before, the Arab philosopher Abu Yusuf Al-Kindi (801-873), working in what is now Iraq, spent his life expounding Greek philosophy to an Islamic audience, and was the world's first cryptanalyst, inventing a way of cracking the letter-substitution cipher that until then had been assumed secure. Some hundred years later, the Iranian Al-Razi (864-930) worked on both medicine and alchemy: the word for one third of science, chemistry, is derived from the Arabic words al kimiya (alchemy).
Mikolaj Kopernik (Copernicus, 1473-1543), whom many don't realise was Polish, and the Italian Galileo Galilei (1564-1642) noticed that the Sun, rather than the Earth, was the centre of what we now call the solar system. Galileo was so forthright in his opinion that he was eventually declared a heretic by the Roman Inquisition and died under house arrest, an insult for which the Roman Catholic Church didn't quite apologise in 1992. Sometimes science isn't just about grant awarding councils and towing the line of dogma.
Revolutionary woman
Before the Secret Brotherhood was disbanded, its members really thought they had grasped the key to the cosmos.
Then everything collapsed. Their whole scheme was destroyed by a fatal discovery, and the Order itself was destroyed by traitors and mob violence. Yet as we retell the somber tale, we will find that it was not a complete tragedy after all, for the Pythagoreans did enjoy their cosmic key briefly. This key was not found in abstract shapes alone nor in music, nor in the stars, but in one factor that-they believed-linked all of these: number.
"Himself" had said it: "Everything is number!"
So they followed Pythagoras teaching that the universe was ruled by whole numbers That did not mean numbers for ordinary counting or calculating. What interested them was the nature of a number itself odd even, divisible, indivisible and the relations between numbers. This was their arithmetike. And they applied it to their other three fields, and found startling number patterns in each.
In music, for instance, a sensational discovery about the relations of whole numbers and musical intervals was attributed to Pythagoras himself.
One legend said that on his long voyages he listened to the music of flapping sails, and the wind whistling and whining through the ship's rigging and playing a melody on the ropes. And that he decided then and there to investigate the connection between the tempest of sounds and the vibrating strings.
Another version said that he was strolling through the village of Croton, deep in thought, listening to the musical sounds of hammers striking anvils in a blacksmith's shop; when suddenly, tripping on a taut string that some children had stretched across the street, he got the inspiration for an experiment.
But the most popular story told that the idea came to him straight from the stringed lyre of his "father" Apollo, who was also the god of music.
Anyway, Pythagoras experimented with stretched strings of different lengths placed under the same tension. Soon he found the relation between the length of the vibrating string and the pitch of the note. He discovered that the octave, fifth, and fourth of a note could be produced by one string under tension, simply by "stopping" the string at different places: at one-half its length for the octave, two-thirds its length for the fifth and three-fourths its length for the fourth!
Other musical innovations were credited to him, such as a one-string apparatus for the study of harmonics. But his great discovery was the tetrachord, where the most important harmonic intervals were obtained by ratios of the whole numbers:
1, 2, 3, 4. The Secret Brotherhood gave this fourfold chord mystical significance and used to say: "What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens."
And the Pythagoreans even used it for their astronomy. In the relation of number and music, they believed they had found the pattern that guided the "wandering" planets through the heavens. They pictured the sun and the planets as geometrically perfect spheres, moving through the visibly circular sky on perfect circular orbits, separated by harmonic ratios-musical intervals! Theirs was a vision of time and space revealed in lines, tones, and mathematical ratios. And they even imagined the brilliant planets emitting harmonious tones, the so-called "music of the spheres."
But it was in the connection of number and geometry, their two completely mathematical subjects, that the Pythagoreans were on surest ground.
Numbers, they had discovered, whole numbers, actually had geometric shapes. There were triangular numbers, square numbers, pentagonal numbers, rectangular numbers, and so on.
This was no wild fantasy, like the singing planets. It was a real mathematical discovery, and came from the circumstance that they did not do their number work by writing the numbers at all. Instead, they placed pebbles on the sand, like the reckoners. But the Pythagoreans placed their pebbles in patterns, adding extra rows for each number. Their two most important series were the square numbers and the triangular numbers.
The most important number of all, to the Pythagoreans, was the fourth triangular number, 10. For it was made up of 1 + 2 + 3 + 4. They called it the "Sacred Tetractys, " swore by it in their oaths, and attached marvelous properties to it, as "the source and root of eternal nature."
Everything fitted perfectly: the Tetractys, the tetrachord, the four regular solids representing the four "elements," inscribed in a dodecahedron representing the celestial sphere. But it was all too pat a jumble of luck, imagination, serious mathematical experiments, and old number magic from the East. just as the Pythagoreans thought they were getting more and more evidence that number was everywhere, the whole system broke down. The entire connection between geometry and number-the foundation of their thinking-was shattered by one disastrous experiment.
Presiding was Hippasus of Metapontum, whose name was to loom dark in their future of the Brotherhood. The idea was simply to find the numbers that matched the sides of the two right triangles with which Pythagoras had first demonstrated his theorem-the Egyptian triangle and the one from the tiled floor.
Of course, the Egyptian rope-triangle worked perfectly: its 3-4-5 sides made a beautiful Pythagorean series. They indicated the intervals with pebbles. Now what about the right triangle from the Greek tile design, where the two sides were equal?
Suppose each side had a length of 1 unit-that would require 1 pebble. Then for the hypotenuse-how many pebbles should they put there? Well, the sum of the squares on the sides would equal the square on the hypotenuse. Therefore,
12 = 1 (square on one side)
and 12 = 1 (square on other side)
and 1 + 1 = 2,
so 2 is the square on the hypotenuse. And the hypotenuse is the square root of 2.
But what was the square root of 2?
It couldn't be a whole number, since there is no whole number between 1 and 2.
Then was it a ratio of whole numbers between 1 and 2? Hopefully, they tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2. We write the answer as 1.4141..., a continuing decimal fraction, but they couldn't do that since they had no concept of zero and of decimals. They could draw the hypotenuse easily, but they could not express its length as a number. It was "unutterable"-"unspeakable"!
Horrified, the Pythagoreans called the square root of 2 an irrational number. After that, they found other irrationals and swore to keep them secret, for the discovery of these "irrationals" wrecked their entire beautifully constructed system of a universe guided by whole numbers. The breakdown in their mystical morale was followed by the breakup of the Secret Brotherhood itself.
In this final demolition, Hippasus played a decisive role, though his own fate is shrouded in mystery. The Order was already in trouble. Bitter resentment had grown up against its secrecy and exclusiveness, and riots of villagers had driven it out of Croton. Pythagoras himself had died on a neighboring island. And now mobs of "democrats" began to attack the aristocratic Pythagorean societies everywhere.
Against this background, Hippasus took a step that was regarded by the conservative members as sheer betrayal. He broke the oath of secrecy and revealed their most closely guarded discoveries-the dodecahedron and the irrationals. When they promptly expelled him, he set himself up as a public teacher of geometry.
The traitor's punishment was swift and terrible. He was very shortly drowned in a mysterious "accident" at sea, and strange rumors circulated. Some said that a storm had struck his ship as a direct vengeance from the gods; others, that he had been pushed overboard by agents of the Secret Brotherhood. But Hippasus' death was to no avail. The harm was already done to the Order of Pythagoreans, though the discovery of irrational numbers eventually worked for the good of mankind.
The remaining secret groups soon collapsed, torn by outer violence and inner dissensions. And more and more "mathematicians" followed Hippasus' example and came out to earn a living as teachers. Pythagoras' idea had been demolished: no longer was there a closed Brotherhood of followers, bound together by a mystical belief in a cosmos ruled by number. Yet his ideals lived on in this broader field. He had pursued knowledge for its own sake, loving wisdom for itself. He knew learning could be shared without diminishing, that it lasts through life and immortalizes the learned after death. And the destruction of the Order gave his legacy to the world.
Geometry was now out in the open-and it was the new Pythagorean geometry. True, mathematics was still mixed with some magic: number mysticism, cosmic ideas about the regular solids. Burt there was, in addition, the famous theorem and its applications, the careful study of shapes, the theory of numbers, and the discovery of irrationals.
Then everything collapsed. Their whole scheme was destroyed by a fatal discovery, and the Order itself was destroyed by traitors and mob violence. Yet as we retell the somber tale, we will find that it was not a complete tragedy after all, for the Pythagoreans did enjoy their cosmic key briefly. This key was not found in abstract shapes alone nor in music, nor in the stars, but in one factor that-they believed-linked all of these: number.
"Himself" had said it: "Everything is number!"
So they followed Pythagoras teaching that the universe was ruled by whole numbers That did not mean numbers for ordinary counting or calculating. What interested them was the nature of a number itself odd even, divisible, indivisible and the relations between numbers. This was their arithmetike. And they applied it to their other three fields, and found startling number patterns in each.
In music, for instance, a sensational discovery about the relations of whole numbers and musical intervals was attributed to Pythagoras himself.
One legend said that on his long voyages he listened to the music of flapping sails, and the wind whistling and whining through the ship's rigging and playing a melody on the ropes. And that he decided then and there to investigate the connection between the tempest of sounds and the vibrating strings.
Another version said that he was strolling through the village of Croton, deep in thought, listening to the musical sounds of hammers striking anvils in a blacksmith's shop; when suddenly, tripping on a taut string that some children had stretched across the street, he got the inspiration for an experiment.
But the most popular story told that the idea came to him straight from the stringed lyre of his "father" Apollo, who was also the god of music.
Anyway, Pythagoras experimented with stretched strings of different lengths placed under the same tension. Soon he found the relation between the length of the vibrating string and the pitch of the note. He discovered that the octave, fifth, and fourth of a note could be produced by one string under tension, simply by "stopping" the string at different places: at one-half its length for the octave, two-thirds its length for the fifth and three-fourths its length for the fourth!
Other musical innovations were credited to him, such as a one-string apparatus for the study of harmonics. But his great discovery was the tetrachord, where the most important harmonic intervals were obtained by ratios of the whole numbers:
1, 2, 3, 4. The Secret Brotherhood gave this fourfold chord mystical significance and used to say: "What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens."
And the Pythagoreans even used it for their astronomy. In the relation of number and music, they believed they had found the pattern that guided the "wandering" planets through the heavens. They pictured the sun and the planets as geometrically perfect spheres, moving through the visibly circular sky on perfect circular orbits, separated by harmonic ratios-musical intervals! Theirs was a vision of time and space revealed in lines, tones, and mathematical ratios. And they even imagined the brilliant planets emitting harmonious tones, the so-called "music of the spheres."
But it was in the connection of number and geometry, their two completely mathematical subjects, that the Pythagoreans were on surest ground.
Numbers, they had discovered, whole numbers, actually had geometric shapes. There were triangular numbers, square numbers, pentagonal numbers, rectangular numbers, and so on.
This was no wild fantasy, like the singing planets. It was a real mathematical discovery, and came from the circumstance that they did not do their number work by writing the numbers at all. Instead, they placed pebbles on the sand, like the reckoners. But the Pythagoreans placed their pebbles in patterns, adding extra rows for each number. Their two most important series were the square numbers and the triangular numbers.
The most important number of all, to the Pythagoreans, was the fourth triangular number, 10. For it was made up of 1 + 2 + 3 + 4. They called it the "Sacred Tetractys, " swore by it in their oaths, and attached marvelous properties to it, as "the source and root of eternal nature."
Everything fitted perfectly: the Tetractys, the tetrachord, the four regular solids representing the four "elements," inscribed in a dodecahedron representing the celestial sphere. But it was all too pat a jumble of luck, imagination, serious mathematical experiments, and old number magic from the East. just as the Pythagoreans thought they were getting more and more evidence that number was everywhere, the whole system broke down. The entire connection between geometry and number-the foundation of their thinking-was shattered by one disastrous experiment.
Presiding was Hippasus of Metapontum, whose name was to loom dark in their future of the Brotherhood. The idea was simply to find the numbers that matched the sides of the two right triangles with which Pythagoras had first demonstrated his theorem-the Egyptian triangle and the one from the tiled floor.
Of course, the Egyptian rope-triangle worked perfectly: its 3-4-5 sides made a beautiful Pythagorean series. They indicated the intervals with pebbles. Now what about the right triangle from the Greek tile design, where the two sides were equal?
Suppose each side had a length of 1 unit-that would require 1 pebble. Then for the hypotenuse-how many pebbles should they put there? Well, the sum of the squares on the sides would equal the square on the hypotenuse. Therefore,
12 = 1 (square on one side)
and 12 = 1 (square on other side)
and 1 + 1 = 2,
so 2 is the square on the hypotenuse. And the hypotenuse is the square root of 2.
But what was the square root of 2?
It couldn't be a whole number, since there is no whole number between 1 and 2.
Then was it a ratio of whole numbers between 1 and 2? Hopefully, they tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2. We write the answer as 1.4141..., a continuing decimal fraction, but they couldn't do that since they had no concept of zero and of decimals. They could draw the hypotenuse easily, but they could not express its length as a number. It was "unutterable"-"unspeakable"!
Horrified, the Pythagoreans called the square root of 2 an irrational number. After that, they found other irrationals and swore to keep them secret, for the discovery of these "irrationals" wrecked their entire beautifully constructed system of a universe guided by whole numbers. The breakdown in their mystical morale was followed by the breakup of the Secret Brotherhood itself.
In this final demolition, Hippasus played a decisive role, though his own fate is shrouded in mystery. The Order was already in trouble. Bitter resentment had grown up against its secrecy and exclusiveness, and riots of villagers had driven it out of Croton. Pythagoras himself had died on a neighboring island. And now mobs of "democrats" began to attack the aristocratic Pythagorean societies everywhere.
Against this background, Hippasus took a step that was regarded by the conservative members as sheer betrayal. He broke the oath of secrecy and revealed their most closely guarded discoveries-the dodecahedron and the irrationals. When they promptly expelled him, he set himself up as a public teacher of geometry.
The traitor's punishment was swift and terrible. He was very shortly drowned in a mysterious "accident" at sea, and strange rumors circulated. Some said that a storm had struck his ship as a direct vengeance from the gods; others, that he had been pushed overboard by agents of the Secret Brotherhood. But Hippasus' death was to no avail. The harm was already done to the Order of Pythagoreans, though the discovery of irrational numbers eventually worked for the good of mankind.
The remaining secret groups soon collapsed, torn by outer violence and inner dissensions. And more and more "mathematicians" followed Hippasus' example and came out to earn a living as teachers. Pythagoras' idea had been demolished: no longer was there a closed Brotherhood of followers, bound together by a mystical belief in a cosmos ruled by number. Yet his ideals lived on in this broader field. He had pursued knowledge for its own sake, loving wisdom for itself. He knew learning could be shared without diminishing, that it lasts through life and immortalizes the learned after death. And the destruction of the Order gave his legacy to the world.
Geometry was now out in the open-and it was the new Pythagorean geometry. True, mathematics was still mixed with some magic: number mysticism, cosmic ideas about the regular solids. Burt there was, in addition, the famous theorem and its applications, the careful study of shapes, the theory of numbers, and the discovery of irrationals.
| QUOTE (Hey Hey @ Sep 06, 09:26 PM) |
| What are numbers and how did we come to discover them? maybe we could explore the meaning of "number" a little more. Or perhaps, think of how we could describe the universe (everything) without numbers. |
the essence of 'number' is 'object'. the essence of 'object' is existence. In 'set' terminology, the set whose only property is existence is an utterly simple object. All sets can be seen to have the property existence, so all sets are comparable in terms of this property of existence. In this sense, all sets are 'countable' as a one-to-one map can be made between any collection of sets and a numerably unique collection of these special 'existence only' sets (the 'cardinal' numbers).
Space Figure
A space figure or three-dimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a polyhedron. A cube and a pyramid are both polyhedrons; a sphere, cylinder, and cone are not.
Volume
Volume is a measure of how much space a space figure takes up. Volume is used to measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length of one of its sides. The volume of a box is the product of its length, width, and height.
Example:
What is the volume of a cube with side-length 6 cm?
The volume of a cube is the cube of its side-length, which is 63 = 216 cubic cm.
Example:
What is the volume of a box whose length is 4cm, width is 5 cm, and height is 6 cm?
The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120 cubic cm.
Surface Area
The surface area of a space figure is the total area of all the faces of the figure.
Example:
What is the surface area of a box whose length is 8, width is 3, and height is 4?
This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces are 8 by 3. Adding the areas of all these faces, we get the surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =
32 + 32 + 12 + 12 +24 + 24=
136.
Cube
A cube is a three-dimensional figure having six matching square sides. If L is the length of one of its sides, the volume of the cube is L3 = L × L × L. A cube has six square-shaped sides. The surface area of a cube is six times the area of one of these sides.
Example:
What is the volume and surface are of a cube having a side-length of 2.1 cm?
Its volume would be 2.1 × 2.1 × 2.1 = 9.261 cubic centimeters.
Its surface area would be 6 × 2.1 × 2.1 = 26.46 square centimeters.
Cylinder
A cylinder is a space figure having two congruent circular bases that are parallel. If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then the volume of the cylinder is L × pi × r2, and the surface area is 2 × r × pi × L + 2 × pi × r2.
Sphere
A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius. Any cross-section of a sphere is a circle.
If r is the radius of a sphere, the volume V of the sphere is given by the formula V = 4/3 × pi ×r3.
The surface area S of the sphere is given by the formula S = 4 × pi ×r2.
Example:
To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?
Using an estimate of 3.14 for pi,
the volume would be 4/3 × 3.14 × 43 = 4/3 × 3.14 × 4 × 4 × 4 = 268 cubic centimeters.
Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 42 = 4 × 3.14 × 4 × 4 = 201 square centimeters.
Cone
A cone is a space figure having a circular base and a single vertex.
If r is the radius of the circular base, and h is the height of the cone, then the volume of the cone is 1/3 × pi × r2 × h.
Example:
What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?
We will use an estimate of 3.14 for pi.
The volume is 1/3 × pi × 32 × 6 = pi ×18 = 56.52, which equals 56.5 cubic cm when rounded to the nearest tenth.
Pyramid
A pyramid is a space figure with a square base and 4 triangle-shaped sides.
Tetrahedron
A tetrahedron is a 4-sided space figure. Each face of a tetrahedron is a triangle.
Prism
A prism is a space figure with two congruent, parallel bases that are polygons.
A space figure or three-dimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a polyhedron. A cube and a pyramid are both polyhedrons; a sphere, cylinder, and cone are not.
Volume
Volume is a measure of how much space a space figure takes up. Volume is used to measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length of one of its sides. The volume of a box is the product of its length, width, and height.
Example:
What is the volume of a cube with side-length 6 cm?
The volume of a cube is the cube of its side-length, which is 63 = 216 cubic cm.
Example:
What is the volume of a box whose length is 4cm, width is 5 cm, and height is 6 cm?
The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120 cubic cm.
Surface Area
The surface area of a space figure is the total area of all the faces of the figure.
Example:
What is the surface area of a box whose length is 8, width is 3, and height is 4?
This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces are 8 by 3. Adding the areas of all these faces, we get the surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =
32 + 32 + 12 + 12 +24 + 24=
136.
Cube
A cube is a three-dimensional figure having six matching square sides. If L is the length of one of its sides, the volume of the cube is L3 = L × L × L. A cube has six square-shaped sides. The surface area of a cube is six times the area of one of these sides.
Example:
What is the volume and surface are of a cube having a side-length of 2.1 cm?
Its volume would be 2.1 × 2.1 × 2.1 = 9.261 cubic centimeters.
Its surface area would be 6 × 2.1 × 2.1 = 26.46 square centimeters.
Cylinder
A cylinder is a space figure having two congruent circular bases that are parallel. If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then the volume of the cylinder is L × pi × r2, and the surface area is 2 × r × pi × L + 2 × pi × r2.
Sphere
A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius. Any cross-section of a sphere is a circle.
If r is the radius of a sphere, the volume V of the sphere is given by the formula V = 4/3 × pi ×r3.
The surface area S of the sphere is given by the formula S = 4 × pi ×r2.
Example:
To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?
Using an estimate of 3.14 for pi,
the volume would be 4/3 × 3.14 × 43 = 4/3 × 3.14 × 4 × 4 × 4 = 268 cubic centimeters.
Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 42 = 4 × 3.14 × 4 × 4 = 201 square centimeters.
Cone
A cone is a space figure having a circular base and a single vertex.
If r is the radius of the circular base, and h is the height of the cone, then the volume of the cone is 1/3 × pi × r2 × h.
Example:
What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?
We will use an estimate of 3.14 for pi.
The volume is 1/3 × pi × 32 × 6 = pi ×18 = 56.52, which equals 56.5 cubic cm when rounded to the nearest tenth.
Pyramid
A pyramid is a space figure with a square base and 4 triangle-shaped sides.
Tetrahedron
A tetrahedron is a 4-sided space figure. Each face of a tetrahedron is a triangle.
Prism
A prism is a space figure with two congruent, parallel bases that are polygons.
Throughout mathematics in Kindergarten, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Kindergarten are developing whole-number concepts and using patterns and sorting to explore number, data, and shape. Students use numbers to count by ones to 100, to order, label, and express quantities (through 9) or determine how many objects are in a set (through 20) as well as use relationships to solve problems and translate informal language into mathematical symbols.
Students use patterns to describe objects, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. They also use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations.
Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 1, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 1 are adding and subtracting whole numbers and organizing and analyzing data.
Students use numbers in ordering whole numbers (through 99), labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical symbols (>, <, =) using sets of concrete objects.
Students use patterns to describe objects, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space.
They learn to model and create addition and subtraction problem situations using concrete models (sums to 18) as well as use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world.
They also begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 2 are comparing and ordering, applying addition and subtraction, and using measurement processes. Students use numbers in ordering, labeling, and expressing quantities (through 999) and relationships to solve problems and translate informal language into mathematical symbols (>, <, =).
Students use patterns to describe objects, name fractional parts of a whole (not to exceed twelfths), express relationships, make predictions, and solve problems (sums to 18) as they build an understanding of number, operation, shape, and space. They also use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 3, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 3 are multiplying and dividing whole numbers, connecting fraction symbols to fractional quantities, and standardizing language and procedures in geometry and measurement. Students use numbers in ordering (through 9,999), labeling, and expressing quantities (through 999,999) and relationships to solve problems and translate informal language into mathematical symbols (>, <, =). Students use patterns to describe objects, name fractional parts of a whole (not to exceed twelfths), express relationships, make predictions, and solve problems (involving whole numbers through 999)) as they build an understanding of number, operation, shape, and space.
Students learn and apply multiplication facts through the tens using concrete models (one-digit multiplier) and use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data, use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 4, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 4 are comparing and ordering fractions and decimals, applying multiplication (with two-digit numbers), and division (involving one-digit divisors), and developing ideas related to congruence and symmetry.
Students use numbers in ordering, labeling, and expressing quantities (through the millions place) and relationships to solve problems and translate informal language into mathematical symbols.
Students use patterns to describe objects to multiply by 10 and 100; name fractional parts of a whole (not to exceed twelfths); and compare decimals and fractions to the hundredths place using concrete and pictorial models, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space.
They recall and apply multiplication facts (through 12 X 12), use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world, and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Additionally, they collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 5, students build a foundation of understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 5 are comparing and contrasting lengths, area and volume of geometric solids; representing and interpreting data in graphs, charts, and tables, and applying whole number operations in a variety of contexts. Students use numbers in ordering, labeling, expressing quantities (through the billions place) and relationships to solve problems, and translate informal language into mathematical symbols.
They use patterns to describe objects to multiply by 10 and 100; name fractional parts of a whole (not to exceed twelfths); comparing and ordering fractions and decimals, apply multiplication facts (with two-digit numbers) and division (involving one-digit divisors); and develop ideas related to congruence and symmetry.
Students compare and order decimals and fractions to the thousandths place, generate equivalent fractions and use equivalent fractions in problem solving situations using a variety of methods (including common denominator); express relationships between objects or numbers; make predictions; and solve problems as they build an understanding of number, operation, shape, and space.
They recall and apply multiplication facts (no more than three digits times two digits without technology) and use division (no more than two-digit divisors and three-digit dividends without technology).
Students also use informal language and observation of geometric properties (parallel, perpendicular and congruent) to describe shapes, solids, and locations in the physical world, and develop measurement concepts as they identify and compare attributes of objects and situations. Additionally, they collect, organize, display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Students use these processes, together with technology and other mathematical tools, such as manipulative materials, to develop conceptual understanding and solve problems as they do mathematics.
Within a well-balanced mathematics curriculum, the primary focal points at Kindergarten are developing whole-number concepts and using patterns and sorting to explore number, data, and shape. Students use numbers to count by ones to 100, to order, label, and express quantities (through 9) or determine how many objects are in a set (through 20) as well as use relationships to solve problems and translate informal language into mathematical symbols.
Students use patterns to describe objects, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. They also use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations.
Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 1, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 1 are adding and subtracting whole numbers and organizing and analyzing data.
Students use numbers in ordering whole numbers (through 99), labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical symbols (>, <, =) using sets of concrete objects.
Students use patterns to describe objects, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space.
They learn to model and create addition and subtraction problem situations using concrete models (sums to 18) as well as use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world.
They also begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 2 are comparing and ordering, applying addition and subtraction, and using measurement processes. Students use numbers in ordering, labeling, and expressing quantities (through 999) and relationships to solve problems and translate informal language into mathematical symbols (>, <, =).
Students use patterns to describe objects, name fractional parts of a whole (not to exceed twelfths), express relationships, make predictions, and solve problems (sums to 18) as they build an understanding of number, operation, shape, and space. They also use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 3, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 3 are multiplying and dividing whole numbers, connecting fraction symbols to fractional quantities, and standardizing language and procedures in geometry and measurement. Students use numbers in ordering (through 9,999), labeling, and expressing quantities (through 999,999) and relationships to solve problems and translate informal language into mathematical symbols (>, <, =). Students use patterns to describe objects, name fractional parts of a whole (not to exceed twelfths), express relationships, make predictions, and solve problems (involving whole numbers through 999)) as they build an understanding of number, operation, shape, and space.
Students learn and apply multiplication facts through the tens using concrete models (one-digit multiplier) and use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data, use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 4, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 4 are comparing and ordering fractions and decimals, applying multiplication (with two-digit numbers), and division (involving one-digit divisors), and developing ideas related to congruence and symmetry.
Students use numbers in ordering, labeling, and expressing quantities (through the millions place) and relationships to solve problems and translate informal language into mathematical symbols.
Students use patterns to describe objects to multiply by 10 and 100; name fractional parts of a whole (not to exceed twelfths); and compare decimals and fractions to the hundredths place using concrete and pictorial models, express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space.
They recall and apply multiplication facts (through 12 X 12), use informal language and observation of geometric properties to describe shapes, solids, and locations in the physical world, and begin to develop measurement concepts as they identify and compare attributes of objects and situations. Additionally, they collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Throughout mathematics in Grade 5, students build a foundation of understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics.
Within a well-balanced mathematics curriculum, the primary focal points at Grade 5 are comparing and contrasting lengths, area and volume of geometric solids; representing and interpreting data in graphs, charts, and tables, and applying whole number operations in a variety of contexts. Students use numbers in ordering, labeling, expressing quantities (through the billions place) and relationships to solve problems, and translate informal language into mathematical symbols.
They use patterns to describe objects to multiply by 10 and 100; name fractional parts of a whole (not to exceed twelfths); comparing and ordering fractions and decimals, apply multiplication facts (with two-digit numbers) and division (involving one-digit divisors); and develop ideas related to congruence and symmetry.
Students compare and order decimals and fractions to the thousandths place, generate equivalent fractions and use equivalent fractions in problem solving situations using a variety of methods (including common denominator); express relationships between objects or numbers; make predictions; and solve problems as they build an understanding of number, operation, shape, and space.
They recall and apply multiplication facts (no more than three digits times two digits without technology) and use division (no more than two-digit divisors and three-digit dividends without technology).
Students also use informal language and observation of geometric properties (parallel, perpendicular and congruent) to describe shapes, solids, and locations in the physical world, and develop measurement concepts as they identify and compare attributes of objects and situations. Additionally, they collect, organize, display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.
Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Students use these processes, together with technology and other mathematical tools, such as manipulative materials, to develop conceptual understanding and solve problems as they do mathematics.
| QUOTE (Hey Hey @ Sep 06, 05:30 PM) |
| Shouldn't "creation" have been "discovery"? What do you think? |
Mathematics is created, not discovered.
Plato and other idealists believed that the forms have an independent existence. However, that's not the case. Each person who learns mathematics needs to build these structures for himself. Writing mathematics in books makes it easier for the student to recreate it, but it still takes a lot of work to rebuild and understand it.
The Sphere
Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm.
The Circle
The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.
The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511...)
It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by those ideas alone.
The Point
At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all dimension. Transcendence of the illusions of time & space result in the point of here and now, our most primal light of consciousness. The proverbial "light at the end of the tunnel" is being validated by the ever-increasing literature on so-called "near-death experiences". If our essence is truly spiritual omnipresence, then perhaps the "point" of our being "here" is to recognize the oneness we share, validating all "individuals" as equally precious and sacred aspects of that one.
Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond.
Almost everywhere we look, the mineral intelligence embodied within crystalline structures follows a geometry unfaltering in its exactitude. The lattice patterns of crystals all express the principles of mathematical perfection and repetition of a fundamental essence, each with a characteristic spectrum of resonances defined by the angles, lengths and relational orientations of its atomic components.
Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm.
The Circle
The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.
The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511...)
It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by those ideas alone.
The Point
At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all dimension. Transcendence of the illusions of time & space result in the point of here and now, our most primal light of consciousness. The proverbial "light at the end of the tunnel" is being validated by the ever-increasing literature on so-called "near-death experiences". If our essence is truly spiritual omnipresence, then perhaps the "point" of our being "here" is to recognize the oneness we share, validating all "individuals" as equally precious and sacred aspects of that one.
Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond.
Almost everywhere we look, the mineral intelligence embodied within crystalline structures follows a geometry unfaltering in its exactitude. The lattice patterns of crystals all express the principles of mathematical perfection and repetition of a fundamental essence, each with a characteristic spectrum of resonances defined by the angles, lengths and relational orientations of its atomic components.
| QUOTE (Rick @ Sep 07, 09:14 AM) | ||
Mathematics is created, not discovered. |
Does not the cosmos (micro and macro/As above, So below) speak in patterns or did we pattern the cosmos?
The forms in nature come from the physics. Not the other way around.
Square, n. [OF. esquarre, esquierre.
1. (Geom.)
(a) The corner, or angle, of a figure. [Obs.]
(
A parallelogram having four equal sides and four right
angles.
2. Hence, anything which is square, or nearly so; as:
(a) A square piece or fragment.
He bolted his food down his capacious throat in
squares of three inches. --Sir W. Scott.
( A pane of glass.
© A certain number of lines, forming a portion of a column, nearly square; -- used chiefly in reckoning the prices of advertisements in newspapers.
(d) (Carp.) One hundred superficial feet.
3. An area of four sides, generally with houses on each side;
sometimes, a solid block of houses; also, an open place or
area for public use, as at the meeting or intersection of
two or more streets.
The statue of Alexander VII. stands in the large
square of the town. --Addison.
4. (Mech. & Joinery) An instrument having at least one right
angle and two or more straight edges, used to lay out or
test square work. It is of several forms, as the T square,
the carpenter's square, the try-square., etc.
5. Hence, a pattern or rule. [Obs.]
6. (Arith. & Alg.) The product of a number or quantity
multiplied by itself; thus, 64 is the square of 8, for 8
[times] 8 = 64; the square of a + b is a^2 + 2ab +
b^2.
7. Exact proportion; justness of workmanship and conduct;
regularity; rule. [Obs.]
They of Galatia [were] much more out of square.
--Hooker.
I have not kept my square. --Shak.
8. (Mil.) A body of troops formed in a square, esp. one
formed to resist a charge of cavalry; a squadron. ``The
brave squares of war.'' --Shak.
9. Fig.: The relation of harmony, or exact agreement;
equality; level.
We live not on the square with such as these.
--Dryden.
10. (Astrol.) The position of planets distant ninety degrees
from each other; a quadrate. [Obs.]
11. The act of squaring, or quarreling; a quarrel. [R.]
12. The front of a woman's dress over the bosom, usually
worked or embroidered. [Obs.] --Shak.
Hollow square (Mil.), a formation of troops in the shape of
a square, each side consisting of four or five ranks, and
the colors, officers, horses, etc., occupying the middle.
Least square, Magic square, etc.
On the square, or Upon the square, in an open, fair
manner; honestly, or upon honor. [Obs. or Colloq.]
On, or Upon, the square with, upon equality with; even
with. --Nares.
To be all squares, to be all settled. [Colloq.] --Dickens.
To be at square, to be in a state of quarreling. [Obs.]
--Nares.
To break no square, to give no offense; to make no
difference. [Obs.]
To break squares, to depart from an accustomed order.
To see how the squares go, to see how the game proceeds; --
a phrase taken from the game of chess, the chessboard
being formed with squares. [Obs.] --L'Estrange.
1. (Geom.)
(a) The corner, or angle, of a figure. [Obs.]
(
A parallelogram having four equal sides and four rightangles.
2. Hence, anything which is square, or nearly so; as:
(a) A square piece or fragment.
He bolted his food down his capacious throat in
squares of three inches. --Sir W. Scott.
(
A pane of glass.© A certain number of lines, forming a portion of a column, nearly square; -- used chiefly in reckoning the prices of advertisements in newspapers.
(d) (Carp.) One hundred superficial feet.
3. An area of four sides, generally with houses on each side;
sometimes, a solid block of houses; also, an open place or
area for public use, as at the meeting or intersection of
two or more streets.
The statue of Alexander VII. stands in the large
square of the town. --Addison.
4. (Mech. & Joinery) An instrument having at least one right
angle and two or more straight edges, used to lay out or
test square work. It is of several forms, as the T square,
the carpenter's square, the try-square., etc.
5. Hence, a pattern or rule. [Obs.]
6. (Arith. & Alg.) The product of a number or quantity
multiplied by itself; thus, 64 is the square of 8, for 8
[times] 8 = 64; the square of a + b is a^2 + 2ab +
b^2.
7. Exact proportion; justness of workmanship and conduct;
regularity; rule. [Obs.]
They of Galatia [were] much more out of square.
--Hooker.
I have not kept my square. --Shak.
8. (Mil.) A body of troops formed in a square, esp. one
formed to resist a charge of cavalry; a squadron. ``The
brave squares of war.'' --Shak.
9. Fig.: The relation of harmony, or exact agreement;
equality; level.
We live not on the square with such as these.
--Dryden.
10. (Astrol.) The position of planets distant ninety degrees
from each other; a quadrate. [Obs.]
11. The act of squaring, or quarreling; a quarrel. [R.]
12. The front of a woman's dress over the bosom, usually
worked or embroidered. [Obs.] --Shak.
Hollow square (Mil.), a formation of troops in the shape of
a square, each side consisting of four or five ranks, and
the colors, officers, horses, etc., occupying the middle.
Least square, Magic square, etc.
On the square, or Upon the square, in an open, fair
manner; honestly, or upon honor. [Obs. or Colloq.]
On, or Upon, the square with, upon equality with; even
with. --Nares.
To be all squares, to be all settled. [Colloq.] --Dickens.
To be at square, to be in a state of quarreling. [Obs.]
--Nares.
To break no square, to give no offense; to make no
difference. [Obs.]
To break squares, to depart from an accustomed order.
To see how the squares go, to see how the game proceeds; --
a phrase taken from the game of chess, the chessboard
being formed with squares. [Obs.] --L'Estrange.
· A universe within the mind. Robert Fludd, a 17th century English physician, grand master of the Prieure de Sion, and inventor, theorized that the mind of man was a universe in miniature.
· Extended and directed distance or volume.
· The unlimited three –dimensional expanse in which all material objects are located and all events occur.
· Extended and directed distance or volume.
· The unlimited three –dimensional expanse in which all material objects are located and all events occur.
Aristotle – 384-B.C.
· Ancient Greek philosopher.
· Was known as the Realist,
· Was the philosopher of living organisms endowed with autonomy and purpose, capable of self-organization and self-replication.
· Hidden structure is stronger than visual structure.
· Each individual has its own built-in specific pattern of development and grows toward proper self-realization as a specimen of its type.
· There is nothing in the intellect that was not first in the senses.
· All things are constantly in a state of flux.
· Potentiality – a block of wood is potentially a statue.
· Actuality – the statue is an actualization of the block of woods potentiality.
· Matter – is what has the potentiality of receiving form: form is what actualizes the potentiality.
· 1st to systemize the logical argument.
· The law of identity – everything is what it is.
· The law of non-contradiction – you cannot be both dead and alive.
· The law of the excluded middle – every statement is either true or not true.
· Ancient Greek philosopher.
· Was known as the Realist,
· Was the philosopher of living organisms endowed with autonomy and purpose, capable of self-organization and self-replication.
· Hidden structure is stronger than visual structure.
· Each individual has its own built-in specific pattern of development and grows toward proper self-realization as a specimen of its type.
· There is nothing in the intellect that was not first in the senses.
· All things are constantly in a state of flux.
· Potentiality – a block of wood is potentially a statue.
· Actuality – the statue is an actualization of the block of woods potentiality.
· Matter – is what has the potentiality of receiving form: form is what actualizes the potentiality.
· 1st to systemize the logical argument.
· The law of identity – everything is what it is.
· The law of non-contradiction – you cannot be both dead and alive.
· The law of the excluded middle – every statement is either true or not true.
| QUOTE (Unknown @ Sep 07, 09:39 AM) |
| The forms in nature come from the physics. Not the other way around. |
This thought needs to be expanded on.
Who? What? Where? When? Why? How?
You know how when water freezes it makes crystals and snowflakes and such? That pattern is embedded in the oxygen atom. The oxygen atom isn't pulling that pattern from some separately existing "world of forms" as Plato thought. All mathematics comes out of the physics. One atom, two atoms, three atoms, etc. Atomic number 1 is hydrogen, 2 is helium, 3 is lithium, etc.
When a student grasps the pythagorean theorem for the first time, he reconstructs it mentally, using his physical brain cells. It doesn't flow to him from some mysterious other world. If it did, learning would be much easier.
When a student grasps the pythagorean theorem for the first time, he reconstructs it mentally, using his physical brain cells. It doesn't flow to him from some mysterious other world. If it did, learning would be much easier.
We discovered its pre-exising existence in nature but had to create a conceptual knowledge (percept, recept, concept) that would allow our perceptual apperati to comprehend its pre-existence via mathematics, geometry, numbers, etc.
Everthing in nature has some sort of mathematical algorithimic explanation.
Everything that we as societies conceptually create has mathematical undertonings.
Take a look around your current settings. What do you see?
I do not disagree. I'm just aiding in ariving at answers to some previously posted questions.
Everthing in nature has some sort of mathematical algorithimic explanation.
Everything that we as societies conceptually create has mathematical undertonings.
Take a look around your current settings. What do you see?
I do not disagree. I'm just aiding in ariving at answers to some previously posted questions.
| QUOTE (Rick @ Sep 07, 09:14 AM) | ||
Mathematics is created, not discovered. |
Is your last statement a flip-flop or a re-iteration of this one?
All of the forms, that is, all of mathematics, software, images, every description of every kind, can all be represented as binary strings. Binary strings can encode the integers. Thus, the set of all forms (all information) is exactly equivalent to the set of all integers.
| QUOTE (Trip like I do @ Sep 07, 11:44 AM) |
| Is your last statement a flip-flop or a re-iteration of this one? |
Sorry for the confusing statement. I meant to be consistent. Let me try this:
Plato was totally wrong! There is no world of forms. There is only the physical world.
Mathematics was created as a precursor in understanding what lies behind and beyond the conceptual knowledge of what is and always has been naturallly occuring in nature.
what is mathematics? Is it something in our heads as a way of making sense of our experience or does it exist outside of us?
Mathematics is something in books and in minds. We infer mathematical expressions of physical law from doing experiments and making measurements.
I pose this:
How is the concept of volume, form, surface, space, etc. as defined and outlined by mathimatical priciples portrayed and expressed?
Visually.
Geometrical forms, numbers, all are meaningless if one cannot visually comprehend and express under these conceptual terms.
One is blind if possession of mathematical concepts of nature and reality itself, is limited or non-existent.
How is the concept of volume, form, surface, space, etc. as defined and outlined by mathimatical priciples portrayed and expressed?
Visually.
Geometrical forms, numbers, all are meaningless if one cannot visually comprehend and express under these conceptual terms.
One is blind if possession of mathematical concepts of nature and reality itself, is limited or non-existent.
| QUOTE (Unknown @ Sep 07, 11:10 AM) |
| I pose this: How is the concept of volume, form, surface, space, etc. as defined and outlined by mathimatical priciples portrayed and expressed? Visually. Geometrical forms, numbers, all are meaningless if one cannot visually comprehend and express under these conceptual terms. One is blind if possession of mathematical concepts of nature and reality itself, is limited or non-existent. |
Sorry, I logged out.
CAD programs display geometric information graphically, yet store and maniuplate that information as binary strings in the computer.
Please expand, for I am unfamiliar with the concept that you describe. What angle are you coming at this from?
My friend Claude Horan, a ceramist in Hawaii, is one of the first to use a computer for graphic art. He started with an Amiga doing layouts for ceramic murals. Now he uses a Pentium and does computer art for itself, using a high quality printer and fade resistent ink.
Machine designers have used computer aided design (CAD) programs for over 50 years now. In the 1970s, Aristides Requicia, now at the Univeristy of Southern California, invented "constructive solid geometry" (CSG) that simulates many of the metal cutting operations of metal working machines. The CSG representations are of solid objects from which material may be added or removed in regular ways by the designer.
Machine designers have used computer aided design (CAD) programs for over 50 years now. In the 1970s, Aristides Requicia, now at the Univeristy of Southern California, invented "constructive solid geometry" (CSG) that simulates many of the metal cutting operations of metal working machines. The CSG representations are of solid objects from which material may be added or removed in regular ways by the designer.
Is this something, again, that we we initially created or discovered, and then implemented?
Where did the conceptual knowledge of any such CAD-like programs initially arise?
The concepts of mathematics that are used to physically describe objects of our existing material reality.
Where did the conceptual knowledge of any such CAD-like programs initially arise?
The concepts of mathematics that are used to physically describe objects of our existing material reality.
Computer graphically designed, although at times aesthetically pleasing bears no resemblence to the essence imbued mathematical art created by man.
Where's the soul?
The computer, after all, is just another tool that we as human beings invented, developed and discovered.
The Hidden Dimension ( ? ) – Edward T. Hall
· Extensions of the human form – computer/brain, telephone/voice, wheel/legs or feet, language/experience in time and space, writing/language.
· We forget that our humanness is rooted in animal nature.
· Man as viewed as an organism that has elaborated and specialized his extensions to such a degree that they have taken over, and rapidly replacing nature.
· Man has created a new dimension, the cultural dimension.
· The relationship between man and the cultural dimension is one in which both man and his environment participate in modelling each other.
Where's the soul?
The computer, after all, is just another tool that we as human beings invented, developed and discovered.
The Hidden Dimension ( ? ) – Edward T. Hall
· Extensions of the human form – computer/brain, telephone/voice, wheel/legs or feet, language/experience in time and space, writing/language.
· We forget that our humanness is rooted in animal nature.
· Man as viewed as an organism that has elaborated and specialized his extensions to such a degree that they have taken over, and rapidly replacing nature.
· Man has created a new dimension, the cultural dimension.
· The relationship between man and the cultural dimension is one in which both man and his environment participate in modelling each other.
| QUOTE (Trip like I do @ Sep 07, 12:42 PM) |
| Where did the conceptual knowledge of any such CAD-like programs initially arise? |
Human intelligence is what drives invention (along with necessity, fun, etc.).
I have to quote one of RTB's posts here because it seems applicable to how this string has developed.
Robert the Bruce Posted: May 25, 12:02 AM Quote
http://www.nytimes.com/2003/03/09/nyregion/09NJCOVER.html
No Ghost-Busting at Princeton, but Skepticism Just the Same
By J. D. REED
RINCETON
DOWN a long cinderblock corridor and past the machine shop in the basement of the engineering quad is a dull brown door bearing a decal of a pear. It is the entrance to the Princeton Engineering Anomalies Research (PEAR) laboratory. And what goes on in this warren of small rooms is most anomalous indeed.
For a quarter of a century, a few Princeton University scientists have been quietly gathering data that they believe prove that the mind can influence the performance of machines. While they do not claim to know why the hair dryer burns out before the big date; why the computer some days seems that it knows what is wanted of it, or why a driver coos, "come on baby," when cranking over the car on a freezing morning - they say such things can be cases of mind over matter.
In fact, they theorize that the resonance of the mind with simple machines and sophisticated electronics can affect kitchen mixers and air traffic control. A newer project spun off from the lab here claims that the collective energy of human consciousness around the world is measurable - and in fact showed a sharp increase during the funeral of Princess Diana in 1997 and the terrorism attacks of Sept. 11.
Such musings have caused a good deal of static in the scientific community, much of which regards the investigations in this bastion of Ivy League science - where Nobel prizes come by the six pack - on a par with table-rapping and spoon bending. And although it has permitted the research to continue for almost 25 years, the university itself appears to be a bit embarrassed by PEAR. The privately financed project is not featured in any of Princeton's brochures, and is difficult to track down on the school's main Internet site.
The lab's startling and controversial work takes place in a suite of rooms that belie the subject matter. The main space resembles a graduate student apartment circa 1975. It is paneled in simulated wood and dominated by a burnt orange couch covered with stuffed animals. One almost expects Timothy Leary or Allen Ginsberg or an electrofrizzed Einstein to wander in.
In smaller rooms nearby, unpaid volunteer "operators" who exhibit no special powers or gifts for ESP sit by a variety of small machines - a tabletop toy robot, a spurting water fountain and a computer screen that displays a flickering line. Some read books; some listen to a Walkman, some just stare into space or wander around.
"They may not look like they're doing much," said Brenda J. Dunne, a developmental psychologist who has been at PEAR since its beginning in 1979 and is currently its manager. "But they're trying to influence the output of these machines with their minds. The ones who get the best results don't think too hard about it."
The devices are all versions of a small box-like rig called a REG, or random event generator, that produces a stream of unpredictable digital bits, either ones or zeros, like a computer does. It has been insulated to filter out electromagnetic signals and influences from the environment, and is calibrated to deviate no more than .05 percent above or below its baseline output. Put another way, the REG is essentially a high-speed electronic coin flipper, making the equivalent of 12,000 tosses a minute with a 50/50 chance of producing a one or a zero.
Without touching the equipment or using wires, microphones or electrodes, operators try to make the fountain spurt higher, the robot turn left or right, or send the line on the screen higher or lower with the powers of thought and feeling.
Over thousands of hours of such experiments, the lab's statistics show a small but verifiable effect of human intention on machines that is not attributable to chance or to any external factor. An operator, notes Ms. Dunn, can influence the outcome 2 or 3 times in 10,000.
"So people aren't rushing off to play roulette in Atlantic City with the power of human intention," she said. "If you bet $10,000 at $1 a spin, you might come home with $10,002. Most people wouldn't be satisfied."
Critics of the lab - and there are many - would not give a nickel for this work. Some think the research is, at best, beside the point; others believe it is plain wrong.
Robert L. Park, a science gadfly and a physics professor at the University of Maryland, spoke for many colleagues when he said: "It's voodoo science. They're kidding themselves. In 300 years of looking, not one of this kind of claim of telekinesis has been confirmed. Some people just can't be happy with a universe that pays no attention to us."
The founder of the PEAR lab, Robert Jahn, a physics professor and dean emeritus of Princeton's School of Engineering and Applied Science, was once reportedly told by the editor of a scientific journal that he would publish a paper of Mr. Jahn's if the scientist would send it to him telepathically.
Philip Anderson, a Nobel Prize-winning physicist at Princeton, became so vitriolic about PEAR that he was, he claims, threatened with a lawsuit.
"I can't talk with my usual candor," Dr. Anderson said in a recent interview. "But I can say that I think I represent 95 percent of the physicists that I know in not believing that this kind of work is sound or worth doing. I don't think the lab has reproduced any effects that have convinced any unbiased observers."
Mr. Jahn, an angular and energetic 73, who still teaches the standard engineering curriculum including his specialized field, the plasma physics of spacecraft propulsion, is used to the criticism, and dismisses Mr. Anderson's assertion.
"I never threatened Anderson with a lawsuit," he said. "In fact, I've never had any direct communications with him. We have had some interactions with other skeptics, of course. We've made it clear that whereas we welcome any informed technical reservations they might offer, any malicious public misrepresentations that unfairly defamed the program or its staff members would not be ignored, and would be corrected by some appropriate form of response, including legal remedy as a last resort. Fortunately, we have never been forced to that extreme.
"There are a lot of charlatans and frauds in a field like this," he said. "Naturally some people are wary of what we do."
Mr. Jahn's spacious second-floor office is lined with stuffed animals, mostly gifts from visitors, and he has collected toy giraffes since childhood. "I could say I prefer stuffed animals to some of my colleagues," he said with a smile, "but I won't."
He turns quite serious about the lab's work. "We are not levitating concrete blocks, here," he said. "We offer empirical data that anyone can access and decide for themselves. Our data may only show a few bits in 10,000 of deviation, but many information processors now work with millions of bits of information in the blink of an eye. Think of where a few bits in 10,000 might be critical in a random processor-in medical technology, in aerospace applications or financial software."
Even after almost 25 years of investigation and data correlation, Mr. Jahn is still stimulated. "The exciting thing in the lab is that we are showing physically tangible effects whose primary correlates are subjective," he said. "They don't depend on what machine you're using, how fast you're running it or what experiment you're doing. They depend on how you're feeling, how you relate to the experience, what sort of resonance you establish with the task.
"If you get your mind in resonance with the processor, it will show a preference for following your desire," Mr. Jahn went on. "And it's not just your computer and other machines that can be affected. When the mind is cooperating with the physical, it creates a step beyond. It is the space in performance that athletes call the zone. It's how great artists paint, how great mathematicians solve equations."
The operators down in the lab are far from artistic or mathematical stars. Ms. Dunn and her colleagues have purposely selected quite ordinary people who must spend at least 10 hours with the experiments. The scientists note that women, about half of the 100 testers who have worked at the lab over the years, get larger results than men, but often in the direction they did not intend. Two people working together do better than a lone operator, and a man and a woman in a relationship produce startling results. When groups of local elementary school children tour the lab and try to influence the toy robot, the outcome, said Ms. Dunne, "is off the charts."
"I go into an alpha state, not too focused," said one operator, a retired information systems manager, who spoke only on the condition of anonymity, "but it doesn't always work."
The woman, who said she has "always been interested in the paranormal," has done 100 hours worth of experimenting over 15 years.
Psychics, mystics, Sufi masters and monks in saffron robes as well as other paranormal investigators have visited the lab and sometimes tried using their minds on the machines. "They don't do any better than our regular operators," said Arnold Lettieri, communications director for the lab. "They're usually quite put out about it."
Some, including Uri Geller of spoon-bending fame, expect money or publicity for exercising their telekinesis at the lab, but PEAR does not pay and keeps the identities of the operators private. After a cursory look around, Mr. Geller decided not to participate.
Outside academia, the lab here is often confused with a kind of ivory tower Ghostbusters.
"We get many calls from people who say their houses are haunted and want us to investigate," said Mr. Lettieri. "And we've had inquiries from parents who say 'My daughter is psychic, we want you to test her.' We tell them that we don't do testing. And we certainly do not track down ghosts."
On the other hand, PEAR's activities have attracted more deliberate attention from government agencies working in intelligence and defense. "You can guess which ones," said Mr. Jahn, "but we did not cooperate because we did not want to compromise our scientific freedom."
The lab was founded in 1979 thanks to one of Mr. Jahn's students. "In the mid 70's a student came to me, and asked if I would supervise her thesis, which was on a topic much like what we now do in the lab," he recalled. "I was skeptical, and told her to take the summer and go through the literature and visit some labs, and if she could show me that there was a there there, she could proceed. Two years later I was looking at a batch of data that just from intellectual honesty I said, 'This deserves serious study.' "
The university would not finance the project, but in a serendipitous meeting with James MacDonnell, the co-founder of MacDonnell-Douglas aircraft who was a Princeton graduate, the scientist found a kindred spirit, and one with deep pockets. Mr. McDonnell was a generous and enthusiastic backer , and quickly became a believer.
Mr. Jahn recalleed: "Old Mr. McDonnell would put his hand on my shoulder and say, 'Bob, I know too much about these consciousness-related anomalies to in good conscience put a young man in the cockpit of my F-18 and presume that all that sophisticated microprocessing equipment is going to continue to function nominally when this kid is in a crisis situation. Can I be sure that all that equipment is ignoring the subjective radiation that is coming from the pilot's mind?' "
After about 500 million trials in the lab and hundreds of kangaroo courts in the scientific community, the question is one that Mr. Jahn and his colleagues can still not answer definitively. The feeling is that the lab has probably gone as far as it can for now.
"Within three years we will have pretty much completed collecting and analyzing our data," said Mr. Jahn. "We may take it offshore, and God knows the university won't be sorry to see us go, or we may just let it be like Camelot and dissolve back into the landscape."
If and when that happens, Mr. Jahn will leave behind an indelible, if curious, legacy. He notes with some pleasure that the field he has more or less pioneered is increasing.
"Science advances funeral by funeral," he said. "You are not likely to change entrenched opinions, scientific or theological, by reason and logic. They change because new generations come along who have not been so indoctrinated, and are willing to think outside the box.
"And the sidewalk is beginning to crack,'' he added. "We have a steady flow of research interns who are uncompensated and receive no academic credit. They are from all over the world and they are spreading the news about human intention on machines. These phenomena are moving from curiosities to real possibilities. The result will be real changes in our views of ourselves and our world."
Mr. Jahn is adamant that his lab has proved that there is at least some empirical evidence that mind affects matter.
"Here is the challenge to the empirical scientific community," he said. "Are you going to be able to consider subjective properties in your equations? Will you do experiments in which the role of the observer and the experience are correlated with the objective effects? Are you going to play, or not?"
PEAR is already responsible for several changes. For one thing, Ms. Dunne has organized about 100 of the lab's far-flung acolytes into a Web network she calls the Pear Tree. For another, the program helped establish a university course, Psychology 322, Human-Machine Interactions. The interdepartmental-taught offering explores human-machine interactions via robotics, information systems, biology and psychology.
PEAR is also a member of the Society for Scientific Exploration which, in addition to questions of consciousness, looks into Cryptobiology (the study of strange creatures like the Yeti), UFO's, cold fusion and other controversial matters. Moreover, the global consciousness project is applying the lab's methods to studying the effects of consciousness on a worldwide scale.
And PEAR may become more familiar to non-scientists in the near future. "We are looking into the possibilities of some toys and other consumer products," said Mr. Jahn.
So, in the most recent analysis, does consciousness influence machines? The answer that Mr. Jahn received from the Dali Lama leaves the answer as enigmatic as ever.
"I was allowed to ask him one question," recalls Mr. Jahn of his meeting with the spiritual leader. "I asked, 'Do machines have a mind, a spirit, a personality?' He thought for a moment, and then he said, 'If we attribute a personality to them, then they have it.' "
Robert the Bruce Posted: May 25, 12:02 AM Quote
http://www.nytimes.com/2003/03/09/nyregion/09NJCOVER.html
No Ghost-Busting at Princeton, but Skepticism Just the Same
By J. D. REED
RINCETON
DOWN a long cinderblock corridor and past the machine shop in the basement of the engineering quad is a dull brown door bearing a decal of a pear. It is the entrance to the Princeton Engineering Anomalies Research (PEAR) laboratory. And what goes on in this warren of small rooms is most anomalous indeed.
For a quarter of a century, a few Princeton University scientists have been quietly gathering data that they believe prove that the mind can influence the performance of machines. While they do not claim to know why the hair dryer burns out before the big date; why the computer some days seems that it knows what is wanted of it, or why a driver coos, "come on baby," when cranking over the car on a freezing morning - they say such things can be cases of mind over matter.
In fact, they theorize that the resonance of the mind with simple machines and sophisticated electronics can affect kitchen mixers and air traffic control. A newer project spun off from the lab here claims that the collective energy of human consciousness around the world is measurable - and in fact showed a sharp increase during the funeral of Princess Diana in 1997 and the terrorism attacks of Sept. 11.
Such musings have caused a good deal of static in the scientific community, much of which regards the investigations in this bastion of Ivy League science - where Nobel prizes come by the six pack - on a par with table-rapping and spoon bending. And although it has permitted the research to continue for almost 25 years, the university itself appears to be a bit embarrassed by PEAR. The privately financed project is not featured in any of Princeton's brochures, and is difficult to track down on the school's main Internet site.
The lab's startling and controversial work takes place in a suite of rooms that belie the subject matter. The main space resembles a graduate student apartment circa 1975. It is paneled in simulated wood and dominated by a burnt orange couch covered with stuffed animals. One almost expects Timothy Leary or Allen Ginsberg or an electrofrizzed Einstein to wander in.
In smaller rooms nearby, unpaid volunteer "operators" who exhibit no special powers or gifts for ESP sit by a variety of small machines - a tabletop toy robot, a spurting water fountain and a computer screen that displays a flickering line. Some read books; some listen to a Walkman, some just stare into space or wander around.
"They may not look like they're doing much," said Brenda J. Dunne, a developmental psychologist who has been at PEAR since its beginning in 1979 and is currently its manager. "But they're trying to influence the output of these machines with their minds. The ones who get the best results don't think too hard about it."
The devices are all versions of a small box-like rig called a REG, or random event generator, that produces a stream of unpredictable digital bits, either ones or zeros, like a computer does. It has been insulated to filter out electromagnetic signals and influences from the environment, and is calibrated to deviate no more than .05 percent above or below its baseline output. Put another way, the REG is essentially a high-speed electronic coin flipper, making the equivalent of 12,000 tosses a minute with a 50/50 chance of producing a one or a zero.
Without touching the equipment or using wires, microphones or electrodes, operators try to make the fountain spurt higher, the robot turn left or right, or send the line on the screen higher or lower with the powers of thought and feeling.
Over thousands of hours of such experiments, the lab's statistics show a small but verifiable effect of human intention on machines that is not attributable to chance or to any external factor. An operator, notes Ms. Dunn, can influence the outcome 2 or 3 times in 10,000.
"So people aren't rushing off to play roulette in Atlantic City with the power of human intention," she said. "If you bet $10,000 at $1 a spin, you might come home with $10,002. Most people wouldn't be satisfied."
Critics of the lab - and there are many - would not give a nickel for this work. Some think the research is, at best, beside the point; others believe it is plain wrong.
Robert L. Park, a science gadfly and a physics professor at the University of Maryland, spoke for many colleagues when he said: "It's voodoo science. They're kidding themselves. In 300 years of looking, not one of this kind of claim of telekinesis has been confirmed. Some people just can't be happy with a universe that pays no attention to us."
The founder of the PEAR lab, Robert Jahn, a physics professor and dean emeritus of Princeton's School of Engineering and Applied Science, was once reportedly told by the editor of a scientific journal that he would publish a paper of Mr. Jahn's if the scientist would send it to him telepathically.
Philip Anderson, a Nobel Prize-winning physicist at Princeton, became so vitriolic about PEAR that he was, he claims, threatened with a lawsuit.
"I can't talk with my usual candor," Dr. Anderson said in a recent interview. "But I can say that I think I represent 95 percent of the physicists that I know in not believing that this kind of work is sound or worth doing. I don't think the lab has reproduced any effects that have convinced any unbiased observers."
Mr. Jahn, an angular and energetic 73, who still teaches the standard engineering curriculum including his specialized field, the plasma physics of spacecraft propulsion, is used to the criticism, and dismisses Mr. Anderson's assertion.
"I never threatened Anderson with a lawsuit," he said. "In fact, I've never had any direct communications with him. We have had some interactions with other skeptics, of course. We've made it clear that whereas we welcome any informed technical reservations they might offer, any malicious public misrepresentations that unfairly defamed the program or its staff members would not be ignored, and would be corrected by some appropriate form of response, including legal remedy as a last resort. Fortunately, we have never been forced to that extreme.
"There are a lot of charlatans and frauds in a field like this," he said. "Naturally some people are wary of what we do."
Mr. Jahn's spacious second-floor office is lined with stuffed animals, mostly gifts from visitors, and he has collected toy giraffes since childhood. "I could say I prefer stuffed animals to some of my colleagues," he said with a smile, "but I won't."
He turns quite serious about the lab's work. "We are not levitating concrete blocks, here," he said. "We offer empirical data that anyone can access and decide for themselves. Our data may only show a few bits in 10,000 of deviation, but many information processors now work with millions of bits of information in the blink of an eye. Think of where a few bits in 10,000 might be critical in a random processor-in medical technology, in aerospace applications or financial software."
Even after almost 25 years of investigation and data correlation, Mr. Jahn is still stimulated. "The exciting thing in the lab is that we are showing physically tangible effects whose primary correlates are subjective," he said. "They don't depend on what machine you're using, how fast you're running it or what experiment you're doing. They depend on how you're feeling, how you relate to the experience, what sort of resonance you establish with the task.
"If you get your mind in resonance with the processor, it will show a preference for following your desire," Mr. Jahn went on. "And it's not just your computer and other machines that can be affected. When the mind is cooperating with the physical, it creates a step beyond. It is the space in performance that athletes call the zone. It's how great artists paint, how great mathematicians solve equations."
The operators down in the lab are far from artistic or mathematical stars. Ms. Dunn and her colleagues have purposely selected quite ordinary people who must spend at least 10 hours with the experiments. The scientists note that women, about half of the 100 testers who have worked at the lab over the years, get larger results than men, but often in the direction they did not intend. Two people working together do better than a lone operator, and a man and a woman in a relationship produce startling results. When groups of local elementary school children tour the lab and try to influence the toy robot, the outcome, said Ms. Dunne, "is off the charts."
"I go into an alpha state, not too focused," said one operator, a retired information systems manager, who spoke only on the condition of anonymity, "but it doesn't always work."
The woman, who said she has "always been interested in the paranormal," has done 100 hours worth of experimenting over 15 years.
Psychics, mystics, Sufi masters and monks in saffron robes as well as other paranormal investigators have visited the lab and sometimes tried using their minds on the machines. "They don't do any better than our regular operators," said Arnold Lettieri, communications director for the lab. "They're usually quite put out about it."
Some, including Uri Geller of spoon-bending fame, expect money or publicity for exercising their telekinesis at the lab, but PEAR does not pay and keeps the identities of the operators private. After a cursory look around, Mr. Geller decided not to participate.
Outside academia, the lab here is often confused with a kind of ivory tower Ghostbusters.
"We get many calls from people who say their houses are haunted and want us to investigate," said Mr. Lettieri. "And we've had inquiries from parents who say 'My daughter is psychic, we want you to test her.' We tell them that we don't do testing. And we certainly do not track down ghosts."
On the other hand, PEAR's activities have attracted more deliberate attention from government agencies working in intelligence and defense. "You can guess which ones," said Mr. Jahn, "but we did not cooperate because we did not want to compromise our scientific freedom."
The lab was founded in 1979 thanks to one of Mr. Jahn's students. "In the mid 70's a student came to me, and asked if I would supervise her thesis, which was on a topic much like what we now do in the lab," he recalled. "I was skeptical, and told her to take the summer and go through the literature and visit some labs, and if she could show me that there was a there there, she could proceed. Two years later I was looking at a batch of data that just from intellectual honesty I said, 'This deserves serious study.' "
The university would not finance the project, but in a serendipitous meeting with James MacDonnell, the co-founder of MacDonnell-Douglas aircraft who was a Princeton graduate, the scientist found a kindred spirit, and one with deep pockets. Mr. McDonnell was a generous and enthusiastic backer , and quickly became a believer.
Mr. Jahn recalleed: "Old Mr. McDonnell would put his hand on my shoulder and say, 'Bob, I know too much about these consciousness-related anomalies to in good conscience put a young man in the cockpit of my F-18 and presume that all that sophisticated microprocessing equipment is going to continue to function nominally when this kid is in a crisis situation. Can I be sure that all that equipment is ignoring the subjective radiation that is coming from the pilot's mind?' "
After about 500 million trials in the lab and hundreds of kangaroo courts in the scientific community, the question is one that Mr. Jahn and his colleagues can still not answer definitively. The feeling is that the lab has probably gone as far as it can for now.
"Within three years we will have pretty much completed collecting and analyzing our data," said Mr. Jahn. "We may take it offshore, and God knows the university won't be sorry to see us go, or we may just let it be like Camelot and dissolve back into the landscape."
If and when that happens, Mr. Jahn will leave behind an indelible, if curious, legacy. He notes with some pleasure that the field he has more or less pioneered is increasing.
"Science advances funeral by funeral," he said. "You are not likely to change entrenched opinions, scientific or theological, by reason and logic. They change because new generations come along who have not been so indoctrinated, and are willing to think outside the box.
"And the sidewalk is beginning to crack,'' he added. "We have a steady flow of research interns who are uncompensated and receive no academic credit. They are from all over the world and they are spreading the news about human intention on machines. These phenomena are moving from curiosities to real possibilities. The result will be real changes in our views of ourselves and our world."
Mr. Jahn is adamant that his lab has proved that there is at least some empirical evidence that mind affects matter.
"Here is the challenge to the empirical scientific community," he said. "Are you going to be able to consider subjective properties in your equations? Will you do experiments in which the role of the observer and the experience are correlated with the objective effects? Are you going to play, or not?"
PEAR is already responsible for several changes. For one thing, Ms. Dunne has organized about 100 of the lab's far-flung acolytes into a Web network she calls the Pear Tree. For another, the program helped establish a university course, Psychology 322, Human-Machine Interactions. The interdepartmental-taught offering explores human-machine interactions via robotics, information systems, biology and psychology.
PEAR is also a member of the Society for Scientific Exploration which, in addition to questions of consciousness, looks into Cryptobiology (the study of strange creatures like the Yeti), UFO's, cold fusion and other controversial matters. Moreover, the global consciousness project is applying the lab's methods to studying the effects of consciousness on a worldwide scale.
And PEAR may become more familiar to non-scientists in the near future. "We are looking into the possibilities of some toys and other consumer products," said Mr. Jahn.
So, in the most recent analysis, does consciousness influence machines? The answer that Mr. Jahn received from the Dali Lama leaves the answer as enigmatic as ever.
"I was allowed to ask him one question," recalls Mr. Jahn of his meeting with the spiritual leader. "I asked, 'Do machines have a mind, a spirit, a personality?' He thought for a moment, and then he said, 'If we attribute a personality to them, then they have it.' "
When I was in the Air Force fixing electronic equipment, there was this guy that we wouldn't let touch the equipment. He could just walk up to a transmitter and stand next to it and it would fail. One time, I saw the doors fall of a rack when he walked up to it. We would physically grab his hands and pull him away from the equipment. He should have been a cook. That was his name, by the say, Sgt. Cook.
This is a "lo-fi" version of our main content. To view the full version with more information, formatting and images, please click here.