forget it.

I have a buddy that I am sure would love to talk to you about this--he is a math major (whatever the hell that means) I am going to direct him to the conversation I personaly am a big fan of the golden ration~ It is apparent to me that just as the golden ration become more and more perfect, life is evolving to better fit the ratio, and in more complex ways. If you look at a conch shell for example, the golden spiral is obvious. If you look at a pretty symetric person, the ratio is still obvious, but in much more complex ways than the conch. Both animals fit the raito, but one is clearly less complex (less evolved). Sorry if this makes no sense what-so-ever I really know nothing about math, but I love talking out my ass and making rash assumptions.

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Thank you for this post~ It is funny, I know nothing about math really, but when I was high off my ass I started thinking about fibonacci and the order of the universe much like you I created a formula for the probability of the universe (I forget the formula now) but when you plugged in X for any number the result was the golden ratio (except when X was 1)...I ultimately determined that the entire workings of the universe can be broken down into something binary. Everything that has happened throughout time can be broken down to on/off just like a circuit. In modern times it seems that this circuit has become increasingly complicated simply due to an increase in functions. This brought me to a beliefe in hard determinism~

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oh I was mainly laughing at myself and my lack of mathmatic ability and the fact that once I got high I thought I would somehow be able to transend my math shortcomings

Keep me updated on any new findings, this is a topic that is very interesting to me~

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it's not that perplexing.




This makes no sense. To claim there is one infinite number in the Fibonacci sequence is to misunderstand the sequence and the meaning of infinity. The Fibonacci sequence does not have its basis in rabbit populations, even though it is often used to model populations. Make a coherent, intelligent post and you'll surely evoke intelligent replies.

lol loc2k I see you got some sit down time U going to rock the turkey today?

I'm the guy Flex is talking about, though I'm sure I am going to disappoint with respect to how illuminating my post will be regarding the golden ratio itself. Firstly, lucid_dream is correct about the ratio's basis in rabbit breeding, that it was an observed mathematical relationship and not that rabbit populations are the very essence of the golden ratio. Next, this whole special 5 thing. Is it no coincidence that phi = (1 + sqrt(5)) / 2? Root 5? No way! Sorry to mock you further, trojan_libido, but half of your discussion might as well be an argument for the existence of God using rigorous logic. The point here is that any arbitrary number of mathematical relationships can be contrived such that the frequency of a particular number is seemingly high. And this is much easier to do with small numbers especially when talking about nature, for how many clovers exist with more or less than 4 leaves, or any plethora of other essentially discrete phenomena?

This is not to say that the ratio is not an amazing constant in a mathematical sense. It has intimate connections with the transcendental constants pi and e. If we express phi in terms of either of these constants, who's to say pi or e is not responsible for the breeding pattern of rabbits instead of phi? Underlying your post is a philosophical question that is a prerequisite for any validity that can come to your claims. That is, is mathematics a conception of man? Or does "Nature", as you say, impose mathematics on the world? As you point out, there are indeed special, minimalistic patterns in nature, such as the shape of the earth, but such patterns like that one occur because of equilibrium. The laws of gravity must be satisfied as well as all other laws of physics, but those laws do not hold ceterus paribus. It is the composition of all the laws, the net relationships we see, that make the universe what it is and give rise to observable things like phi. Once there was the geocentric (earth-centered) model of the universe before we discovered that the correct model is the heliocentric (sun-centered). Both models gave mathematically useful descriptions of our solar system, and yet we know the heliocentric to be correct. Any model less observably correct than the heliocentric is as superfluous as the geocentric. Similarly, I reiterate that there are an infinitude of mathematical relations that can be observed in our Natural world, yet only a small few are fundamental to however we would define "truth".

It is difficult to deduce that the golden ratio, and the pentagram for that matter, is anything more than beautiful mathematics without a some philosophical--and likely metaphysical--allusions. That's just the way it is when one tries to compile rules for the complex world we live in.

Most definitely I will be rocking the turkey harder than ever. Which reminds me... I need to get my hands on Guitar Hero Deuce when my funds allow!

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Perhaps we misunderstand each other. Your use of the term "infinite fraction" is unconventional. The mainstream mathematics community would construe that as meaning continued fraction, or infinite continued fraction, which is an entirely different thing than what you mean. What you mean to say is infinite decimal expansion. Also, the "complexity" of a fraction needs to be defined. For example, the variance of the distribution of numbers in an expansion can be a measure of complexity. I will elaborate on what I have said earlier.

I do not endorse the statement by lucid_dream that implied that your topic post was unintelligent, so sorry if that's what you thought. However, the fact that no one knows how the process of life began doesn't justify believing, say, in flying pigs. I believe there is such a thing as objective falsehood, and we as people of reason should strive to connect concrete evidence with our hypotheses.

A few things on my statement about phi. Firstly, there existing more n-gons having relations with the Ratio, however complex those relations are, means that the choice of pentagon is arbitrary (as arbitrary as phi being expressed in terms of sqrt(5)). Secondly, asking why mathematical relations are the way they are is not a meaningful question. Why does 1 = 1? Why is the sum of the distance from the foci of an ellipse to a point on an the ellipse the same regardless of the choice of point? The only meaningful alternative is why, philosophically, does phi occur in so many identifiable places. To this end, you have cited history and culture, evidence which is utterly unscientific.

I didn't literally mean that we were arguing God or any sort of creator. I meant, as I have stated, that resolving your questions is asking for a proof of the unprovable. We often use the prominence of something in the universe to infer its significance, which by one definition of significant is valid, but we cannot then pursue why it is significant if ubiquity is our sole reason. Why is hydrogen seen in so many places and objects in the universe? When we question why things are the way they are, which is what we are doing when we ask why the Golden Ratio has seemingly coincidental relations to the penta-gon/-gram, we are questioning things axiomatic in nature. The things you discuss are simply ontological. Again, if you believe my personal religious beliefs play a role in my response to your post, then I am not the attacker.

The point about equilibrium was a response to your statement about the earth being coincidentally round, which you use to support the fact that phi is an equally amazing coincidence. By appealing to equilibrium, I was trying say that the causal forces behind the coincidences that we see are interactions of infinitely many objects in space, objects which by virtue of occupying space have intrinsic relations to every other object. The totality of these relations provides a structure for simplistically beautiful relations. I was trying to achieve that there is no such thing as coincidence in the "special" way that you ascribe to phi. This is especially true when your premises consist of historical/cultural references. A probabilistic argument shows that given a sufficient lapse of time, there are bound to be cultures that make use of "special" numbers.

Now about AI. I have heard before the theory of the co-evolution of artificial intelligence and organic species. With respect to knowledge (information), humans and AI would co-evolve, provided that such AI is benevolent. But all of this talk is highly theoretical. You have essentially presupposed that AI is possible, which in accordance with your opinion is your right, but in that direction I disagree--biological science at its current state is far too primitive to promise with any degree of confidence even the prospect of AI. Math will evolve indeed, but have its foundations and fundamentals not been fully explored? Furthermore, is there a limit to what we can know? Certainly problems in the realm of P/NP completeness, the primes, etc. by there very nature are unpromising for proof.

Again, I am not attacking you. If you feel so, you have misinterpreted what I have said. I am, as lucid_dream would want, attempting to bring coherence to your words.

I am familiar with the mathematics of the Egyptians, the time of Pythagoras, and Fibonacci, but your references are again cultural/historical and perhaps even archetypal. There do seem to be some satisfying philosophical answers to why Nature is the way it is. For example, the shape of an egg is near-optimal for its resilience. If shells and petals which conform to the Fibonacci sequence or the Golden Ratio have some environmental advantages due to their conformity, then that sense of "why" is answerable but in the end far from knowable.

The difference between your belief and mine is the concreteness of our respective evidences.

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What the heck is a mechanism for propagation?

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Sierpinski Triangle:

http://ejad.best.vwh.net/java/fractals/sierpinski.shtml

hey trojan... can you post the images you used in this thread again... i think i may be onto something. thanks.

Lol I didn't know you were still somewhat active on here~ See you soon (you are coming out to the Meg and Dia/Daphne Loves Derbie dates right?)

haha, completely forgot about this for a while. yeah, i was gonna come out to 7 of them, but my car got trashed. i'm going to saratoga for sure. i'm looking for a backup plan for the other shows. hopefully i can jack a car somewhere =).

wow, i just noticed trojan completely wiped out his posts for this thread. what a coward. you hear me, trojan!!?

I'm not a coward, but you brought nothing old or new to my thoughts. You said maths is beautiful and not a lot else. I wiped out my posts simply because I do not want to have my ideas online, I only wanted a sample of response. Granted it was only a small slice, and one that confused those not into philosophy and ancient cultures. I suppose I have only myself to blame.

I think you mathematicians should get together and decide what your stance is on reality. I've been watching the Christmas lectures (not sure if its only on in UK), and the first lecture was about the prime numbers. He explained how some certain insects have a life cycle of 7, 13, or 17 years. They are dormant underground for almost all their lives until the appropriate year, then they all come out at once and make such a racket with their cricket style buzzing that people have to move away. The current theory is that they have evolved to use prime numbers in the life cycle to avoid periodic predators.

I really don't feel that mathematics is just a dry technical subject, and I really feel that you lack a little passion and spirituality for your chosen path.

I didn't know you still read this thread. I've heard of prime life cycles before. The fact that nature can do that shows that there is a physical mechanism in place and therefore suggests that the primes are distributed a certain way (contrary to intuition). At best, we have the Prime Number Theorem and an on-going pseudo-induction on the Riemann Hypothesis, the latter of which thus far has been unsurmountable. Nevertheless, nature is ever-suggestive of mathematical relations, and mathematics developed through a demand from physics. Math is even deeply applicable in fields in which relations were not seen until the math was developed (as in applying group theory to chemistry). The point here is that mathematics is, although formed from the real world around us, no longer just descriptive, which is why through understanding this idea I don't think I lack passion at all, though we must know when in fact math--in nature--is just a descriptive thing as opposed to a relational thing.

That said, I don't think it's fair to say that mathematicians should get together to do such and such. The fact is: they are. As far as spirituality, if you equate that with passion, you are sadly mistaken. I am passionate in all my mathematical research, none of which I have remotely revealed to you. If you mean spirituality in the sense in which it is typically used, that is an unfair charge. I believe math draws conclusions from physical/temporal space, and not that math is put in place by something or due to some spiritual reason.

For your information, the reason I brought this topic back to light is I believe I have found an answer for your original question, or at least an arrow in the right direction. It reminds me of a question Flex asked me a couple months ago about the Fibonacci sequence converging to certain constants. If you have ever read about probabilistic ways to derive pi (the famous needle and parallel lines experiment), you know that there are methods for the geometric convergence of constants. I believe such is the case for the golden ratio with respect to n-grams. There is a regular pattern in the series of n-grams. It is important to note that there may be more than one way to form any given n-gram (there are up to 2, at least for small n). For example, the 7-gram can be formed by vertices at k*2pi/7, k in K = {1, 2, ... , 7} (K congruent mod 7), with each line connecting vertices at a*2pi/7 to those at (a+2)*2pi/7, and it can also be formed by connecting vertices at a*2pi/7 to those at (a+3)*2pi/7, for a in K. It is easy to see the reason for this (isomorphism to cyclic groups). I point this out in case you have not covered the second (or perhaps higher) cases for each n-gram. Now the direction I'm suggesting is that we derive a convergence to phi based on this scheme. You noted that the 5-gram has special properties involving phi. Since the circle in which all n-grams are inscribed relates deeply to pi, the strategy is to find a n-gram sequence that converges to phi (using the fact that pi and phi relate). If this is accomplished, we have the "why" answer for the relationship between phi and pi through a strictly geometric, and thus physical, model. We may be able to make a separate argument using polar coordinates to avoid non-elementary functions (then apply a transformation). I'll post when/if I think of something.

At this point, I take back what I said about 5 being arbitrary. This is very speculative, but since I mentioned groups above, the intimately related Galois theory may apply to your hypothesis about the 5-gram. Abel's impossibility theorem proves there is no closed-form general equation for 5th degree polynomials. (In fact, the quartic equation is the highest degree in which there is a closed-form general solution.) It is probably the case that we have coincidental 5's, but I am not yet ready to reject the 5-gram as special with regard to phi. This is interesting because if indeed a convergence to phi can be derived by the aforementioned method from n-grams, the 5-gram case should not be unique (although this is hard to prove). In other words, my feeling is that the method above contradicts your hypothesis, but at this point it's ambitious to prove rigorously.

Exactly, although I'm more spiritually inclined to ponder on the beginning and how it must have started from a simple process. This is why im intrigued by phi, Fibonacci and the regular n-gons.




Granted passion and spirituality may be perceivably different to you, but I believe you may be confusing religion and spirituality. To see a possible truth and be passionate about its attainment is the belief in yourself and all that mathematics stands for. To believe God made you do it, is a completely separate notion. I believe you are spiritually motivated, you defend your belief with conviction.



Nice work! Are you saying that if you connect the first vertex to the third and fourth, then continue to apply that logic, that you end up recreating the original ngon? I think this is what you are saying, although I'm rusty with my maths - just starting a University course to brush up. I drew a lot of shapes manually with my daughter, to physically check phi and other ngons. I noticed the ngon did repeat, but almost always lines would be drawn through the new ngon.



I'm glad you are attempting to disprove what I originally proposed through your speciality and some new passion has been fostered. I merely spoke philosophically and am unable to say with confidence whether 5th degree polynomials have anything to do with what I spoke of. I will probably have difficultly appreciating what you put forward, but I am working on that.



Its clear that the ngons prior to 5 have no relationship to phi. After 5 its a different story completely. But what I said is that IF this was a mathematical symptom of the force passing information forwards, the force behind DNA, and IF it is indeed infinite, then it would forever reappear after the point of its first appearance. However you are probably not the one to have this conversation with.

I'd be interested to know any unusual fact or fiction about Phi you have learned on your path, as its quite a grey area to research, as well as any findings on this matter.