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Old 05-08-2010, 09:36 PM   #1
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The golden ratio

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In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887
Golden ratio - Wikipedia, the free encyclopedia


Interesting. I remember Douglas Adams' hinting at this in his book "Dirk Gently's Holistic Detective Agency".
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Old 05-08-2010, 09:55 PM   #2
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You can play some really fun games with the golden ratio,

Take a piece of A4 paper (ratio of √2 : 1), if you fold it from top to bottom you create a smaller piece of paper with the same ratio, keep folding in the same manner and the ratio stays the same.

Very fun.
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Old 05-08-2010, 10:00 PM   #3
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Quote:
Originally Posted by A_Wanderer View Post
You can play some really fun games with the golden ratio,

Take a piece of A4 paper (ratio of √2 : 1), if you fold it from top to bottom you create a smaller piece of paper with the same ratio, keep folding in the same manner and the ratio stays the same.

Very fun.
It is also interesting that Fibonnaci analysis, used by technical traders, seemingly occurs in nature:

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Fibonacci numbers in nature

Sunflower head displaying florets in spirals of 34 and 55 around the outsideFibonacci sequences appear in biological settings,[39] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[40] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.[41] In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves.[42] The Fibonacci numbers are also found in the family tree of honeybees.[43]

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[44]

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.[45] This has the form


where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.[46]

Fibonacci number - Wikipedia, the free encyclopedia
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Old 05-08-2010, 10:03 PM   #4
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Originally Posted by A_Wanderer View Post
You can play some really fun games with the golden ratio,

Take a piece of A4 paper (ratio of √2 : 1), if you fold it from top to bottom you create a smaller piece of paper with the same ratio, keep folding in the same manner and the ratio stays the same.

Very fun.
Hold on. So what?
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Old 05-08-2010, 10:58 PM   #5
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It's a fun way of learning about the golden mean, even if the A size paper has been intelligently designed.
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Old 05-08-2010, 11:39 PM   #6
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It's a fun way of learning about the golden mean, even if the A size paper has been intelligently designed.
So, folding a paper in half folds it in half.
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Old 05-09-2010, 12:56 AM   #7
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But that second half is a perfectly scaled down version of the original piece.
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Old 05-09-2010, 07:50 PM   #8
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It is also interesting that Fibonnaci analysis, used by technical traders, seemingly occurs in nature:
And in the human body. I posted a few videos about this recently in the LOGOS thread. Very cool indeed.
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Old 05-09-2010, 07:54 PM   #9
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It's in the Shroud of Turin, too.
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Old 05-09-2010, 09:40 PM   #10
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From what I've read, instances of the Golden Ratio popping up everywhere in nature are somewhat embellished. Its interesting, none the less
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Old 05-09-2010, 10:37 PM   #11
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Fractals are more common in nature. And a lot cooler too.
Probably off the subject but I just think fractals are simple, elegant and amazing.

Isn't that how they make CGI graphics in all the newest Hollywood films?
They basically start with fractals and then simulate mountains (or whatever) with them.
Something like that...
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Old 05-09-2010, 10:43 PM   #12
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I could sit and watch one of those Mandelbrot Set animations for hours
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Old 05-09-2010, 10:44 PM   #13
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And apparently Tool's Lateralus is full of Fibonacci sequences
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Old 05-09-2010, 11:10 PM   #14
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Fractals are more common in nature. And a lot cooler too.
Probably off the subject but I just think fractals are simple, elegant and amazing.
More broadly the subject is mathematical patterns in the universe so it's hard not to meander.

I mentioned McKenna's novelty theory (timewave) in the LOGOS thread as well - it's the fractal nature of it that makes it so cool.

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It predicts the ebb and flow of novelty in the universe as an inherent quality of time.

...

The theory proposes that the universe is an engine designed for the production and conservation of novelty. Novelty, in this context, can be thought of as newness, or extropy (a term coined by Max More meaning the opposite of entropy). According to McKenna, when novelty is graphed over time, a fractal waveform known as "timewave zero" or simply the "timewave" results. The graph shows at what time periods, but never at what locations, novelty increases or decreases and is supposed to represent a model of history's most important events.
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Old 05-10-2010, 11:36 AM   #15
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Fractals are more common in nature. And a lot cooler too.
Probably off the subject but I just think fractals are simple, elegant and amazing.
They fascinate me. I'm believe the universe is just one massive fractal, with out universe just being one level of it.
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