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Old 05-08-2010, 08:36 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, 08:55 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, 09:00 PM   #3
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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.
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, 09:03 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, 09:58 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, 10:39 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 05-08-2010, 11:56 PM   #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, 06:50 PM   #8
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Originally Posted by financeguy View Post
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, 06:54 PM   #9
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It's in the Shroud of Turin, too.
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Old 05-09-2010, 08:40 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, 09:37 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, 09:43 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, 09:44 PM   #13
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And apparently Tool's Lateralus is full of Fibonacci sequences
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Old 05-09-2010, 10:10 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, 10:36 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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Old 05-10-2010, 10:37 PM   #16
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This is two years old, so take it FWIW.

Galaxy map hints at fractal universe - space - 25 June 2008 - New Scientist

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Is the matter in the universe arranged in a fractal pattern? A new study of nearly a million galaxies suggests it is - though there are no well-accepted theories to explain why that would be so.

Cosmologists trying to reconstruct the entire history of the universe have precious few clues from which to work. One key clue is the distribution of matter throughout space, which has been sculpted for nearly 14 billion years by the competing forces of gravity and cosmic expansion. If there is a pattern in the sky, it encodes the secrets of the universe.

A lot is at stake, and the matter distribution has become a source of impassioned debate between those who say the distribution is smooth and homogeneous and those who say it is hierarchically structured and clumpy, like a fractal.

Nearly all physicists agree that on relatively small scales the distribution is fractal-like: hundreds of billions of stars group together to form galaxies, galaxies clump together to form clusters, and clusters amass into superclusters.

The point of contention, however, is what happens at even larger scales. According to most physicists, this Russian doll-style clustering comes to an end and the universe, on large scales, becomes homogeneous.


But a small team of physicists, including Francesco Sylos Labini of the Enrico Fermi Centre in Rome and Luciano Pietronero of the University of Rome argue that the data shows the opposite: the universe continues to look fractal as far out as our telescopes can see.
3D maps

The best data for looking at the galaxy distribution comes from the Sloan Digital Sky Survey (SDSS), which is constructing the largest 3D map of the universe. When completed, it will map the positions of about a million galaxies and quasars.

When SDSS data was released in 2004, physicists David Hogg of New York University and Daniel Eisenstein of the University of Arizona, both in the US, published an analysis of 55,000 luminous red galaxies suggesting that the fractal pattern smoothed out at scales over 200 million light years.

But Sylos Labini and Pietronero were not convinced. They believed that the apparent smoothing was an illusion caused by weak statistics - the smoothing seemed to occur at the largest scales the survey was capable of studying, where there were too few large regions to be able to reliably compare their densities, they said. Only a bigger map could resolve the debate.

Now, SDSS has released its sixth round of data, which plots the locations of roughly 800,000 galaxies and 100,000 quasars, bright objects powered by violent supermassive black holes.
Huge scales

According to their latest paper, which has been submitted to Nature Physics, Sylos Labini and Pietronero, along with physicists Nikolay Vasilyev and Yurij Baryshev of St Petersburg State University in Russia, argue that the new data shows that the galaxies exhibit an explicitly fractal pattern up to a scale of about 100 million light years.

And they say if the universe does become homogeneous at some point, it has to be on a scale larger than a staggering 300 million light years across. That's because even at that scale, they still observe large fluctuations - a cluster here, a void there - in the matter distribution.

Most cosmologists interpret such fluctuations as being no more significant than small waves on the surface of the sea, but Sylos Labini and colleagues say that these are more like tsunamis.
No model

Many cosmologists find fault with their analysis, largely because a fractal matter distribution out to such huge scales undermines the standard model of cosmology. According to the accepted story of cosmic evolution, there simply hasn't been enough time since the big bang nearly 14 billion years ago for gravity to build up such large structures.

What's more, the assumption that the distribution is homogeneous has allowed cosmologists to model the universe fairly simply using Einstein's theory of general relativity - which relates the shape of space to the distribution of matter.

Modelling a fractal universe with general relativity is possible in theory, but in reality it would be devilishly complicated. That would leave cosmologists without a working model, like acrobats without a net.
Relic radiation

To support the homogeneity assumption, cosmologists point to the smoothness of the cosmic microwave background (CMB), relic radiation from the nascent universe. The CMB is perfectly uniform up to one part in 100,000, suggesting the early universe was nearly homogeneous.

"The standard picture of a homogeneous universe on large scales is holding up very well when tested with very large-scale observations like those mapping the cosmic background radiation, X-rays and radio galaxies," says physicist Neil Turok of Cambridge University in the UK.

"If the observations of galaxies in optical surveys don't agree, there may be a number of possible explanations, without resorting to an extremely inhomogeneous, fractal universe," he told New Scientist.
I only quoted page 1 of 2 pages, for length.
So if you guys are interested check out the link.
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