The Golden Ratio

Mighty_Emperor

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June 2004

Good stories, pity they're not true

The enormous success of Dan Brown's novel The Da Vinci Code has introduced the famous Golden Ratio (henceforth GR) to a whole new audience. Regular readers of this column will surely be familiar with the story. The ancient Greeks believed that there is a rectangle that the human eye finds the most pleasing, and that its aspect ratio is the positive root of the quadratic equation:

x2 - x - 1 = 0

You are faced with this equation when you try to determine how to divide a line segment into two pieces such that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter. The answer is an irrational number whose decimal expansion begins 1.618.

Having found this number, the story continues, the Greeks then made extensive use of the magic number in their architecture, including the famous Parthenon building in Athens. Inspired by the Greeks, future generations of architects likewise based their designs of buildings on this wonderful ratio. Painters did not lag far behind. The great Leonardo Da Vinci is said to have used the Golden Ratio to proportion the human figures in his paintings - which is how the Golden Ratio finds its way into Dan Brown's potboiler.

It's a great story that tends to get better every time it's told. Unfortunately, apart from the fact that Euclid did solve the line division problem in his book Elements, there's not a shred of evidence to support any of these claims, and good reason to believe they are completely false, as University of Maine mathematician George Markowsky pointed out in his article "Misconceptions About the Golden Ratio", published in the College Mathematics Journal in January 1992. But with such a wonderful story, which marries some decidedly accessible pure mathematics with aethestics, architecture, and painting - a high school math teacher's dream if ever there were one - the facts have had little impact.

But being aware that few people will take note of what I say has never stopped me before. (I was, after all, a department chair in a college mathematics department for four years and a college dean for another eight.) So let's try to separate the fact from the fiction.

First, what do we know for sure about the Golden Ratio? As mentioned above, Euclid showed how to calculate it, but his interest seemed more that of mathematics than visual aesthestics or architecture, for he gave it the decidedly unromantic name "extreme and mean ratio". The term "Divine Proportion," which is oten used to refer to GR, first appeared with the publication of the three volume work by that name by the 15th century mathematician Luca Pacioli. Calling GR "golden" is even more recent: 1835, in fact, in a book written by the mathematician Martin Ohm (whose physicist brother discovered Ohm's law).

It is also true that the Golden Ratio is linked to the pentagram (five-pointed star), to the five Platonic solids, to fractal geometry, to certain crystal structures, and to Penrose tilings. So far so good.

The oft repeated claim (actually, all claims about GR are oft repeated) that the ratios of successive terms of the Fibonacci sequence tend to GR is also correct. The Fibonacci sequence, you may recall, is generated by starting with 0, 1 and repeatedly applying the rule that each new number is equal to the sum of the two previous numbers. So 0+1 = 1, 1+1 = 2, 1+2 = 3, 2+3 = 5, etc., giving the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... The sequence of successive ratios of the numbers in this sequence, namely 1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666... ; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615...; 34/21 = 1.619...; 55/34 = 1.6176...; 89/55 = 1.6181; ..., does indeed tend to GR. As I'll explain momentarily, this is a key part of the explanation of why the Fibonacci numbers keep appearing in flowers and plants - which they do.

For instance, if you count the number of petals in most flowers you will find that the answer is a Fibonacci number. For example, an iris has 3 petals, a primrose 5, a delphinium 8, ragwort 13, an aster 21, daisies 13, 21, or 34, and Michaelmas daisies 55 or 89 petals. All Fibonacci numbers.

Again, if you look at a sunflower, you will see a beautiful pattern of two spirals, one running clockwise, the other counterclockwise. Count those spirals, and for most sunflowers you will find that there are 21 or 34 running clockwise and 34 or 55 counterclockwise, respectively - all Fibonacci numbers. Less common are sunflowers with 55 and 89, with 89 and 144, and even 144 and 233 in one confirmed case. Other flowers exhibit the same phenomenon; the wildflower Black-Eyed Susan is a good example. Similarly, pine cones have 5 clockwise spirals and 8 counterclockwise spirals, and the pineapple has 8 clockwise spirals and 13 going counterclockwise.

Finally, if you take a close look at the way leaves are located on the stems of trees and plants, you will see that they are located on a spiral that winds around the stem. Starting at one leaf, count how many complete turns of the spiral it takes before you find a second leaf directly above the first. Let P be that number. Also count the number of leaves you encounter (excluding the first one itself). That gives you another number Q. The quotient P/Q is called the divergence of the plant. (The divergence is characteristic for any particular species.) If you calculate the divergence for different species of plants, you find that both the numerator and the denominator are usually Fibonacci numbers. In particular, 1/2, 1/3, 2/5, 3/8, 5/13, and 8/21 are all common divergence ratios. For instance, common grasses have a divergence of 1/2, sedges have 1/3, many fruit trees (including the apple) have a divergence of 2/5, plantains have 3/8, and leeks come in at 5/13.

Although many of these observations were made a hundred year or more ago, it was only recently that mathematicians and scientists were finally able to figure out what is going on. It's a question of Nature being efficient.

For instance, in the case of leaves, each new leaf is added so that it least obscures the leaves already below and is least obscured by any future leaves above it. Hence the leaves spiral around the stem. For seeds in the seedhead of a flower, Nature wants to pack in as many seeds as possible, and the way to do this is to add new seeds in a spiral fashion.

As early as the 18th century, mathematicians suspected that a single angle of rotation can make all of this happen in the most efficient way: the Golden Ratio (measured in number of turns per leaf, etc.). However, it took a long time to put together all the pieces of the puzzle, with the final step coming in the early 1990s.

The worst kind of angle for efficient growth would be a rational number of turns, eg. 2 turns, or 1/2 a turn, or 3/8 of a turn, since they will soon lead to a complete cycle. Mathematically, a turn through an irrational part of a circle will never cycle, but in practical terms it could eventually come close. What angle will come least close to a cycle? Maximum efficiency will be achieved when the angle is "furthest away" from being a rational. But what exactly does that mean? The appropriate way (via s vis plant growth) to measure how far removed from being rational an irrational number is, is to look at its continued fraction expansion. For GR, this is:

Rational numbers have a finite continued fraction. That unending, constant sequence of 1's in the continued fraction for GR says that, measured in terms of continued fractions, GR is the irrational number furthest removed from being rational. And that's the mathematical reason why Nature favors GR as her growth ratio. The Fibonacci numbers appear because the number of leaves, spirals, etc. are whole numbers, and (because of the ratio limit property mentioned above) the Fibonacci numbers are the best whole number approximations to a GR growth.

For other examples of the appearance of the Golden Ratio in Nature, the growth of the Nautilus shell is governed by the Golden Ratio, as is the path followed by a Pergrine falcon when it swoops down to catch its pray. In these cases, the explanation is that the GR is closely related to the logarithmic spiral, the spiral that turns by a constant angle along its entire length, making it everywhere self-similar.

As the Nautilus grows, it has repeated need to enlarge its living quarters. Since the creature does not change shape, rather simply grows larger, the most efficient way to do this is for its shell to grow in the self-similar form of a logarithmic spiral.

The falcon must keep the prey in its sight all the time, but, although its eyes are razor sharp, they are fixed in its head, one on either side. So what the creature does is swivel its head to one side, by an angle of about 40o, and fix the prey in one eye. Keeping its head fixed at that 40o angle, the falcon then dives in a way that keep the prey in view in that one eye. The fixed angle of the head results in the bird following an equi-angular spiral path that converges on the prey.

So much for the good (i.e., true) stuff. Now for those many, many myths about GR that continue to do the rounds. The issue here is not whether you can find GR somewhere. If you look hard enough you will be able to find any (reasonably sized) number almost anywhere. The question is whether there is more to it than mere numerology. Is there a good scientific explanation to show why GR appears (as with the examples from Nature mentioned above), or is there definite evidence that, say, a particular artist made deliberate use of GR in his or her work? If not, all you have is an unsubstantiated belief. You may as well believe in fairies.

First of all, whether or not the ancient Greeks felt that the Golden Ratio was the most perfect proportion for a rectangle, many modern humans do not. Numerous tests have failed to show up any one rectangle that most observers prefer, and preferences are easily influenced by other factors. As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the Golden Ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle - usually red or yellow - is drawn and ask yourself: why put it exactly there and why make the lines so thick?)

Another claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense. First of all, you won't get exactly the number GR. You never can; GR is irrational, remember. But in the case of measuring the human body, there is a lot of variation. True, the answers will always be fairly close to 1.6. But there's nothing special about 1.6. Why not say the answer is 1.603? Besides, there's no reason to divide the human body by the navel. If you spend a half an hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6, or 1.5, or whatever you want.

Then there is the claim that Leonardo Da Vinci believed the Golden Ratio is the ratio of the height to the width of a "perfect" human face and that he used GR in his Vitruvian Man painting. While there is no concrete evidence against this belief, there is no evidence for it either, so once again the only reason to believe it is that you want to. The same is also true for the common claims that Boticelli used GR to proportion Venus in his famous painting The Birth of Venus and that Georges Seurat based his painting The Parade of a Circus on GR.

Painters who definitely did make use of GR include Paul Serusier, Juan Gris, and Giro Severini, all in the early 19th century, and Salvador Dali in the 20th, but all four seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason. Also, the Cubists did organize an exhibition called "Section d'Or" in Paris in 1912, but the name was just that; none of the art shown involved the Golden Ratio.

Then there are the claims that the Egyptian Pyramids and some Egyptian tombs were constructed using the Golden Ratio. There is no evidence to support these claims. Likewise there is no evidence to support the claim that some stone tablets show the Babylonians knew about the Golden Ratio, and in fact there is good reason to conclude that it's false.

Turning to more modern architecture, while it is true that the famous French architect Corbusier advocated and used the Golden Ratio in architecture, the claim that many modern buildings are based on Golden Rectangles, among them the General Secretariat building at the United Nations headquarters in New York, seems to have no foundation. By way of an aside, a small (and not at all scientific) survey I carried out myself a few months ago revealed that all architects polled knew about the GR, and all believed that other architects used the GR in their work, but none of them had ever used it themselves. Make whatever inference you wish.

Music too is not without its GR fans. Among the many claims are: that some Gregorian chants are based on the Golden Ratio, that Mozart used the Golden Ratio in some of his music, and that Bartok used GR in some of his music. All those claims are without any concrete support. Less clear cut is whether Debussy used the Golden Ratio in some of his music. Here the experts don't agree on whether some GR patterns that can be discerned are intended or spurious.

Poetry too is not immune, but here there is a refreshing surprise in store for us. Whereas the claim that the Roman poet Vergil based the meter of his poem Aeneid on the Golden Ratio has no support, it really is true that some 12th Century Sanskrit poems have a meter based on the Fibonacci sequence (and hence related to the Golden Ratio).

I could go on, as there are many more examples, ranging from the sacred (eg. the dimensions of the Ark of the Covenant) to the profane (such as, predicting the behavior of the stock market), all of which, on close examination, turn out to be without any supporting evidence whatsoever. Despite the lack of evidence, however, and in some cases in the face of evidence to the contrary, each claim seems to attract its own band of devotees, who will not for a moment entertain the possibility that their cherished beliefs are not true. Consequently, not only is GR a very special number mathematically - all of its genuine appearances in mathematics and Nature show that - it also has enormous cultural significance as the number that most people have the greatest number of false beliefs about. Now there's a GR fact that has plenty of supporting evidence.

----------------------------
For more details on the Golden Ratio, including evidence to support many of the claims I have made above, see the Markowsky article mentioned earlier, as well as the excellent book The Golden Ratio: The Story of PHI, the World's Most Astonishing Number, by Mario Livio. Also worth a visit is Ron Knott's excellent website Fibonacci Numbers and the Golden Section, at the University of Surrey in England.

-----------------------------
Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book is Sets, Functions, and Logic: an Introduction to Abstract Mathematics (Third Edition), published by Chapman and Hall in 2003.

http://www.maa.org/devlin/devlin_06_04.html
 
A

Anonymous

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As an artist i find this topic very interesting. The article says that the GR is mainly "nature being efficient" - never disagreed with that. Still, we may have a better chance of understanding the beauty surrounding us, by understanding the mechanisms behind it - and maybe to reproduce some of that beauty by using those mechanisms to our own ends?

"Music too is not without its GR fans."

to those interested in music composed utilizing the fibonacci numbers:

Magnus Lindberg: "action - situation - signification" (1988, fazer music inc.)

Mikko Heiniö: "Duo per violino e pianoforte op. 34 (1979)

and on a lighter note: played a little with the ratio:

http://members.fortunecity.com/rattus777/leikkaus.html
(the same image duplicated over itself and rotated a number of times)

:)

cheers, rattus
 

austen27

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The proportion 1 to root 2 is quite common in Georgian architecture as it is naturally created in the diagonals.

Le Corbusiers "Modulor" used fibonacci numbers to proportion parts rather than the whole.
 
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Austen said:
The proportion 1 to root 2 is quite common in Georgian architecture as it is naturally created in the diagonals.

This is also the ratio of the length to width in A-series paper sizes (e.g. A4)- it has the property that halving the rectangle gives two rectangles of the same proportions as the original one.

So you can get two sheets of A5 by cutting one sheet of A4 in two, and the proportions of A4 and A5 are the same.
 

phi23

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On their album, Geogaddi, Boards of Canada have at track called "a is to be as b is to c" (which is the golden ratio) and they have mentioned in interviews that they have used the golden ratio in the production of some of their music.
 

rjmrjmrjm

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I remember seeing a poster of this in a A level maths room. I seem to recall it had a human body, the UN building and a leaf as examples of the Golden Rectange.

Evidence of intelligent design?
 

ramonmercado

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Another view on the Golden Ratio.

Researcher explains mystery of golden ratio
http://www.physorg.com/print180531747.html
December 21st, 2009 in Other Sciences / Mathematics


This is Adrian Bejan of Duke University. Credit: Duke University


The Egyptians supposedly used it to guide the construction the Pyramids. The architecture of ancient Athens is thought to have been based on it. Fictional Harvard symbologist Robert Langdon tried to unravel its mysteries in the novel The Da Vinci Code.

"It" is the golden ratio, a geometric proportion that has been theorized to be the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. Now, a Duke University engineer has found it to be a compelling springboard to unify vision, thought and movement under a single law of nature's design.

Also know the divine proportion, the golden ratio describes a rectangle with a length roughly one and a half times its width. Many artists and architects have fashioned their works around this proportion. For example, the Parthenon in Athens and Leonardo da Vinci's painting Mona Lisa are commonly cited examples of the ratio.

Adrian Bejan, professor of mechanical engineering at Duke's Pratt School of Engineering, thinks he knows why the golden ratio pops up everywhere: the eyes scan an image the fastest when it is shaped as a golden-ratio rectangle.

The natural design that connects vision and cognition is a theory that flowing systems -- from airways in the lungs to the formation of river deltas -- evolve in time so that they flow more and more easily. Bejan termed this the constructal law in 1996, and its latest application appears early online in the International Journal of Design & Nature and Ecodynamics.

"When you look atwhat so many people have been drawing and building, you see these proportions everywhere," Bejan said. "It is well known that the eyes take in information more efficiently when they scan side-to-side, as opposed to up and down."

Bejan argues that the world - whether it is a human looking at a painting or a gazelle on the open plain scanning the horizon - is basically oriented on the horizontal. For the gazelle, danger primarily comes from the sides or from behind, not from above or below, so their scope of vision evolved to go side-to-side. As vision developed, he argues, the animals got "smarter" by seeing better and moving faster and more safely.

"As animals developed organs for vision, they minimized the danger from ahead and the sides," Bejan said. "This has made the overall flow of animals on earth safer and more efficient. The flow of animal mass develops for itself flow channels that are efficient and conducive to survival - straighter, with fewer obstacles and predators."

For Bejan, vision and cognition evolved together and are one and the same design as locomotion.The increased efficiency of information flowing from the world through the eyes to the brain corresponds with the transmission of this information through the branching architecture of nerves and the brain.

"Cognition is the name of the constructal evolution of the brain's architecture, every minute and every moment," Bejan said. "This is the phenomenon of thinking, knowing, and then thinking again more efficiently. Getting smarter is the constructal law in action."

While the golden ratio provided a conceptual entryway into this view of nature's design, Bejan sees something even broader.

"It is the oneness of vision, cognition and locomotion as the design of the movement of all animals on earth," he said. "The phenomenon of the golden ratio contributes to this understanding the idea that pattern and diversity coexist as integral and necessary features of the evolutionary design of nature."

In numerous papers and books over past decade, Bejan has demonstrated that the constructal law (www.constructal.org) predicts a wide range of flow system designs seen in nature, from biology and geophysics to social dynamics and technology evolution.

Provided by Duke University
 

ramonmercado

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You wait 5 1/2 years for a golden ratio story to come along and then you get 2 in the same day.

Researchers discover new 'golden ratios' for female facial beauty
http://www.physorg.com/news180195066.html
December 16th, 2009 in Medicine & Health / Psychology & Psychiatry

Subjects were shown faces with the same features but with different distances between the eyes and between the eyes and mouth. Faces with an average length or width ratio - which were chosen as most attractive - are framed in black. (Photo courtesy of Pamela Pallett, UC San Diego.)

(PhysOrg.com) -- Beauty is not only in the eye of the beholder but also in the relationship of the eyes and mouth of the beholden. The distance between a woman's eyes and the distance between her eyes and her mouth are key factors in determining how attractive she is to others, according to new psychology research from the University of California, San Diego and the University of Toronto.

Pamela Pallett and Stephen Link of UC San Diego and Kang Lee of the University of Toronto tested the existence of an ideal facial feature arrangement. They successfully identified the optimal relation between the eyes, the mouth and the edge of the face for individual beauty.

In four separate experiments, the researchers asked university students to make paired comparisons of attractiveness between female faces with identical facial features but different eye-mouth distances and different distances between the eyes.

They discovered two "golden ratios," one for length and one for width. Female faces were judged more attractive when the vertical distance between their eyes and the mouth was approximately 36 percent of the face's length, and the horizontal distance between their eyes was approximately 46 percent of the face's width.

Interestingly, these proportions correspond with those of an average face.

"People have tried and failed to find these ratios since antiquity. The ancient Greeks found what they believed was a 'golden ratio' - also known as 'phi' or the 'divine proportion' - and used it in their architecture and art. Some even suggest that Leonardo Da Vinci used the golden ratio when painting his 'Mona Lisa.' But there was never any proof that the golden ratio was special. As it turns out, it isn't. Instead of phi, we showed that average distances between the eyes, mouth and face contour form the true golden ratios," said Pallett, a post-doctoral fellow in psychology at UC San Diego and also an alumna of the department.

"We already know that different facial features make a female face attractive - large eyes, for example, or full lips," said Lee, a professor at University of Toronto and the director of the Institute of Child Study at the Ontario Institute for Studies in Education. "Our study conclusively proves that the structure of faces - the relation between our face contour and the eyes, mouth and nose - also contributes to our perception of facial attractiveness. Our finding also explains why sometimes an attractive person looks unattractive or vice versa after a haircut, because hairdos change the ratios."

The researchers suggest that the perception of facial attractiveness is a result of a cognitive averaging process by which people take in all the faces they see and average them to get an ideal width ratio and an ideal length ratio. They also posit that "averageness" (like symmetry) is a proxy for health, and that we may be predisposed by biology and evolution to find average faces attractive.

More information: The research is published by the journal Vision Research.


Provided by University of Toronto
 

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Disclaimer: I'd forgotten that I wrote this back at the beginning of January while dosing myself with laphroaig (for medicinal purposes, of course). It may be a bit stream of consciousness...and badly punctuated...and a bit rubbish, but I haven't got the energy to edit it just now.

James_H2 said:
The Golden Ratio in the popular 'Amen Break'? This seems like nonsense.

Not necessarily. The golden section was common to all elements of the Pythagorean Quadrivium, including music, and I don't think it's a matter for argument that mathematics and music are fundamentally connected in all sorts of ways - again, Pythagoras (although I believe that the legend of the blacksmiths hammers has been debunked there's not much doubt that he found a relationship between musical harmony and mathematics). Admittedly, you do have to indulge in a little triage when researching the subject via the net. It's unfortunate that mathematical truth is occasionally lost under layers of woo-woo - Dan Brown has a lot to answer for.

Allegedly Bartok and Sibelius consciously used the golden section in structuring composition, and there are claims that it appears in the work of many other composers, including Haydn, Chopin and Schubert (consciously or otherwise, I'm not sure - I've never seen a proper analysis). Unfortunately the example which, in my experience, is most often illustrated - in some of Dufay's motets - and which I'd always assumed was an undisputed example, has been somewhat demolished by Charles B Madden in his book Fib and Phi in Music. (However, Madden offers other examples of the connection between maths and music that I'd never heard of).

Of course, if you follow the theory that the golden section represents both a mathematical and a natural truth (ie - something we naturally recognise as harmonious even if we are not aware of the mathematical theory behind it) then there's no reason why The Winstons should have consciously been using the golden section when they constructed the Amen Break.

I find the subject fascinating and love the idea of the golden section, but I'm not sure it doesn't get over attributed. Madden points out a tendency to find patterns by employing weak structural points - which, it strikes me, could be equally true in many subjects which are based on the discovery of underlying patterns - from earth mysteries to conspiracy theories.

There's quite a nice little piece on the Amen Break here. There's some interesting stuff on ownership in there as well, which seems very timely.

Now, some people are going to claim that the amen break itself is overattributed: how can such a short sequence of drumbeats be 'read' into so many later works? (And it strikes me that the Amen Break is therefore in it's way an echo of the Golden Section itself). However, it's worth pointing out that there's no argument that many people who use the amen break do so very consciously and without ambiguity and are aware of it's history.
 

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I know that artists have been consciously using the golden section for some time, as the notion of a 'higher geometry' has long been popular. Especially with architects, but many composers, painters and sculptors etc see parallels between their work and architecture.

But what this guy is doing is finding ratios in the image of the waveform, starting and ending his scale where he likes.
 

rynner2

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Audio/Video File:

The Golden Ratio: Possibly the best rectangle in the world
Duration: 01:55

Find out more about how the proportions of the Golden Ratio are consistently found in nature and have replicated by artists and architects for thousands of years. Voiced by Harry Shearer. Scripted by Nigel Warburton.

http://www.bbc.co.uk/programmes/p02bllqm

:D
 

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rynner2 said:
Audio/Video File:

The Golden Ratio: Possibly the best rectangle in the world
Duration: 01:55

Find out more about how the proportions of the Golden Ratio are consistently found in nature and have replicated by artists and architects for thousands of years. Voiced by Harry Shearer. Scripted by Nigel Warburton.

http://www.bbc.co.uk/programmes/p02bllqm

:D

What annoys me about those Beeb animations is that they don't say who did the cartoons. They look a lot like the work of my fave artist Hunt Emerson, but there's no signature or credit anywhere.
 

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and as if by magic....

Is space-time shaped like a SPIRAL? Universe has a 'golden ratio' that keeps everything in order, researchers claim
South African researchers say the universe is governed by a 'golden ratio'
They say space-time itself is defined by this mathematical constant
The ratio - 1.618 - is found across nature in plants, hurricanes and more
But the researchers say it is also ever-present in the universe
This means it might make up space-time itself
Some have suggested our universe may have been the only one in the multiverse theory to have this ratio that allowed it to form

A cosmic constant known as the ‘golden ratio’ is said to be found in the shape of hurricanes, elephant tusks and even in galaxies.

Now researchers say this ratio is also seen in the topology of space-time, affecting the entire universe as a whole.

And they say this number can be used to link everything in the universe together, from space-time to chemistry to biology.


The golden ratio, represented by the Greek letter 'phi', is said to be is a mathematical connection between two aspects of an object.

It can be artificially used – for example, some 20th century artists used it for the rectangular shape of their portraits from the long side to the short side.

They believed that the ratio created an aesthetically pleasing appearance.

But the ratio is not just artificially created – it is apparently found through nature in the stems of plants, skeletons of animals and so on.

And the shape of spirals also seem to follow the golden ratio. This suggests that geometric shapes in the universe ultimately succumb to this mathematical property.

‘A convincing case for assuming a cosmic character of the golden ratio can be made based on the ubiquity of logarithmic spirals,’ the researchers write.

‘Spectacular examples include the Whirlpool Galaxy (M51), ammonites, the shape of Nautilus shells, Hurricane Katrina and the distribution of planets, moons, asteroids and rings in the solar system.’

The researchers suggest that the reason that this ratio is so ubiquitous is that it is actually a property of space-time.

‘The argument that this amazing consilience (self-similarity) arises from a common environmental constraint, which can only be an intrinsic feature of curved space-time, is compelling,’ they write.

‘The time has come to recognise that relativity and quantum theories can be integrated, and linked numerically to the value of a mathematical constant - whether in the context of space-time or biology’

Quite why the universe follows this rule, however, is not known.

Some think that our fine-tuned universe is simply a lucky coincidence and, under the multiverse theory, there are an infinite number of other universes that were not quite so lucky.

http://www.dailymail.co.uk/sciencetech/ ... claim.html
 

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Anonymous-50446

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What annoys me about those Beeb animations is that they don't say who did the cartoons. They look a lot like the work of my fave artist Hunt Emerson, but there's no signature or credit anywhere.

His "Ryhme of the Ancient Mariner" is quite wonderful.
 

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Thanks to @GroenMNG for proving the golden ratio can be applied to this pic: pic.twitter.com/Fa1EYSV6ih

1:51 PM - 1 Jan 2016
CXo57HFWEAAhRe0.jpg:large
 

JamesWhitehead

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Peripart

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This is also the ratio of the length to width in A-series paper sizes (e.g. A4)- it has the property that halving the rectangle gives two rectangles of the same proportions as the original one.

So you can get two sheets of A5 by cutting one sheet of A4 in two, and the proportions of A4 and A5 are the same.

Whereas, of course, a rectangle with sides differing by the golden ratio, would be one where, of you cut the biggest possible square from it, what remained would still be the same shape*. The square root of 5 comes into it, but not the root of 2. Sorry for quoting a post from 2004!

*As neatly illustrated in the image in post #18
 

rynner2

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The gallery loads in Firefox too. Good to see the interweb getting interesting and creative for a change. :)
 

EnolaGaia

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This newly-published article provides a (-nother) overview of the famous Golden Ratio and explains why it is arguably the most irrational of all irrational numbers.
The Most Irrational Number

The golden ratio is even more astonishing than Dan Brown and Pepsi thought.

One of the great charms of number theory is the existence of irrational numbers—numbers like the square root of 2 or π that can’t be expressed as the ratio of any two whole numbers, no matter how large. The legend goes—probably false, but hey, it makes a point—that the discovery of the irrationality of√2 was so disconcerting to the Pythagoreans, who wanted all numbers to be rational, that they threw the discoverer into the ocean.

Among the mysteries of the irrationals, one number holds a special place: the so-called golden ratio. The golden ratio’s value is about 1.618 ...

People have been making a fuss over this number for centuries. ...

There’s been a miasma of mysticism around the golden ratio for a long time. The number theorist George Ballard Mathews was already complaining about it in 1904, writing that “the ‘divine proportion’ or ‘golden section’ impressed the ignorant, nay even learned men like Kepler, with a sense of mystery, and set them a dreaming all kinds of fantastic symbolism.” Figures with lengths in golden proportion to one another are sometimes said to be inherently the most beautiful, though the claims that the Great Pyramid of Giza, the Parthenon, and the Mona Lisa were all designed on this principle aren’t well substantiated. An influential 1978 paper in the Journal of Prosthetic Dentistry suggests that a set of false teeth, for maximum smile appeal, should have the central incisor 1.618 times the width of the lateral incisor, which should in turn be 1.618 times as wide as the canine. There’s a small but persistent school of financial analysis which holds that the golden ratio governs fluctuations in the stock market ...
FULL STORY: https://slate.com/technology/2021/06/golden-ratio-phi-irrational-number-ellenberg-shape.html
 
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