It is a lovely paradox that the most interesting thing about the golden ratio is that it isn't a ratio. This article is based on a talk in Budd's ongoing Gresham College lecture series see video above. You can see other articles based on the talk here. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.
He has co-written the popular mathematics book Mathematics Galore! Sangwin, and features in the book 50 Visions of Mathematics ed. Sam Parc. The claim about the golden ratio in music actually refers to form, not to frequency though that doesn't stop people from making music with tunings related to the golden ratio, but anyway. The claim is that if you have some work with an AB form, the A and B sections will ideally have durations in the golden ratio, etc. I think it's claimed that this proportion can be specifically found in the music of Mozart.
Golden Ratio is widely practised in the variois drum beats in Carnatic music. They follow rule called 'Hemachandra series - about byears prior to Fibonacci himself'. Also see Melakarta rules structure in Carnatic music. Would love to having a discussion on it. The smaller part goes into the larger as the larger goes into the whole. So the small portion is a ratio of the larger Portion in the same ratio that the larger portion goes into a whole.
Great article that exposes the whole "golden ratio" baloney! One comment, though: you cannot say "degrees are measured in radians", as in the next-to-last paragraph of the section titled "Spirals, Golden and Otherwise"; it is like saying "meters are measured in feet". Note that in Lego bricks the golden ratio is to be found in several aspects, including the relationship of the studs and tubes.
It has been claimed as a significant contributor to their commercial success, although it may be that the system was in part inspired by Le Corbusier architectural designs. Can't we use 2 consecutive Fibonacci numbers in the higher range to approximate the golden ratio? Yes you can, and in fact these are the best rational approximations to the golden ratio.
However, they are still poor approximations and even if you take numbers in the higher range the approximation is not good, and only slowly converges to the correct value. This is in contrast to a transcendental number such as pi which you can approximate much better by using a fraction. You assume the golden ratio is strictly limited to exact formations Its not and no one ever claimed it was.
The fact that you took the generalization that holds try when averaged then complained about individual examples not matching it perfectly shows a gross lack of understanding of the mathematical applications. Explain to me then why life conforms to the golden ratio? Why do flowers have 3 or 5 petal increments? Ratio x2 rounded or x4 rounded. You ignore that this is a living number and that life rounds things as you cant have a partial petal or head as a standard.
Your pantheon example somehow measures the front? When the golden ratio is applied to the base The greatest issue is you clearly think that math is hard unflinching fact. Math is our description of the world. For example if i say no tree is taller than ten increments in height. Then we defining that increment does not change the height of the trees. And most horrifying of all You are using an incorrect equation to calculate the golden ratio.
Golden ratio is a mathematical feed back loop. When your equation is flawed and you are attempting to force it to work perfectly in a science that is widely recognized as variable aka soft science. And relies on generalization and not unbendable law. You also face the issue that your convinced somehow that the ratio is a two dimensional rectangle being applied to three dimensional shapes I have.
When you apply mathematics to soft sciences that vary aka anything that is being used to measure living things. Your argument quite bluntly disproves all medical sciences because no two people have identical hearts and since its not identical mathematically it must mean the theory behind heart disease is wrong because EVERY heart is not identical. To disprove a theory such as this you cannot disprove it as you have.
That's not how science works. You have to attempt to prove it and find what breaks it so completely that is disproven. Please site an example that none of them matched not just a majority. You seem to have made many test but refuse to examine the averages that fall almost exactly on the ratio. No ones ratio will be exactly 1. But the more times you test it the closer to 1.
The golden ratio is not an exact answer. But put your results through calculus graphing. You'll find like limits, the more samples you take the closer to the limit 1. You have not disproven or even dented the golden ratio argument. You have only argued variable facts that supporters of the golden ratio theories have already successfully explained the apparent discrepancies of.
If you simply ignore the fact that math is the measurement of patterns in nature and that the golden ratio is the continuously occuring pattern in life You did not disprove the golden ratio, you disproved math and science. When math disproves itself it typically means the person was wrong as math cannot disprove itself by its nature. I believe that your mis-printing of the ratio fraction is the primary cause.
Your equation is basically converting pi to 3. That kind of early conversion of a fixed number causes massive discrepancies in your math.
I assume you expecting exacting answers in a soft science, refusing to view all of your result objectively via scatter plot or averages led to this extreme inconsistencies in your attempted disprovable of this theory. The nature of life is its an approximation, a rounding of a universal equation we have not unlocked. But the golden ratio is ultimately part of it as the golden ratio is the closest number we can get to infinite. Two methods for the same results. Just 15! For the Voyager spacecraft this imprecision equates with an error of its position of only 1.
PatronDemon, having read the Wikipedia article on the Golden Ratio, I do require now some proof of your horror. That is also the definition and derivation quoted by the author. The Wikipedia article also lists your continued fraction as an alternate formulation for the same number.
Where's the horror? This is a direct copy without the link. I agree with some of your points at a high level, but at the very least they do have the equation correct. It's not an approximation -- the irrationality is derived from the square root of 5. The golden ratio symbol is the Greek letter "phi" shown at left is a special number approximately equal to 1.
This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it? Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape. Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.
The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number , and I will tell you more about it later. The square root of 5 is approximately 2. This is an easy way to calculate it when you need it.
Attempts to find phi in the human body also succumb to similar fallacies. A recent study claimed to find the golden ratio in different proportions of the human skull. And while phi is said to be common in nature, its significance is overblown. Flower petals often come in Fibonacci numbers, such as five or eight, and pine cones grow their seeds outward in spirals of Fibonacci numbers.
But there are just as many plants that don't follow this rule as those that do, Keith Devlin, a mathematician at Stanford University, told Live Science. People have claimed that seashells, such as those of the nautilus, exhibit properties in which phi lurks. But as Devlin points out on his website , "the nautilus does grow its shell in a fashion that follows a logarithmic spiral, i.
But that constant angle is not the golden ratio. Pity, I know, but there it is. While phi is certainly an interesting mathematical idea, it is we humans who assign importance to things we find in the universe. An advocate looking through phi-colored glasses might see the golden ratio everywhere. But it's always useful to step outside a particular perspective and ask whether the world truly conforms to our limited understanding of it.
Adam Mann is a journalist specializing in astronomy and physics stories. He has a bachelor's degree in astrophysics from UC Berkeley. He lives in Oakland, California, where he enjoys riding his bike. Live Science. Adam Mann.
See all comments 4. Many people still think the golden ratio is found all over nature and represents perfect beauty - that is a myth. Even so, phi is a pretty cool math concept. For instance, it's related to the Fibonacci sequence: If you take the ratio of successive Fibonacci numbers, you get closer and closer to phi. Also, like Pi, the golden ratio is irrational and goes on forever! The Golden Ratio, also called Divyank Ratio, is the most economical algorithm of Nature with which the perfect and most beautiful objects of the universe and Nature are designed.
It is designated as Phi. To comprehend the fundamental of the Divyank Ratio, let us contemplate on the following. Fibonacci sequence: It is represented as, and so forth. It is primarily observed in the plant kingdom, like, the branches of a tree, the arrangement of leaves, flowers, fruits, seeds of pineapples, and the pine cone etc.
It is also observed in the family tree of honey-bees and rabbits etc. The Golden Ratio: It is a linear number and represents the two dimensions of an object. It is also an irrational number with never-ending infinite numbers of digits, 1. It is calculated with the help of the following man-made mathematical formula. Hence, there should be limited numbers of digits. This confusion is resolved by Divyank Ratio of
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