So it is indeed possible to see the golden ratio in the garden, and there are very good mathematical reasons for this. Fibonacci thought of his sequence when considering the population growth of idealised rabbits. See this article to find out more. But perhaps even more interesting are the many fascinating mathematical properties of the golden ratio.
These are explored in various Plus articles , but I would like to point out one that is particularly fascinating and which really sets the golden ratio apart from other numbers: its extreme irrationality. Irrational numbers are numbers that can't be represented by fractions and that have an infinite decimal expansion that doesn't end in a repeating block.
This very fact means that it is hard to observe irrational numbers in nature. The golden ratio has the amazing property of being the most irrational number of them all.
This means that not only is it not possible to represent it exactly as a fraction, it isn't even possible to approximate it easily by a fraction. See this article for the mathematical details. The difficulty of approximating the golden ratio by a fraction makes it a very useful number to mathematicians and scientists studying the process of synchronisation.
This occurs when a system with a natural frequency is forced by one of a different frequency, and adopts the forcing frequency. One example is the synchronisation of the human body to the daily frequency of sunlight. A second example is the Earth's climate which synchronises to the natural cycles of the orbit around the Sun.
However, synchronisation can itself be a problem, leading to unwanted resonances in a system such as a suspension bridge vibrating severely if a marching band walks over it. By choosing two frequencies in the ratio of we can avoid synchronisation due to the extreme irrationality of the golden ratio. This very useful property appears to be exploited by the brain and insect species as well as climate scientists and even people who manufacture aircraft.
So the golden ratio does have a starring role, but not one that you often read about in the mythology associated with it. This is a great pity! 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.
For example, if you overlay the Golden Spiral on an image, you can make sure that the focal point is in the middle of the spiral. Designer Kazi Mohammed Erfan even challenged himself to create 25 new logos entirely based on the Golden Ratio. The result? Simple, balanced, and beautiful icons. Here are five tools to help you use the Golden Ratio in your designs:.
Look at your hands. Even your fingers follow the Golden Ratio. The human eye is used to seeing this magical number and we subconsciously react positively to it. As designers, we can use this number to our advantage. Even small tweaks to the way you crop an image or develop a layout can dramatically improve how your users interact with your design.
Watch it now. Emily has written for some of the top tech companies, covering everything from creative copywriting to UX design. When she's not writing, she's traveling the world next stop: Japan! Design A guide to the Golden Ratio for designers 4 min read. Link copied to clipboard. Some artists and architects believe that the golden ratio makes the most beautiful shapes. As a result the ratio can be found in many famous buildings and artworks, such as those by Leonardo da Vinci.
Click below to explore the golden ratio in some of Leonardo's most renowned pieces:. Text Block:. What is the golden ratio?
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