The Mysteries of the Golden Ratio Explained by Math and Sunflowers

This number, approximately 1.62, has fascinated mathematicians and artists for centuries—but perhaps not for the reasons you think.


The golden ratio is an almost mythical number that you may have heard of in various areas of architecture or design. For example, many claim the . Others have said that the .

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These claims are rough approximations of the golden ratio at best and pseudoscience at worst. The truth is much more awesome.

The wonderful YouTube channel recently spoke to Ben Sparks, a mathematician working at the University of Bath, to reveal the true nature of the golden ratio: 1 plus the square root of 5 over 2, or approximately 1.62, represented in mathematics by the Greek letter phi. In geometry, this number produces some fascinating patterns, such as a . (If you take the shorter length of a golden rectangle and make a square with that length, and then remove the area of that square from the golden rectangle, you are left with another, smaller golden rectangle.)

As the video below explains, the golden ratio can also be considered the "most irrational" of all irrational numbers. An irrational number is one that cannot be expressed by a fraction of integers, or whole numbers. Pi, for example, is an irrational number. It is almost 22/7, but not quite.

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A mathematical method for exploring irrational numbers is to play a kind of game. The idea is that you have a flower, and you are trying to place seeds on the face of the flower in such a way that you can fit as many as possible. If you place a seed, and then rotate the flower face a certain amount, and place another seed, and then repeat this process, what would be the ideal amount to rotate the flower face?

For example, if you place a seed, rotate the flower 1/2 turn, place another seed and repeat, then you will get two parts of the flower with all the seeds. If you do 1/3 of a turn then you will get 3 lines of seeds, and if you do 1/10 of a turn you get 10 lines.

The best number to rotate the flower for optimal seed placement is the golden ratio, where the entire face gets covered more or less evenly without the seeds clustering in any one spot. As the video demonstrates, this is because the golden ratio is a highly irrational number, meaning it is not very well approximated by any whole number or even any fraction of whole numbers.

As it turns out, when you map the seeds of a sunflower on the face of the flower, it's as though the flower were playing this very game, placing seeds to near mathematical perfection. Make sure to watch the video for a visual explanation and more information about the awesome power of the golden ratio—and why it is the most irrational of numbers.

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