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Unformatted text preview: Hemachandra Numbers (and the Golden Ratio) in Nature! The Magic of Numbers February 18, 2010 The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! Hemachandra numbers and the Golden Ratio in Nature We saw last time that the Hemachandra numbers and the Golden Ratio appear everywhere in nature in a very important way. The Golden Ratio φ = 1+ √ 5 2 ≈ 1 . 618 ... is arguably the most impor tant ratio in the natural world. Objects (such as seeds and leaves) that grow in spirals make frequent use of this ratio to achieve packing efficiency and optimal sunlight and rainwater. Why the Golden Ratio? (rather than say 1 / 2, or π , or 1 . 48, or . . . ) The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! How seed spirals grow When a plant (such as a sunflower) grows, it produces seeds at the center of the flower and these push the other seeds outward. Each seed settles into a location that turns out to have a specific constant angle of rotation relative to the previous seed. It is this rotating seed placement that creates the spiraling patterns in the seed pod. The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! The seeds in a sunflower head The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! Summary The seeds in a flower head grow in a spiral with a constant angle. The flowers wish to choose this angle of rotation in order to pack the seeds into the head as efficiently (tightly) as possible. Question : What rotating angle should these flowers choose, in order to pack in the seeds as efficiently as possible? The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! An easy example Suppose this rotating angle is 180 ◦ = 360 ◦ (1 / 2). Then we get a seed arrangement that looks kind of like this: Not terribly efficient for packing seeds! The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! Another easy example Suppose this rotating angle is instead 45 ◦ = 360 ◦ (1 / 8). Then we get a seed arrangement that looks roughly like this: Better, but still not very efficient! The Magic of Numbers Hemachandra Numbers (and the Golden Ratio) in Nature! Slightly better example Clearly a plant shouldn’t use such a simple fraction like 1 / 8 of 360 ◦ ....
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This note was uploaded on 03/14/2010 for the course MAT 190 taught by Professor Bhargava during the Spring '10 term at Princeton.
 Spring '10
 Bhargava

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