The Eagle, the Shell, and the Sunflower

The Golden Spiral appears all over nature. /

Editor’s note: This is the third in a series on the Fibonacci sequence. You’ll get the most out of it if you read parts one and two first.

“To see a world in a grain of sand.
And Heaven in a wild flower.
Hold infinity in the palm of your hand.
And eternity in an hour.” —William Blake, “Auguries of Innocence”

If the language of the universe is math, then there’s one poem engraved across scientific fields as disparate as botany, ornithology, and meteorology: the Golden Spiral.

To understand the Golden Spiral, we have to start with the Fibonacci numbers. I explained in previous articles that when we arrange the Fibonacci numbers arithmetically, we obtain the Golden Ratio. What happens when we arrange them geometrically instead?

Let’s get to building: We’ll start by taking squares with side lengths the size of the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13…). We will rotate the squares counterclockwise around each other, building each square off of a side formed by the combination of the previous squares. We can follow the process of the Fibonacci sequence in the increasing sizes of the squares.

This “Golden Rectangle” (so named because the ratio between the length and the width of the rectangle approaches The Golden Ratio with each iteration) is a geometric manifestation of the Fibonacci numbers.

Now go back to the interior squares (1 unit by 1 unit) and connect the vertices of the squares using successive quarter-circles with radii that have corresponding Fibonacci numbers. The interconnected series of quarter-circles is called the “Golden Spiral.” The Golden Spiral is another geometric manifestation of the Fibonacci numbers, and it’s ...

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Also in this Issue

Issue 50 / June 9, 2016
  1. Editor's Note from June 09, 2016

    Issue 50: Lightning bugs, beating hearts, and golden spirals. /

  2. The Most Spectacular Firefly

    We’re drawn to animals that shine their own light. For one, it’s a group effort. /

  3. Mysteries of a Beating Heart

    We don’t really know how a heartbeat sparks to life. /

  4. Bright Angels

    “When I was growing up, / they were lightning bugs.” /

  5. Wonder on the Web

    Issue 50: Links to amazing stuff.

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