0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,…

The Fibonacci sequence, found by adding two consecutive terms together to get the following term, is found everywhere in nature, art, and mathematics.

If we look at ratios of terms of the Fibonacci sequence, 2/1, 3/2, 5/3, 8/5, 13/8, 21/8, 34/21, 55/34, etc., the terms of the resulting sequence get closer and closer to what we call the “Golden Ratio,” which is approximately 1.618 and commonly referred to in mathematics with the lower-case Greek letter phi. Rectangles with sides in this ratio are said to be in “ideal proportion.”

If we take a “golden rectangle” and split it into a square with side length the smaller dimension of the rectangle, and a smaller rectangle, the resulting rectangle will also be a golden rectangle. If we keep dividing these smaller golden rectangles into squares and smaller golden rectangles, we can connect the vertices to form a “golden spiral.”

Image courtesy Wikimedia Commons.

Perhaps the best way to really grasp the essence of the golden ratio is to construct a golden rectangle. Call the shorter side of the rectangle (the side length of the square) x, and the shorter side of the smaller inner rectangle 1. Then the sides of the larger rectangle are in the ratio (1+x)/x, and since the smaller rectangle is also golden, its side length ratio of x/1 is equal to (1+x)/x. If we take this equality (1+x)/x = x and solve for x (using the quadratic formula – negative a plus or minus the square root of b squared minus 4ac all over 2a), we get the exact value of the golden ratio – (1+sqrt5)/2.

One of the most obvious examples of the golden ratio in nature is in phyllotaxis, the arrangement of leaves on a plant stem. These ratios are almost always ratios of consequtive Fibonacci numbers. If we divide the circumference of a circle into two arcs in golden ratio to each other, the angle subtending the smaller arc is approximately 137.5 degrees and referred to as the “golden angle.” The seeds of a sunflower are arranged according to this angle.

Image courtesy Wikimedia Commons.

My recent painting “Fibonacci Octopus” (on display at the ASMS Gallery until August 23 as part of my show Visual Mathematics) explores these properties of the golden ratio and more.

The spacing of the panels: 1, 1, 2, 3.

There are 5 panels.

An octopus has 8 arms.

My octopus’s tentacles form golden spirals.

Image courtesy artnotapathy.com.

In the piece I show a few other interesting formulas and expressions that lead us to the same exact number phi that is approximately 1.618. For example the continued fraction 1+1/(1+1/(1+1/(1+…))), when iterated “infinitely” approaches phi.

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