The golden ratio appears in the shape of seashells, the proportions of the human body, and the arms of spiral galaxies. In the spirit of show don’t tell:
The golden ratio:
The golden ratio represented as a line divided into two segments a and b, such that the entire line is to the longer a segment as the a segment is to the shorter b segment.
A golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left) combine to form a similar golden rectangle with long side a + b and short side a.
The golden spiral then is created, by bending a spiral through the corners of an infinite amount of golden ratio rectangles, stacked next to each other:
Golden spirals are self-similar. The shape is infinitely repeated when magnified.
A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares derived from the Fibonacci sequence.
An example of a spiral galaxy, the Pinwheel Galaxy (also known as Messier 101 or NGC 5457)
