In each step, a square the length of the rectangle's longest side is added to the rectangle. A Fibonacci spiral starts with a rectangle partitioned into 2 squares. Īnother approximation is a Fibonacci spiral, which is constructed slightly differently. The result, though not a true logarithmic spiral, closely approximates a golden spiral. The corners of these squares can be connected by quarter- circles. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares.
This rectangle can then be partitioned into a square and a similar rectangle and this rectangle can then be split in the same way. įor example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. The next width is 1/φ², then 1/φ³, and so on. For a square with side length 1, the next smaller square is 1/φ wide.
The length of the side of a larger square to the next smaller square is in the golden ratio.
Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral.
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