from Curious and Interesting Geometry by David Wells
If a star pentagon is inscribed in a regular pentagon, the golden ratio naturally appears. The same ratio appears in the dodecahedron and the icosahedron, which Euclid constructed using the division of a line in the 'extreme and mean ratio', as he called it.
(Figure of pentagon ABCDE with inscribed star, not shown on this web site)
Each of the ratios AQ/QD, AP/PQ, and AD/BC is equal to ½(1+ √5), about 1.618. This is usually denoted by the Greek letter φ (or sometimes τ).
This ratio has the property that φ = 1/(φ - 1) or, expressed in another way, φ² = φ + 1
a 'golden rectangle' whose sides are in this ratio can therefore be dissected into a square and another rectangle of the same shape. The process can be repeated ad infinitum.
(Figure of golden rectangle not shown on this web site)
An equilangular spiral can be drawn through these vertices. A sequence of circular quadrants is a good approximation to the spiral. The true spiral does not actually touch the rectangles.
(Figure of equilangular spiral inscribed within golden rectangle not shown on this web site)
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