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from Curious and Interesting Geometry by David Wells##
Golden Ratio, |

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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