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99.22 The golden ratio and equilateral triangles

The golden ratio arises in a wide range of geometrical situations [1], possibly the most well-known of which concerns a regular pentagon of edge-length 1 unit. Indeed, is equal to the length of any diagonal of such a pentagon. Slightly less familiar might be the fact that can be constructed from an equilateral triangle and a square [2] or from an equilateral triangle, a hexagon and a circle [3]. We show here how also arises from a simple recursive geometric construction involving equilateral triangles. x2 − x − 1 = 0 lim n → ∞ Fn + 1 / Fn Fn n th