Yet another encounter with the golden ratio: balancing laminar bodies on the edge

摘要

If one removes a regular even sided polygon from a larger self-similar polygon then the excised polygon can be balanced on the edge provided the ratio of the sides of the larger to the smaller polygon is the golden ratio. Such an excision can be carried out in two ways: (i) vertex excision, where the vertices of the two polygons coincide; and (ii) mid-side excision, where the mid points of the edges of two polygons coincide. We also show why such an exercise with an odd sided polygon does not yield a golden ratio. We briefly discuss the case of the circle which is a limiting case of a regular polygon with an infinitely large number of sides. The results when generalised to other dimensions lead to an interesting pattern of equations.