Harmonic ratios of the regular polygons, definitionsA regular polygon is an equilateral and equiangular polygon: the equilateral triangle, the square, the equilateral pentagon, hexagon, etc. As proved in Euclid's Elements, Book IV, each regular polygon (with number of sides 3, 4, 5, 6, 8, 10, 12, 15) has a unique center, and two unique circles around that center: the inscribed circle, which touches each side once in its midpoint, and the circumscribed circle, which passes through each vertex. In fact, Euclid constructs these circles using straight edge and compass only.The harmonic ratio of a regular polygon is the ratio of the radius of the circumscribed circle to that of the inscribed circle. Rev'd 17 jan 2002 by ralph abraham |