# Platonic Solids

21 20 round and round lesser circles Any navigator will tell you that the shortest distance between two points on a spheres surface is always an arc of a great circle. When a polyhedrons edges are projected onto its circumsphere the result is a set of great circle arcs known as a radial projection. Opposite, the left hand column shows the radial projections of the Platonic Solids with their great circles shown in dotted line. A spherical circle smaller than a great circle is called a lesser circle. Tracing a circle around all the faces of the Platonic Solids set in their circumspheres generates the patterns of lesser circles shown in the middle column. Book XIV of Euclids Elements proves that when set in the same sphere, the lesser circles around the dodecahedrons faces fourth row are equal to the lesser circles around the icosahedrons faces fifth row. The same is true of the cube second row and the octahedron third row as a pair. Shrink the lesser circles in the middle column until they just touch each other to define the five spherical curiosities in the right hand column. Many neolithic carved stone spheres have been found in Scotland carved with the same patterns as the first four of these arrangements. The dodecahedral carvings of twelve circles on a sphere, some 4000 years old, are the earliest known examples of man made designs with icosahedral symmetry. Large lesser circle models can be made from circles of willow, or cheap hulahoops, lashed together with wire, string or tape.