Some observations on the excavated forms of the Platonic and Archimedean Solids
The ratios between the edge lengths and the radii of these solids are expressed in rational and irrational numbers. In the case of the cuboctahedron which stands halfway between the cube and octahedron, the ratio between edge length and radius is 1:1.
The full width of the arms from fingertip to fingertip is the same as the height. The ratio is 1:1.
(H0: To an accuracy of 3 significant figures with a sufficiently large sample of adults.)
In the case of the icosidodecahedron which stands halfway between the dodecahedron and icosahedron, the ratio between edge length and radius is 1: Φ or 1 : (√5+1)/2, the golden ratio.
The distance from the top of the head to the navel and the distance from the navel to the soles of the feet is also in the golden ratio, as is the distance from the navel to the soles of the feet to the whole height.
Quite why these simple ratios should be instantiated in the human form remains to be explained. Pythagoreans believed that ‘All is Number and Ratio’ so perhaps this fact would have made them happy.
Chirality is also found in the Archimedean solids. The snub cube and the snub dodecahedron both have left-handed and right-handed forms. So that is another point in common between the simple solids – generated by imposing some basic conditions on Euclidean space – and the human form.
