# Properties Of The Platonic Solids

 Posts 53
Casual Member
Myrtone
Casual Member
Joined: Mar 22 2010, 08:08 AM
I've found out about an interesting property of three of the platonic solids. If s is the edge of a platonic solid (whether a tetrahedron, a cube or octahedron) and d is the diameter of the circumscribing sphere then s*s/d*d is a dozenal number. For a tetrehedron it is 0;8, 0;6 for an octahedron, and 0;4 for a cube. And the ratios are exactly these numbers, not just these numbers to one dozenal place. Only one of them, by contrast, is a decimal number, the other two having repeating decimal expansions.
Furthermore the square of the sine of the angle between faces of a tetrahedron is simply 0;Ð–8, exactly, not just eight ninths to two dozenal places.

That aside, both the number of corners of a tetrahedron and the number of faces (these being the same number), as well as the number of edges, are both factors of twelve, neither being factors of ten, only the former being a factor of the short hundred. Being factors of twelve, they are also factors of a gross, as are the numbers of faces, sides and edges of both a cube and octahedron, these being dual.

 Posts 237
Regular
jrus
Regular
Joined: Oct 23 2015, 12:31 AM
For me, 8/9 is a clearer representation than 0;Ð–8. YMMV.

More generally, if you start computing the sine squared of various angles constructed from icosahedral symmetries, you end up getting various rational combinations of 1 and the golden ratio, with 5 and 15 in the denominator. These donâ€™t have any nice â€œdozenalâ€ expansion.

(e.g. the angle between two lines both passing through the center of an icosahedron and through adjacent vertices is ~63.4349Â°, with sine squared of 4/5.)