I recently ran across references to these dice having been made as long ago as Ptolemaic Egypt out of stone, bone and glass. Making a hollow version out of twenty triangles, or molding one of clay is easy enough;, but how do you cut and polish an accurate solid version without advanced math like trig? Specifically, I was wondering if there was a way to start with an easy to make solid like a cube, and then follow some rule to plane it down to an icosahedron.
You can plane a dodacahedron into an icosahedron…but I don’t think that really helps.
Once you’ve made one, through any method, you could easily use it as a template for planing down others, no math needed. You could even turn a poor approximation of a template into a better approximation on the second one, by repeatedly rotating your template in various ways.
There is a geometrical connection between icosohedra and cubes. If you can make a cube accurately, mark its faces accurately, you should then be able to plane down to an icosohedron.
I should add that the rectangles shown in the first cut-the-knot graphic are golden rectangles; which means that they are constructible using classical methods. All of the platonic solids are therefore constructible.
I believe that all the archimedian solids are also constructible since they are derived from the platonic solids. However, it would not surprise me if the snub cube and the snub dodecahedron posed more of a problem. They are the quirky members of that family.
So far so good; but given that, how would you go about actually planing the cube? Can it be done solely by truncating vertices, edges and faces, or would you have to mark golden ratio reference points on the cube and use those as guides?
All 12 vertices of an icosohedron are represented by points on the surface of teh cube. So, I would begin by marking these clearly.
Better than that, six of the 30 edges can be marked – one on each of the cube faces.
Assuming I was able to plan a flat – well plane, (and probably my skills are not up to it) it is simply a matter of planing down to those marked points. The desired shape is after all convex and so provided I don’t dish out the surface or go past the marks I have made then I should be ok.
From a practical standpoint, how to physically hold the material at a suitable angle while I plane it – I guess that is another problem. If I was making it out of bone or some similar material I guess I would be holding it in my hand while I laborously rubbed it against a flat rock.