Well, the number of faces you get depends on how you join them together. This post has nothing to do with truestone, or much of anything else, really, but my fondness for geometry won't let this comment pass unremarked.
If you join 3 congruent equilateral triangles at the "top" vertex, then you get a tetrahedron by sticking a fourth congruent triangle on the bottom.
If you join 4 at the top vertex, you get a "four-sided pyramid" (albeit with a somewhat steeper slope than historical pyramid builders used), which can be completed by putting a square on the bottom, or by matching two of them up base-to-base, which makes a d8 (octahedron). (So perhaps pyramids are actually octahedrons buried up to their waists?)
If you join 5, you get a pyramid with a pentagonal base, which doesn't appear anywhere on its own, but if you continue joining 5 together at every vertex, you will eventually end up with an icosahedron (d20)!
If you join 6, you get a hexagon -- a *flat*, planar hexagon. Continue joining 6 at every vertex, and you tesselate the plane.
If you join 7 or more at every vertex, you make a hyperbolic surface, that has all sorts of odd bumps and buckles everywhere, and can never be flattened out (negative Gaussian curvature).
Some of the above holds for congruent equilateral triangles only. You can have tetrahedra where no side is equilateral (the one we know and love as a d4 is a "regular" tetrahedron, and the d6, d12 and d20 are the other specially-named regular polyhedra, the so-called "Platonic solids"). If you work with congruent isosceles triangles, sticking 3 or more of them together at the apex still works, and still makes pyramids, flat polygons, or bits of hyperbolic surface, but you cannot then easily repeat the process (although d10s and some variant d20s and d30s I've seen, the ones that spin like tops, are essentially the octahedron idea above applied to these: stick two n-faced pyramids base-to-base to made a d(2n)). In this case, the number of triangles needed to make the surface flat or hyperbolic instead of a pyramid varies with apex angle: if you use right triangles, then 3 are a pyramid, but 4 are flat and 5 are hyperbolic; if the apex angle is only 18 degrees, then 3-19 form a pyramid, 20 are flat, and 21+ are hyperbolic; for 17 degrees, no combination is flat -- you get either pyramids or hyperbolic surfaces.
(This digression brought to you by the letters pi and theta, and the number e).
> The neat thing about this is that there are three otherworlds, all
> joined together by the underworld, with the mortal realm somewhere
> between them all. Just like the three faces of the pyramid all
> joined by one base. This would make the inside analagous to the
> mortal world. However I've never seen any suggestion that the
> spike was the other place that the three otherworlds met....
> although who knows, I'm sure somebody could interpret it that way..
This I really like! A cube inside a tetrahedron, with perhaps a (hemi?)sphere inside to denote the Dome of the Sky... now we're getting into good old Pythagorean numerology of the orbits of the planets... much to play with there.
Ryan Caveney
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