Quote:Original post by Erzengeldeslichtes
Now, I like the dodecahedron, it uses pentagons, with the pentagons meeting on the edges, but is it possible to use pentagons to make a larger sphere?

You can always chop up the faces of a polyhedron to make "sides" with whatever number of faces you want, but if you want it to be a regular polyehedron, with just pentagons, the answer is no.

If you put regular pentagons edge-to edge, you get this:

http://mathworld.wolfram.com/p1img1955.gif

When you fold that together, and keep adding more pentagons (12 total), you eventually close the polyhedron and get an dodecahedron:

http://en.wikipedia.org/wiki/Dodecahedron

There's no way to add any more regular pentagons. You can chop each pentagon into 5 triangles (cut corner to centre), but this has the same problems as the icosahedron.

My suggestion is to let units move to directly adjacent triangles only, on a map built by subdividing a regular polyhedron built from equilateral triangles into four smaller equilateral triangles. Start with a tetrahedron, octahedron or icosahedron. This way every triangle will always have 3 adjacent triangles, no matter how much you subdivide.

You do still pay a price for this though, in that if you keep subdividing, the triangles near the original vertices of the polyhedron become distorted when you "blow out" the new vertices you make to the surface of the sphere you're trying to approximate. It'll look weird, but it'll be a uniform set of movement options... though distance calculations might be a little funky. The icosahedron is the least affected by this problem, so you should probly start with that. Octahedrons are nice, though, in that they have a built in equator and perpendicular meridians.