DW> If you're not concerned with convexity, can't you, say, glue platonic solids
together at their faces? For example, join two dodecahedra at a face to get a 22-face pentagonahedron(?). This can be done ad infinitum to make tree-shaped hedra.
Right, nonconvex is easy. E.g., that 3D "Snowflake". Or worse: flat triangular grids on the faces of a tetrahedron. But hey, isn't the following a fairly ball-shaped 360hedron?: "Start" with a pentakis dodecahedron = 60 equilateral triangles (which are not quite equilateral when projected onto the circumsphere). Then "triakis" that with very shallow triangular pyramids. Finally perpendicularly bisect the long sides and lift the midpoints even more slightly. Since not all vertices are on the sphere, it's hard to claim moral superiority over the dipyramid, except perhaps on volume. Rejecting that, there's "DisdyakisTriacontahedron", which looks to me like it 120-sects its circumsphere. (It's hard to tell without rebooting this stupid laptop, which stubbornly refuses to reset to 1440x900.) rwg>If you unite each hexagon with three nonconsecutive
neighbor triangles, you get equilateral triangles enclosing the pentagons in an appealing camera-shutter arrangement that maybe we could sell to a soccer ball manufacturer.
I think you can open and close the twelve shutters and continuously morph between an icosahedron and a dodecahedron. --rwg ANODISED ADENOIDS