[Happy Monkey]

Originally Posted by BigV

It looks like two different creations to me.

HM, have you ever considered stuff like Uniform Tilings when doing work like this?

Not explicitly, though I assume that that type of math is involved when new polyhedra are discovered.

I think of a dodecahedron, a regular polyhedron made up of triangular faces. Then, I see regular pentagons formed from your groups of five equilateral triangles and I wondered if those pentagons could be used to uniformly tile a sphere.

Actually, a dodecahedron IS a pentagon-tiled sphere - 12 pentagons. The one with 20 triangles is an icosahedron.

It makes me wonder if you do all of this freestyle, just making a shape to connect to a previous shape, until you get to the last ones where you'd have to do some thinking, like a 3D game of Nim.

That most describes the process when I made this one (shown in progress):

Originally Posted by Happy Monkey

I made a corner, by making an icosahedron with 3 pentagons removed and replaced by rings of triangles. It looked like it might make something, so I made more corners until they met. The math isn't quite right, so as I got to the end, I had to use the flexibility of the material to pull it into shape, and certain interior parts look crushed.

There's another one I made by trial and error, which I'll post soon.

Usually, I have a polyhedron in mind from the web or a book, and I replace non-triangle faces with a pyramid made of triangles. That's how I made my favorite shape:



I found a polyhedron pattern, and replaced pentagons with concave pentagonal pyramids, and triangles with convex triangular pyramids, and it turned out that the faces of those lined up, creating diamond-shaped faces. And the resulting shape is actually the logo of Wolfram Alpha.