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Origami polyhedra
Here's a page on origami polyhedron modules. The one I recently learned how to work with is the very simple Sonobe module, which my coworker Bill Barnert showed us how to produce one day at lunch (Bill had a big pile of leftover handbills for a Lions Club carnival that we used for the pieces).
With the exception of the "Epcot ball", which has plane tesselations on the large faces, the shapes on that second page are all cumulations of Platonic solids with triangular faces, where the pyramid used for the cumulation is a cube-corner. You need one module for every edge in the uncumulated polyhedron, so the little cube (which is really a cumulated tetrahedron) uses 6, the octahedron uses 12, and the icosahedron 30. The icosahedron is an interesting toy, since you can push some of the vertices inward to make it undergo startling transformations, including one form that looks like a saucer-shaped cluster of cubes.
I figured out that you can also make flat square faces by linking together four of the modules instead of three (though they pull apart easily, so shapes that use them hold together best if they've also got pyramidal faces). You can actually, I think, make cubes of at least three different sizes: that little one that is a cumulated tetrahedron, a bigger one made from 12 modules with the flat square faces, and an even bigger one from 24 modules that is a cuboctahedron with the triangular faces cumulated. (Even bigger flat-faced cubes would require square tesselations on the faces and would probably be extremely flimsy.)
You can go beyond that by using other non-Platonic polyhedra as a basis. A simple, strange-looking one I did was the gyrobifastigium, which I've known about ever since Kibo taught me that it's fun to say "gyrobifastigium". What I really want to make someday is the 60-module Sonobe version of the snub cube, which I suspect would be awesome, kind of like the aftermath of six cubes violently colliding in hyperspace.
It's also interesting to think about the minimal nontrivial shapes made from these Sonobes. You can hook two of them together to make a flat (but thick) square, and three to make a sort of tiny, squat triangular dipyramid that is the cumulation of a flat triangle (or, I suppose, really two flat triangles face to face).
With the exception of the "Epcot ball", which has plane tesselations on the large faces, the shapes on that second page are all cumulations of Platonic solids with triangular faces, where the pyramid used for the cumulation is a cube-corner. You need one module for every edge in the uncumulated polyhedron, so the little cube (which is really a cumulated tetrahedron) uses 6, the octahedron uses 12, and the icosahedron 30. The icosahedron is an interesting toy, since you can push some of the vertices inward to make it undergo startling transformations, including one form that looks like a saucer-shaped cluster of cubes.
I figured out that you can also make flat square faces by linking together four of the modules instead of three (though they pull apart easily, so shapes that use them hold together best if they've also got pyramidal faces). You can actually, I think, make cubes of at least three different sizes: that little one that is a cumulated tetrahedron, a bigger one made from 12 modules with the flat square faces, and an even bigger one from 24 modules that is a cuboctahedron with the triangular faces cumulated. (Even bigger flat-faced cubes would require square tesselations on the faces and would probably be extremely flimsy.)
You can go beyond that by using other non-Platonic polyhedra as a basis. A simple, strange-looking one I did was the gyrobifastigium, which I've known about ever since Kibo taught me that it's fun to say "gyrobifastigium". What I really want to make someday is the 60-module Sonobe version of the snub cube, which I suspect would be awesome, kind of like the aftermath of six cubes violently colliding in hyperspace.
It's also interesting to think about the minimal nontrivial shapes made from these Sonobes. You can hook two of them together to make a flat (but thick) square, and three to make a sort of tiny, squat triangular dipyramid that is the cumulation of a flat triangle (or, I suppose, really two flat triangles face to face).