After I finished my hyperbolic plane, I kept playing. After a few minutes, I ended up with a cuboctahedron:
The cuboctahedron is an Archimedean solid - one that has several types of regular polygonal faces (in this case, squares and triangles) that meet at identical vertices (each with two squares and two triangles, alternating).
It can be constructed by cutting off the corners of an octahedron:
..or of a cube (mine is not a perfect cube; I'd need shorter pieces):
Cubes have the interesting property that by choosing four vertices so that none of them touch, you get a tetrahedron:
The cuboctahedron can be sliced into two halves. If you rotate one of the halves by one-sixth of a revolution, you get a triangular orthobicupola. It's not quite regular like the cuboctahedron; instead, it's something called a Johnson solid.
Both the cuboctahedron and the triangular orthobicupola have a very similar internal structure. It's not technically part of the polyhedra, but it fits almost exactly. It has three layers of rods - 3, then 6, then 3:
There's only one other way to symmetrically fit twelve rods onto a single vertex: a 1-5-5-1 arrangement:
...which happens to almost be a solid internal structure for an icosahedron:
Unfortunately, it doesn't work perfectly, because an icosahedron is just a tiny bit different from being twenty tetrahedrons put together - a few percentage points different in internal dimensions.
Launch Report 2017-2 - LDRS-36
2 weeks ago