Internally Subdividing Polyhedra

We can generate embedded polyhedra by a simple technique. Between any two vertices which make an edge, we create a new vertex for the embedded polyhedra in the middle of that edge. Joining those new vertices generates the new solid. For the case of the cube, this process immediately results in a truncated cube, the cuboctahedron. Curiously enough, we can also generate the cuboctahedron by two stages of dividing the tetrahedron. The end of the first stage results in an octohedron. Dividing the octahedron results in a cuboctahedron.

Subdivided Tetrahedron

Geomags are perfect for building these geometrical objects.

Rendered Tetrahedron

In the rendered image, I've colored the three planes of the internal octohedron red, green and blue.

Rendered Octohedron

Here is the isolated octohedron. Again, notice the orthogonal bisecting planes.



Now we take the octohedron, magnify by two, and create nodes at the edge midpoints. Eliminate the original nodes, and use the central node for bracing gives us a rigid cuboctahedron.



While the tetrahedron and octohedron are stiff structures, the cube and cuboctahedron are not. Internally bracing the cuboctahedron with a twelve point star does achieve a stiff structure with cartesian orientation and stackability.

Rendered images, of course, don't care about mechanical stability. Here is the cuboctahedron rendered, with four bisecting planes with hexagon cross sections, colored red, green, blue and yellowish.

Rendered Cuboctohedron