Some Solid Connections, Page 3: The Platonic Solids and the Cuboctahedron

Each of the three solids just looked at above (generated from 3, 4, and 6 great circles dividing the sphere evenly) has a very special relation to one, or a specific pair, of a most regular class of solids, the “platonic” solids (where the solid is composed solely of a single, regular shape for each of its faces, and all its vertices lie on a single sphere).  Here, for reference, are each of the five platonic solids, and each mapped onto a sphere.

 

OCTAHEDRON:

CUBE:

ICOSAHEDRON:

DODECAHEDRON:

TETRAHEDRON:

A very beautiful ordering is shown, that the solids generated by dividing the sphere with great circles in the least arbitrary way (i.e. the most regular division in terms of great circles), have an intimate relation to the solids which are generated by dividing the sphere into the same regular shapes (i.e. most regular in terms of faces), as we will explore this relation below.

We are only dipping into a much broader field of different regular and quasi-regular divisions of the sphere, and the corresponding solids; the connection to our investigation comes through a specific type of “truncation” of these platonic solids: taking the midpoint of each edge of a solid, and connecting these midpoints with rectilinear lines, generating a new solid (as a “truncation” of the first, as if we were chopping off sections of the first).  Because of the regular nature of the platonic solids we are truncating, the resulting solids will, like the platonic solids, have all their corresponding edges be equal, but the faces will not necessarily all be the same shape.

The bisected-edge truncation of the octahedron:

The bisected-edge truncation of the cube:

Notice, the bisected-edge truncation of the cube and of the octahedron produce the same solid: the cuboctahedron (in the animation below where the cube and octahedron share the same cuboctahedron, the cube and octahedron will not be from the same sized sphere; for our investigation here, this is just worth noting).

The reason why the octahedron and cube produce the same solid, when truncated in this way, is that they (the cube and octahedron) are “duals.” This dual relationship can be known by connecting the centers of the faces of a platonic solid, this will generate its dual, another platonic solid.  Again, it is necessary that the reader is to construct these, both on the sphere, and in the rectilinear form (preferably on the sphere first).

The octahedron is exposed as the dual of cube:

And the cube, likewise, as the dual of the octahedron:

The duals shown together, including their spherical arcs:

Now it gets interesting; here are the spherical arcs that correspond to the edges of the cube and the octahedron- and their bisected-edge truncation the cuboctahedron: