Here is the same with the pentagramma, but without the sphere (play with the sliders, and try to get a sense of how the octahedra interconnect):

And here are just the five octahedra:

The five octahedra together define what is called a “compound” solid, and, as a compound, exist as an artifact of the pentagramma mirificum. Play with the animations above, and make one for your self.

(*Note that this is not *the *compound of five octahedra (the author knows a different compound of five octahedra, and there likely could be others), but this is a very interesting compound of five octahedra, as we will see below; we can call it the *compound mirificum*)

Take note of how the different octahedra connect. Begin by looking (on the animations above, or the one you made) at a corner of the pentagon of the pentagramma mirificum: at each of these five corners of the pentagon, two octahedra come together, and, as should be evident from the work above, the same two octahedra will also come together at a vertex on the opposite side of the sphere. Thus, each of octahedron shares two opposite vertices, and an axis, with another octahedron; this actually occurs twice for each octahedron. In the picture below: the red octahedron connects to the brown and wood colored octahedra at the indicated places (which are corners of the pentagon of the pentagramma).

However, the red octahedron does not connect to the white or silver octahedra; the wood and brown do connect to the silver and white octahedra though. So traveling along the edge of the red, one could, through crossing at a shared vertex, pass to the wood, and from the wood travel to the white, across the vertex they share. So all five are interconnected in this chain-like fashion. Looks complex, at least as it appears visually; but we could, as we did in another way in part one, develop a map. How many faces, edges and vertices does an octahedron have? (If you haven't built one, build one and find out.) The following image will suffice as a planar map of one octahedron, where we have the proper number of sides, vertices, and faces, and they all connect properly, though the sizes are distorted (note that, for this planar map, we treat the largest, outer triangle (EFD) as one of the faces; then, along with the other 7 irregular triangles, the 8 faces of an octahedron are accounted for):

In the compound mirificum, one octahedron connects to another by sharing one pair of opposite vertices, so we could take two such planar maps, and place them vertex to vertex, and we have one of the connections:

However, a connected pair of octahedra share two vertices; no worry, we just have to find the second set of vertices which are be connected, and we can represent this second connection with a dashed line like so:

Thus, the map of the compound of five octahedra will look like this (the colors correspond to the one pictured above; also note that when two dashed lines cross they *do not* represent a connection, any one dashed green line is only connecting two vertices of two octahedra):

What may, initially, look to be a jumbled mess, now, having its relations mapped out, shows a beautiful symmetry; but what can we make of this compound mirificum now that we have this map? The specific way the octahedra connect to one another (with any one octahedron not being connected to all the others directly, only to two specific others) is shown clearly here. An investigation of these interconnections leads to the question of “topology.”