The 8 labeled cube sponges

Shown below are the eight possible isogonal labelings of the cube linked to their VRML models, using three different display variations. The cube edges have been marked using the three colors red, blue, and green, representing the a, b, and c edges, respectively. You can see that the cube looks identical no matter which of the eight corners you choose to view it from. (Keep in mind that a vertex and its reflection are considered identical. This is because there is no way to decide on a "correct" version (original or reflected) so mathematicians consider them basically the same as each other.) But when a vertex has been reflected in the new perspective, observe that every other corner is also a reflection from its previous perspective.

One thing to note is that the red edges are always paired with other red edges. Suppose instead that red edges were always paired with blue edges. This would mean, of course, that every blue edge would have to be paired with a red edge, leaving the green edges only to be paired with other green edges. The end result would be that the only thing that has changed is that a different color for the 'a' edge (green) has been chosen instead of red. In other words, nothing has changed mathematically. Therefore these eight labelings do truly represent all that are possible.

Another way to think about this is as follows. Suppose each edge is a different color (red, blue, and green) and that (as above) red edges always attach to other red edges. One possibility, now, is for the blue edges to attach only to blue edges and green only to green. The other possibility is for blue to attach to green (and vice versa). This gives two variations. But the cube can be sliced down the middle between the paired red edges leaving two halves. The halves can then be put together as they were, or with one half first reflected and then reassembled. This produces four possible variations. If only two colors are used (red and blue) then this is the same as simply changing the green edges to blue. The red edges must still connect only to red edges (because there aren't enough red edges to go with every blue edge) so there is only one way to color the cube. And, finally, if every edge is red there is obviously only one possible coloring. (This would be all there were except that the full labeling scheme involves more than just edge color.)

The first display variation is a wire-frame of the twelve edges with each edge of the vertex colored half-way along the edge from the vertex to its adjacent vertex. The second variation is a solid box with colored beads attached to the edges. And, finally, the third variation is a cube constructed from 48 colored right triangles that surround the edges with their assigned colors. Due to the many different capabilities of 3-D display software out there, at least one of these variations should be able to show you the eight different labelings. (Note, though, that the last three sponges cannot be distinguished by edge colors alone, so they all look alike.)