Today I was trying to draw the Cayley diagram for symmetry groups of a regular tetrahedron. (A tetrahedron is a like a pyramid, but with a triangular base.) What I ended up with, surprised me. I had started with the idea that I would end up with something that was tetrahedron like - or atleast suggestive of the fact that it started from a tetrahedron.

Cayley diagram for a symmetry group of the tetrahedron.

Now, this is not the only group one can generate from a tetrahedron. And even for this group, this certainly not the only Cayley diagram as well. We can visualize the above diagram as a 3d object If I were to redraw this slightly, we get -

And now if I imagine this as 3d object, I get something looks like this.

front and back respectively.

It is as if each triangle has triangle facing in the opposite direction behind it. And there are 4 such perspectives, each corresponding to a vertex of the tetrahedron. Cute. I wonder if I can make choices for a different set of rotations such that the triangles are not inverted? 

After some playing around with it, I got this. I realized that following any sequence of red-blue-red-blue arrows always leads back to the starting point. If I were to compose every pair of red-blue arrows into a single arrow, lets color this green, then we get a different diagram. This diagram is indeed suggestive of the shape of the tetrahedron. The same shape would result if we were to include the blue arrows instead of the red ones.

Cayley diagram for a symmetry group of the tetrahedron

This shape is interesting because you can imagine that looks almost like a large green tetrahedron with the each tip sliced of to reveal a small red triangle. Here is a picture I found on the web:

This diagram also lets itself to a nice explanation - each triangle at a vertex corresponds to the sub-group for rotation about that vertex.