# Visualizing $180^\circ$ rotational symmetries of a tetrahedron

I am trying to learn about the symmetries of a regular tetrahedron.

I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, $(123),(132),(243),(234),(134),(143),(124),(142)$ but I cannot at all understand how to visualize the so-called $180^\circ$ rotations, i.e., $(13)(24)$ , $(14)(23)$ , $(12)(34)$.

Can anyone suggest anything for this?

## 3 Answers

Here's a Schlegel diagram of the tetrahedron:

The axis of rotation is in red, and it goes through opposite edges of the tetrahedron.

In general, the rotational symmetries of any Platonic Solid come in three flavors: Those with rotations axes

• Through the centers of opposite faces,

• Through the midpoints of opposite edges, and

• Through opposite vertices.

The tetrahedron is unlike the other solids in that it's not centrally-symmetric: It doesn't have opposite faces and opposite vertices. Instead, across from every vertex, there's the center of a face. So the first and last kind of rotations above collapse into one, in a sense.

Only that you can place a regular tetrahedron in a regular cube. Vertices, for example, at $$(1,1,1); \; \; (1,-1,-1); \; \; (-1,1,-1); \; \; (-1,-1,1).$$ Each pair should disagree in two coordinates, agree in one.

Represents rotation on axis through midpoints of two disjoint edges, see from 1:32 https://www.youtube.com/watch?v=qAR8BFMS3Bc