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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?

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Here's a Schlegel diagram of the tetrahedron: Tetrahedron Rotation

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.

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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.

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

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