I'm working on an applet that will calculate the product of two symmetries. (It's unfinished but here's a link to the project if you're curious.) I want the applet to show visuals to help the user understand what's happening for each symmetry action -- for example: for rotations, it displays the axis of rotation; for reflections, it displays the plane of reflection.
There are six symmetries which I can't figure out a visualization for. I believe they're called "roto-rotations" or "inversions," but I can't find much information online about them (in my applet they're currently labeled $φ_1$ through $φ_6$). They are the following elements of $S_4$:
(2341), (2413), (3142), (3421), (4123), (4312)
Can anybody help me figure out how these elements can be visualized on the tetrahedron through axes of rotation/planes of reflection/some other visual?
To expand my comment:
We've embedded a regular tetrahedron as "alternate vertices" in an axis-oriented cube centered at the origin, and labeled the cube's vertices as shown, with vertical edges $AA'$, $B'B$, $CC'$, and $D'D$.
The indicated quarter-turn about the vertical axis cyclically permutes the edges $(AA'\ B'B\ CC'\ D'D)$ and maps our tetrahedron $ABCD$ to the "other" tetrahedron $A'B'C'D'$.
Reflection in the horizontal plane exchanges each primed-unprimed label pair, again swapping $ABCD$ and $A'B'C'D'$.
The composition is therefore a symmetry of $ABCD$, and effects the cyclic permutation $(A\ B\ C\ D)$ of vertices.
The rotation and reflection commute, and the composite transformation has block-diagonal standard matrix. Here, for example, the composite transformation has standard matrix $$ \left[\begin{array}{@{}rr|r@{}} 0 & -1 & 0 \\ 1 & 0 & 0 \\ \hline 0 & 0 & -1 \\ \end{array}\right]. $$ There are six such matrices because we have three choices for the rotation axis, and then two choices of quarter-turn. In terms of entries, we can change the sign of the $2 \times 2$ block, and simultaneously permuts rows and columns cyclically. This agrees with the count of six $4$-cycles in the symmetric group.
No such matrix is a rotation (the determinant of the standard matrix is $-1$, i.e., the transformation reverses orientation), and neither is it a reflection (because the transformation is not its own inverse). Unlike the situation in the plane, not every orthogonal transformation of three-space is a rotation or a reflection.