# Symmetries of the Tetrahedron

• M. A. Armstrong
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

How much symmetry has a tetrahedron? Consider a regular tetrahedron T and, for simplicity, think only of rotational symmetry. Figure 1.1 shows two axes. One, labelled L, passes through a vertex of the tetrahedron and through the centroid of the opposite face; the other, labelled M, is determined by the midpoints of a pair of opposite edges. There are four axes like L and two rotations about each of these, through 2π/3 and 4π/3, which send the tetra­hedron to itself. The sense of the rotations is as shown: looking along the axis from the vertex in question the opposite face is rotated anticlockwise. Of course, rotating through 2π/3 (or 4π/3) in the opposite sense has the same effect on Tas our rotation through 4π/3 (respectively 2π/3). As for axis M, all we can do is rotate through π, and there are three axes of this kind. So far we have (4 × 2) + 3 = 11 symmetries. Throwing in the identity symmetry, which leaves T fixed and is equivalent to a full rotation through 2π about any of our axes, gives a total of twelve rotations.