# 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.

## Keywords

Rotational Symmetry Opposite Edge Opposite Sense Regular Tetrahedron Opposite Face
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.