Groups and Symmetry pp 1-5 | Cite as

# Symmetries of the Tetrahedron

## 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 *T*as 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.

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