ABSTRACT
We’d now like to turn to symmetries of three-dimensional objects, in particular of the regular tetrahedron and the cube. By way of analogy with plane figures, it should be clear that we should be concerned with rigid motions of three-dimensional space (R3) taking the given object onto itself. However, we should make clear what is meant by a rigid motion of R3. We mean more than just a one-to-one onto distancepreserving function. Consider R3 equipped with a coordinate system with the usual right-hand rule orientation as depicted below (where ~i×~j = ~k):
The function β : R3 → R3 defined by β(x, y, z) = (x,−y, z) (that is, reflection through the xz-plane) is a one-to-one onto distance-preserving function that cannot be accomplished by moving R3 in R3. We can see this because it changes the set of vectors~i,~j,~k, which has right-handed orientation, into the set ~i,−~j,~k, which has left-handed orientation. The pertinent comparison is to the reflection (x, y) 7→ (x,−y) in the
plane, which can’t be accomplished in the plane. It is an accident of physics that we inhabit three-dimensional space. Hence, we are happy with reflections through a line in the plane, because they can be accomplished in three-dimensional space. We are not so happy with reflections through a plane in space (even though it turns out they can be accomplished as a motion in four dimensions). The key here is to restrict ourselves to one-to-one onto distance-preserving functions that preserve the right-handed orientation of our coordinate system; we (This should jibe
and
is restrictive because it excludes functions like β, which can be realized as mirror reflections. Many people would call such functions symmetries too, but we will not do so here.
