Symmetries of the cube – kinds of possible isometries

Before we take up symmetries of the cube we need to have a round-up of isometries of space, so we know what we are looking for.

1. Plane reflections. We have talked quite a bit about this. They are a lot like line reflections. If M is reflection in a plane m, then M² = I and M is the perpendicular bisecting plane of AM(A) for any A not on m.
2. Double plane reflections. Let M and N denote reflection in two distinct planes m and n. If A is a point in space, we want to know about NM(A). As for line reflections in the plane, this depends on how the planes are arranged. We can figure this out pretty well from the plane case by using the following trick: let p be a plane through A that is perpendicular to both planes m and n. Then m and n intersect p in lines m' and n'. M(A) = reflection of A in plane p across line m' and N(M(A)) = = reflection of M(A) in plane p across line n'. So the whole story takes place in plane p, where we know the answers.
1. Two parallel planes. MN = translation in the direction perpendicular to m and n by twice the distance between the two planes (the translation vector is in the direction pointing from m towards n.
2. Two intersecting planes. The line of intersection is o. Then plane P is perpendicular to o and intersects the line at point O. Then M(N(A)) is the rotation of A with center O by angle twice the angle between the lines m' and n', which angle = the dihedral angle between m and n.
3. Triple plane reflections. For the cube, let's stick to only those cases where the three planes m, n, p pass through a single point O. There are some others as well that we may take up later.
1. Triple reflection in 3 planes all intersecting in a single line o through O. This is a plane reflection in another plane through o. It follows from the plane case.
2. Triple reflection in 3 planes through O, each perpendicular to the other 2 planes. This triple reflection is point reflection with center O.
3. Triple reflection in 3 planes through O, with plane p perpendicular m and n. This triple reflection is a new isometry called a rotoreflection. We will study it in lab. If o is the line of intersection of m and n, then PNM is rotation with axis o followed by reflection in the plane through O that is perpendicular to o.
4. Triple reflection in 3 planes through O in general. This is also a rotoreflection, but the proof requires the sort of mirror adjustment that was used in the plane to prove that the product of 3 line reflections is a glide reflection.
4. Quadruple plane reflections for 4 planes through O. This case is the composition of two rotations, which turns out to be a rotation. But it is not so obvious that the product of rotations with center ) is a rotation with center O.