Hello Thigle and all,
If any of you have access to John Stillwell's new book
The Four Pillars of Geometry, pp. 163-166, that'd be better. However, I'll do my best to paraphrase what I believe John Stillwell's saying...
Basically, John Stillwell defines a mapping between the rotational symmetries of the tetrahedron and the 24-cell using quaternions of the form:
For this, he positions a tetrahedron inside a cube with each vertex of the tetrahedron coinciding with a vertex of the cube:
The red axis represents a tetrahedron rotation of pi through opposite edges. These correspond to +/- i,j,k. The red axis depicted in the figure above is the rotational axis for k. Obviously, the axes corresponding to the other two pairs of edges goes through the i and j axes.
Notice that the quaternion gets pi/2, since θ here equals pi. Also notice we get 6 vertices of the 24-cell for 3 rotations of the tetrahedron because of the +/-. Further, the identity rotation maps to +/-1. (I think... with -1...?)
The blue axis corresponds to the tetrahedron rotations of 2pi/3 through a vertex and its opposing face, with coordinates (1/sqrt(3)*(+/-i +/-j +/-k). (1/sqrt(3) is a scaling factor to give the quaternions absolute value 1) These 8 tetrahedron rotations, along with the identity, are again doubled by +/- to correspond to the 16 vertices, as 8 pairs of opposites, of the 24-cell +/-(1/2), +/-(i/2), +/-(j/2), +/-(k/2). Again, the quaternions here have an angle of 2pi/6 since θ equals 2pi/3.
First of all, did I depict all that in the figure correctly?
My questions here now are... Can a similar mapping be made between the 3 rotational symmetries of the triangle, and say, the vertices of the octahedron in 3-space?
Further, is there always some similar mapping between the n-simplex and some (n+1)-d polytope? When is there, and when is there not such a mapping between the n-simplex and some (n+1)-d polytope?
Also... is a reflection in dimension n always going to be expressible as a rotation in dimension n+1?