Rotation Axes of the Faces of the Regular 4-Polytopes and the n-fold Axes of Their Rotation Groups

Every regular 4-polytope has flat polygon faces, and each face sits in a plane through the origin of 4-dimensional space. This paper attaches two unit vectors in ordinary 3-dimensional space, called a and b, to each such face. They are the two halves of the face’s plane, obtained by splitting its six coordinates into a “self-dual” and an “anti-self-dual” part. Computed with quaternions, a is the rotation axis of the face from the author’s earlier zero-octonion work. The main result is that rotating by one step around a face (2π/p) splits into two simple rotations, one about a and one about b. Because of this, the vectors a, taken over all faces, are exactly the p-fold rotation axes of a familiar 3-dimensional symmetry group, the icosahedral or the octahedral one. This gives the same sets of directions for all six polytopes (the vertices of an icosahedron, dodecahedron, octahedron or cube). It also explains why the 120-cell and 600-cell, which are dual to each other, give the icosahedron and the dodecahedron. The two self-dual polytopes (5-cell and 24-cell) give only one solid each, so no hidden dual solid is needed. The paper also shows that faces with the same a (or the same b) lie in Clifford-parallel planes, and that for the 120-cell and 600-cell every pair (a, b) occurs, with 10 and 6 oriented faces per pair. The group classification of the polytopes is taken from the literature, not re-derived. A few further observations rest on computation only and are listed as open.