Left quaternion multiplication as an action on S3 of unit quaternions. This depiction of S3 shows Hurwitz quaternions and the 2-sphere of unit 3-vectors (quaternions with zero scalar part), visible as the yellow plane. The action shown is v ↦ 1 + i + j + k/2v.
The projection of S3 to the affine 3-space is quasi-stereographic with the pole at −i + j + k/√3, but generally irregular. It should be a stereographic projection on the 2-sphere of 3-vectors, it is a gnomonic projection in four overlapping 9- and 7-points regions around 1, 1 + i + j + k/2, −1 + i + j + k/2, and −1, and it is unspecified in the surrounding space of S3 (from where some curved arrows are mapped). Because of this piecewise compromise projection octahedral cells of the polychoron look distorted.
24 unit Hurwitz quaternions form vertices of a 24-cell and 4 cyclic orbits (grey, blue, purple, orange) of 6 elements each. All 24 elements are depicted (as balls), but only 59 polychoron’s edges of 96 total are visible as straight line segments, and yet 13 show as curved lines and arrows. 24 edges are completely missing, although 2 of them can be inferred if one closes white and orange cycles.

The red roundabout is the invariant great circle of 3-vectors: see w:Talk:Versor for details. The 2-sphere of 3-vectors features a rhombic dodecahedron (green edges, yellow faces) formed by the polychoron’s faces and cells respectively. 9 its faces of 12 are visible: in six of them the invariant circle passes, whereas three inner rhombi near the vertical axis look as a Reuleaux triangle (due to the stereographic projection most “straight” segments on S2 project to circle arcs of various curvature).
Own work
Incnis Mrsi