[dzyomchb] symmetrically distributing points on a hypersphere

the vertices of a geodesic polyhedron distribute points evenly and symmetrically around a sphere (2-sphere, embedded in 3 dimensions).  it is pretty.

is there an analogous family of pretty distributions of points on the 3-sphere, embedded in 4 dimensions?

when there are many vertices on a 2-sphere, the vertex figure around most vertices is a hexagon, reflecting the optimality of hexagonal close packing of discs on a plane.

on the 3-sphere, we analogously expect most vertices to be packed as if at the centers of spheres in a fcc or hcp close packing.  assuming fcc because it has more symmetry, most Voronoi cells around vertices will be approximate rhombic dodecahedra.

on the 2-sphere, for any geodesic polyhedron based on the icosahedron, there are 12 special vertices surrounded by only 5 triangles, not 6.  the Voronoi cell around such a vertex is a pentagon.

what is the maximum symmetry the analogous 4D geodesic polytope can have?  is it the symmetry of the 600-cell?  or the tesseract?  what polyhedron is the Voronoi cell around weird vertices, and how does that polyhedron connect to the rhombic dodecahedra that make up most of the cells?  it probably needs quadrilateral faces.

maybe it has to be the symmetry of the tesseract, and the weird vertices are surrounded by cubes, and the rhombic faces of the rhombic dodecahedra get distorted into squares to meet the faces of the cubes.  all this is wild speculation.

what happens on 4-spheres (in 5D) and beyond?  we remain interested only in pretty distributions of points.  high density lattice packings of identical hyperspheres in Euclidean spaces of various (low) dimensions are known.  for large numbers of points on an N-sphere, distortions of these sphere packings will form "most" of the points on the hypersphere.

in higher dimensions, regular polytopes become boring: only the symmetry of the hypercube is available.  but maybe asking that the symmetry of a point distribution have the symmetry of a regular polytope asks too much.  for example, the E8 polytope and its associated symmetry are not regular.