3-D Polyhedra

Tetrahedron (and its dual)

Cube/Octahedron

Dodecahedron/Icosahedron

4-D Polytopes (Polychora)

4-cell simplex: analogue of tetrahedron	8-cell (tesseract) hypercube: analogue of cube	16-cell orthoplex: analogue of octahedron
24-cell	120-cell analogue of dodecahedron	600-cell analogue of icosahedron

Note: the polychora can be plotted in 3-D because a hypersphere in 4-D is really 3-dimensional, just as a sphere in 3-D is a surface. In fact, a sphere (without a point) can be plotted as a disk, either via stereographic projection (which covers the whole plane), or by mapping to the open unit disk by preserving the areas of spherical disks. In the 3-D case, the hypersphere (minus a point) is mapped to the open unit ball by preserving volumes, that is, a point (x,y,z,t) is mapped to (x, y, z)(1/2 + ArcSin(t)/Pi + t Sqrt(1 - t²)/Pi)^1/3/Sqrt(1 - t²). Thus the "north pole" of the 3-D hypersphere corresponds to the center of the ball, and the point at the "south pole" is smeared as the surface of the ball.


©Joseph Muscat