This was uplifted from a 2008 post about the same topic.
So I have considered the envelopes (the outermost convex covering of a polytope assuming orthogonal projection) of the six regular polychora. Here is the list:
5-cell
Cell/Vertex: Tetrahedron
Edge/Face: Equilateral-triangular bipyramid (the convex hull of a unit triangle and a perpendicular edge of length 1)
Tesseract
Cell: Cube
Face: Square prism (height is sqrt(2) times the square edge length)
Edge: Nonuniform regular-hexagonal prism (height is sqrt(3/2) times the hexagon edge length)
Vertex: Rhombic dodecahedron
16-cell
Cell: Cube
Face: Regular-hexagonal bipyramid (the convex hull of a unit hexagon and a perpendicular edge of length sqrt(6))
Edge: Square bipyramid (the convex hull of a unit square and a perpendicular edge of length 1)
Vertex: Octahedron
24-cell
Cell: Cuboctahedron
Face: Regular-hexagonal biantiprism (two hexagonal antiprisms joined at each other by a large hexagonal face)
Edge: Hexakis regular-hexagonal prism
Vertex: Rhombic dodecahedron
120-cell
Cell: Chamfered dodecahedron (or order-5 truncated rhombic triacontahedron)
Face: something with D10h symmetry (two decagons, which represent dodecahedra, straddle on the axis of symmetry; the rest are pentagons)
Edge: something with D6h symmetry (it has two dodecagons (again representing dodecahedra) with all other sides being pentagons)
Vertex: something with Oh symmetry (it is composed of 12 decagons in the planes of the cuboctahedron and 72 pentagons)
600-cell
Cell: Hexakis chamfered cube (or hexakis order-4 truncated rhombic dodecahedron)
Face: something with D6h symmetry (this object has 12 trapezoids, the rest are triangles)
Edge: Decakis order-10 truncated order-10 decakis regular-decagonal biantiprism (a biantiprism is composed of two antiprisms with different-sized faces joined at a larger face)
Vertex: Pentakis icosidodecahedron
I will do some research on the envelopes of the uniform polychora. Stay tuned!
Mercurial