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