Runcinated 24-cells

The Cartesian coordinates of the runcinated 24-cell having edge length 2 is given by all permutations of sign and coordinates of:

(0, 0, √2, 2+√2)

(1, 1, 1+√2, 1+√2)

The permutations of the second set of coordinates coincide with the vertices of an inscribed cantellated tesseract.

The regular skew polyhedron, {4,8|3}, exists in 4-space with 8 square around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed. The dual regular skew polyhedron, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell.

The Cartesian coordinates of an origin-centered runcitruncated 24-cell having edge length 2 are given by all permutations of coordinates and sign of:

(0, √2, 2√2, 2+3√2)

(1, 1+√2, 1+2√2, 1+3√2)

The permutations of the second set of coordinates give the vertices of an inscribed omnitruncated tesseract.

The dual configuration has coordinates generated from all permutations and signs of:

(1,1,1+√2,5+√2)

(1,3,3+√2,3+√2)

(2,2,2+√2,4+√2)

A half-symmetry construction of the runcitruncated 24-cell (or runcicantellated 24-cell), as , also called a runcicantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. Like the snub 24-cell, it has symmetry [3⁺,4,3], order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons. The difference can be seen in the vertex figures:

Runcic snub 24-cell
Schläfli symbol	s3{3,4,3}
Coxeter diagram
Cells	240	24 {3,5} 24 t{3,3} 96 (4.4.3) 96 tricup
Faces	960	576 {3} 288 {4} 96 {6}
Edges	1008
Vertices	288
Vertex figure
Symmetry group	[3+,4,3], order 576
Properties	convex

A related 4-polytope is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s₃{3,4,3}, . It is not uniform, but it is vertex-transitive and has all regular polygon faces. It is constructed with 24 icosahedra, 24 truncated tetrahedra, 96 triangular prisms, and 96 triangular cupolae in the gaps, for a total of 240 cells, 960 faces, 1008 edges, and 288 vertices. Like the snub 24-cell, it has symmetry [3⁺,4,3], order 576.[1]

The vertex figure contains one icosahedron, two triangular prisms, one truncated tetrahedron, and 3 triangular cupolae.

Orthographic projections			Net

The 48 truncated cuboctahedral cells are joined to each other via their octagonal faces. They can be grouped into two groups of 24 each, corresponding with the cells of a 24-cell and its dual. The gaps between them are filled in by a network of 192 hexagonal prisms, joined to each other via alternating square faces in alternating orientation, and to the truncated cuboctahedra via their hexagonal faces and remaining square faces.

The Cartesian coordinates of an omnitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:

(1, 1+√2, 1+2√2, 5+3√2)

(1, 3+√2, 3+2√2, 3+3√2)

(2, 2+√2, 2+2√2, 4+3√2)

Nonuniform variants with [3,4,3] symmetry and two types of truncated cuboctahedra can be doubled by placing the two types of truncated cuboctahedra on each other to produce a nonuniform polychoron with 48 truncated cuboctahedra, 144 octagonal prisms (as ditetragonal trapezoprisms), 192 hexagonal prisms, two kinds of 864 rectangular trapezoprisms (288 with D_2d symmetry and 576 with C_2v symmetry), and 2304 vertices. Its vertex figure is an irregular triangular bipyramid.

Vertex figure

This polychoron can then be alternated to produce another nonuniform polychoron with 48 snub cubes, 144 square antiprisms, 192 octahedra (as triangular antiprisms), three kinds of 2016 tetrahedra (288 tetragonal disphenoids, 576 phyllic disphenoids, and 1152 irregular tetrahedra), and 1152 vertices. It has a symmetry of [[3,4,3]⁺], order 1152.

Vertex figure

Vertex figure for the omnisnub 24-cell

The uniform snub 24-cell is called a semi-snub 24-cell by John Horton Conway with Coxeter diagram within the F₄ family, although it is a full snub or omnisnub within the D₄ family, as .

In contrast a full snub 24-cell or omnisnub 24-cell, defined as an alternation of the omnitruncated 24-cell, cannot be made uniform, but it can be given Coxeter diagram , and symmetry [[3,4,3]]⁺, order 1152, and constructed from 48 snub cubes, 192 octahedrons, and 576 tetrahedrons filling the gaps at the deleted vertices. Its vertex figure contains 4 tetrahedra, 2 octahedra, and 2 snub cubes. It has 816 cells, 2832 faces, 2592 edges, and 576 vertices.[2]