Cantellated 24-cells

24-cell	Cantellated 24-cell	Cantitruncated 24-cell
Orthogonal projections in F4 Coxeter plane

In four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell.

There are 2 unique degrees of cantellations of the 24-cell including permutations with truncations.

Cantellated 24-cell

Cantellated 24-cell
Type	Uniform 4-polytope
Schläfli symbol	rr{3,4,3} s2{3,4,3}
Coxeter diagram
Cells	144	24 (3.4.4.4) 24 (3.4.3.4) 96 (3.4.4)
Faces	720	288 triangles 432 squares
Edges	864
Vertices	288
Vertex figure	Wedge
Symmetry group	F4, [3,4,3], order 1152
Properties	convex
Uniform index	24 25 26

Net

The cantellated 24-cell or small rhombated icositetrachoron is a uniform 4-polytope.

The boundary of the cantellated 24-cell is composed of 24 truncated octahedral cells, 24 cuboctahedral cells and 96 triangular prisms. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.

Construction

When the cantellation process is applied to 24-cell, each of the 24 octahedra becomes a small rhombicuboctahedron. In addition however, since each octahedra's edge was previously shared with two other octahedra, the separating edges form the three parallel edges of a triangular prism - 96 triangular prisms, since the 24-cell contains 96 edges. Further, since each vertex was previously shared with 12 faces, the vertex would split into 12 (24*12=288) new vertices. Each group of 12 new vertices forms a cuboctahedron.

The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:

(0, √2, √2, 2 2√2)

(1, 1 √2, 1 √2, 1 2√2)

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

The dual configuration has all permutations and signs of:

(0,2,2 √2,2 √2)

(1,1,1 √2,3 √2)

Structure

The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra via their axial square faces, and to the triangular prisms via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.

Cantic snub 24-cell

A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.

The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:

Images

orthographic projections

Coxeter plane	F4
Graph
Dihedral symmetry	[12]
Coxeter plane	B3 / A2 (a)	B3 / A2 (b)
Graph
Dihedral symmetry	[6]	[6]
Coxeter plane	B4	B2 / A3
Graph
Dihedral symmetry	[8]	[4]

Schlegel diagrams

Schlegel diagram

Showing 24 cuboctahedra.

Showing 96 triangular prisms.

Related polytopes

The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Cantitruncated 24-cell

Cantitruncated 24-cell
Schlegel diagram, centered on truncated cuboctahedron
Type	Uniform 4-polytope
Schläfli symbol	tr{3,4,3}
Coxeter diagram
Cells	144	24 4.6.8 96 4.4.3 24 3.8.8
Faces	720	192{3} 288{4} 96{6} 144{8}
Edges	1152
Vertices	576
Vertex figure	sphenoid
Symmetry group	F4, [3,4,3], order 1152
Properties	convex
Uniform index	27 28 29

Net

The cantitruncated 24-cell or great rhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24-cell, 24 truncated cubes corresponding with the cells of the dual 24-cell, and 96 triangular prisms corresponding with the edges of the first 24-cell.

Coordinates

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

(1,1 √2,1 2√2,3 3√2)

(0,2 √2,2 2√2,2 3√2)

The dual configuration has coordinates as all permutations and signs of:

(1,1 √2,1 √2,5 2√2)

(1,3 √2,3 √2,3 2√2)

(2,2 √2,2 √2,4 2√2)

Projections

orthographic projections

Coxeter plane	F4
Graph
Dihedral symmetry	[12]
Coxeter plane	B3 / A2 (a)	B3 / A2 (b)
Graph
Dihedral symmetry	[6]	[6]
Coxeter plane	B4	B2 / A3
Graph
Dihedral symmetry	[8]	[4]

Stereographic projection

Related polytopes

24-cell family polytopes
Name	24-cell	truncated 24-cell	snub 24-cell	rectified 24-cell	cantellated 24-cell	bitruncated 24-cell	cantitruncated 24-cell	runcinated 24-cell	runcitruncated 24-cell	omnitruncated 24-cell
Schläfli symbol	{3,4,3}	t0,1{3,4,3} t{3,4,3}	s{3,4,3}	t1{3,4,3} r{3,4,3}	t0,2{3,4,3} rr{3,4,3}	t1,2{3,4,3} 2t{3,4,3}	t0,1,2{3,4,3} tr{3,4,3}	t0,3{3,4,3}	t0,1,3{3,4,3}	t0,1,2,3{3,4,3}
Coxeter diagram
Schlegel diagram
F4
B4
B3(a)
B3(b)
B2

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 25, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora)". x3o4x3o - srico, o3x4x3o - grico