3. Convex uniform polychora based on the icositetrachoron (24-cell)
Symmetry group of all polychora in this section except #26, 27, 30, and 31: [3,4,3], the diploid icositetrachoric group, of order 1152
(o)----o-----o-----o
4
- Icositetrachoron [22]
Alternate names:- 24-cell (most frequently used)
- Ikosatetrahedroid or 24-hedroid (Henry Parker Manning)
- Rectified 16-cell (Norman W. Johnson)
- Rectified hexadecachoron
- Rectified [four-dimensional] cross polytope
- Rectified [four-dimensional regular] orthoplex
- Polyoctahedron (John H. Conway)
- Four-dimensional rhombic-dodecahedron-analogue
Schläfli symbol: {3,4,3}, also t0{3,3,4}, t3{4,3,3}, or {31,1,1}
Elements:- Cells: 24 octahedra
- Faces: 96 triangles (all joining octahedra to octahedra)
- Edges: 96
- Vertices: 24
Vertex figure:- Cube, edge length 1
o----(o)----o-----o
4
- Disicositetrachoron [23]
Alternate names:- Rectified 24-cell (Norman W. Johnson)
- Rectified icositetrachoron
- Rectified polyoctahedron
- Cantellated 16-cell (Norman W. Johnson)
- Cantellated hexadecachoron
- Cantellated [four-dimensional] cross polytope
- Cantellated [four-dimensional regular] orthoplex
Schläfli symbol: r{3,4,3}, also t1{3,4,3}, t2{3,4,3}, t0,2{3,3,4}, t1,3{4,3,3}, t0,2{3,31,1}, or t1{31,1,1}
Elements:- Cells: 24 cuboctahedra, 24 cubes
- Faces: 96 triangles (joining cuboctahedra to cuboctahedra), 144 squares (joining cuboctahedra to cubes)
- Edges: 288
- Vertices: 96 (located at the midpoints of the edges of an icositetrachoron, or at the centroids of its faces)
Vertex figure:- Right equilateral-triangular prism: base an equilateral triangle, edge length sqrt(2); height 1
(o)---(o)----o-----o
4
- Truncated icositetrachoron [24]
Alternate names:- Truncated 24-cell
- Truncated polyoctahedron
Schläfli symbol: t{3,4,3}, also t0,1{3,4,3}, t2,3{3,4,3}, t0,1,2{3,3,4}, t1,2,3{4,3,3}, t0,1,2{3,31,1}, or t0,1{31,1,1}
Elements:- Cells: 24 truncated octahedra, 24 cubes
- Faces: 144 squares (joining truncated octahedra to cubes), 96 hexagons (joining truncated octahedra to truncated octahedra)
- Edges: 384
- Vertices: 192
Vertex figure:- Equilateral-triangular pyramid (or triangular spike): base an equilateral triangle, edge length sqrt(2); all 3 lateral faces isosceles triangles, edge lengths sqrt(2), sqrt(3), sqrt(3)
(o)----o----(o)----o
4
- [Small] prismatodisicositetrachoron [25]
Alternate names:- Cantellated 24-cell (Norman W. Johnson)
- Cantellated icositetrachoron
- Cantellated polyoctahedron
Schläfli symbol: t0,2{3,4,3} or t1,3{3,4,3}
Elements:- Cells: 24 rhombicuboctahedra, 24 cuboctahedra, 96 triangular prisms
- Faces: 288 triangles (96 joining rhombicuboctahedra to rhombicuboctahedra, 192 joining cuboctahedra to triangular prisms), 432 squares (144 joining rhombicuboctahedra to cuboctahedra, 288 joining rhombicuboctahedra to triangular prisms)
- Edges: 864
- Vertices: 288
Vertex figure:- Rectangular right sphenoid (or rectangular wedge): pentahedron with rectangular base, edge lengths 1, sqrt(2); wedge edge of length 1 parallel to long edges of base and centered above it; lateral faces are 2 trapezoids, edge lengths 1, sqrt(2), sqrt(2), sqrt(2), alternating with isosceles triangles, edge lengths 1, sqrt(2), sqrt(2)
(o)----o-----o----(o)
4
- [Small] prismatotetracontaoctachoron [26]
Alternate names:- Runcinated 24-cell (Norman W. Johnson)
- Runcinated icositetrachoron
- Runcinated polyoctahedron
Symmetry group: [[3,4,3]], the extended icositetrachoric group, of order 2304
Schläfli symbol: t0,3{3,4,3}
Elements:- Cells: 48 octahedra, 192 triangular prisms
- Faces: 384 triangles (all joining octahedra to triangular prisms), 288 squares (all joining triangular prisms to triangular prisms)
- Edges: 576
- Vertices: 144
Vertex figure:- Elongate square antiprism: bases both squares, edge length 1; 6 lateral faces all isosceles triangles, edge lengths 1, sqrt(2), sqrt(2)
o----(o)---(o)----o
4
- [Truncated-cubic] tetracontaoctachoron [27]
Alternate names:- Bitruncated 24-cell (Norman W. Johnson)
- Bitruncated icositetrachoron
- Bitruncated polyoctahedron
Symmetry group: [[3,4,3]], the extended icositetrachoric group, of order 2304
Schläfli symbol: 2t{3,4,3}, also t1,2{3,4,3}
Elements:- Cells: 48 truncated cubes
- Faces: 192 triangles (all joining truncated cubes to truncated cubes), 144 octagons (all joining truncated cubes to truncated cubes)
- Edges: 576
- Vertices: 288
Vertex figure:- Elongate disphenoid: tetrahedron with 2 opposite edges length 1; all 4 lateral edges length sqrt(2+sqrt(2))
(o)---(o)---(o)----o
4
- Great prismatodisicositetrachoron [28]
Alternate names:- Cantitruncated 24-cell (Norman W. Johnson)
- Cantitruncated icositetrachoron
- Cantitruncated polyoctahedron
Schläfli symbol: t0,1,2{3,4,3} or t1,2,3{3,4,3}
Elements:- Cells: 24 great rhombicuboctahedra, 24 truncated cubes, 96 triangular prisms
- Faces: 192 triangles (all joining truncated cubes to triangular prisms), 288 squares (all joining truncated cuboctahedra to triangular prisms), 96 hexagons (all joining truncated cuboctahedra to truncated cuboctahedra), 144 octagons (all joining truncated cuboctahedra to truncated cubes)
- Edges: 1152
- Vertices: 576
Vertex figure:- Bilaterally symmetric tetrahedron: one face an isosceles triangle, edge lengths 1, sqrt(2), sqrt(2), joined to another isosceles triangle, edge lengths 1, sqrt(2+sqrt(2)), sqrt(2+sqrt(2)); other 2 faces congruent scalene triangles, edge lengths sqrt(2), sqrt(3), sqrt(2+sqrt(2)), joined so that edges length 1 and sqrt(3) are opposite
(o)---(o)----o----(o)
4
- Diprismatodisicositetrachoron [29]
Alternate names:- Runcitruncated 24-cell (Norman W. Johnson)
- Runcitruncated icositetrachoron
- Runcitruncated polyoctahedron
Schläfli symbol: t0,1,3{3,4,3} or t0,2,3{3,4,3}
Elements:- Cells: 24 truncated octahedra, 24 rhombicuboctahedra, 96 triangular prisms, 96 hexagonal prisms
- Faces: 192 triangles (all joining rhombicuboctahedra to triangular prisms), 720 squares (144 joining truncated octahedra to rhombicuboctahedra, 288 joining rhombicuboctahedra to hexagonal prisms, 288 joining hexagonal prisms to triangular prisms), 192 hexagons (joining truncated octahedra to hexagonal prisms)
- Edges: 1440
- Vertices: 576
Vertex figure:- Trapezoidal pyramid: base a trapezoid with edges length 1, sqrt(2), sqrt(2), sqrt(2); 4 lateral faces are (1) isosceles triangle, edges length 1, sqrt(2), sqrt(2) and (2) isosceles triangle, edges length sqrt(2), sqrt(3), sqrt(3), alternating with (3 and 4) congruent isosceles triangles, edges length sqrt(2), sqrt(2), sqrt(3)
(o)---(o)---(o)---(o)
4
- Great prismatotetracontaoctachoron [30]
Alternate names:- Omnitruncated 24-cell (Norman W. Johnson)
- Omnitruncated icositetrachoron
- Omnitruncated polyoctahedron
Symmetry group: [[3,4,3]], the extended icositetrachoric group, of order 2304
Schläfli symbol: t0,1,2,3{3,4,3}
Elements:- Cells: 48 truncated cuboctahedra, 192 hexagonal prisms
- Faces: 864 squares (576 joining truncated cuboctahedra to hexagonal prisms, 288 joining hexagonal prisms to hexagonal prisms), 384 hexagons (all joining truncated cuboctahedra to hexagonal prisms), 144 octagons (all joining truncated cuboctahedra to truncated cuboctahedra)
- Edges: 2304
- Vertices: 1152
Vertex figure:- Chiral tetrahedron: two faces isosceles with edges sqrt(2), sqrt(2), sqrt(3), other two scalene with edges sqrt(2), sqrt(3), sqrt(2+sqrt(2)), arranged so the sqrt(2) edges form a chain and the sqrt(3), sqrt(2+sqrt2(2)), sqrt(3) edges in that order form the complementary chain passing through all 4 vertices; dextro and laevo versions each occur at 576 vertices
( )---( )----o-----o
4
- Snub icositetrachoron [31]
Alternate names:- Snub 24-cell
- Snub polyoctahedron
Symmetry group: [3+,4,3], the ionic diminished icositetrachoric group, of order 576
Schläfli symbol: s{3,4,3}, also s{31,1,1}
Elements:- Cells: 24 icosahedra, 120 tetrahedra
- Faces: 480 triangles (96 joining icosahedra to icosahedra, 96 joining tetrahedra to tetrahedra, 288 joining icosahedra to tetrahedra)
- Edges: 432
- Vertices: 96 (located along each edge of a unit regular icositetrachoron at a distance (sqrt(5)1)/2 from one of the edges ends)
Vertex figure:- Tridiminished icosahedron: a regular icosahedron, edge length 1, from which 3 pentagonal pyramids of triangles are removed and replaced by regular pentagons, yielding a polyhedron whose 8 faces are the 3 pentagons and the 5 remaining equilateral triangles
Click on the underlined text to access various portions of the Convex Uniform Polychora List:
Four Dimensional Figures Page: Return to initial page
Nomenclature: How the convex uniform polychora are named
List Key: Explanations of the various List entries
Section 1: Convex uniform polychora based on the pentachoron (5-cell): polychora #19
Section 2: Convex uniform polychora based on the tesseract (hypercube) and hexadecachoron (16-cell): polychora #1021
Section 4: Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell): polychora #3246
Section 5: The anomalous non-Wythoffian convex uniform polychoron: polychoron #47
Section 6: Convex uniform prismatic polychora: polychora #4864 and infinite sets
Section 7: Uniform polychora derived from hyperspherical tetrahedron B4: all duplicates of prior polychora