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
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:Triangular Prism
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
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
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
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
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
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
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
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 edge’s ends)

Vertex figure:Tridiminished Icosahedron
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 #1–9

Section 2: Convex uniform polychora based on the tesseract (hypercube) and hexadecachoron (16-cell): polychora #10–21

Section 4: Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell): polychora #32–46

Section 5: The anomalous non-Wythoffian convex uniform polychoron: polychoron #47

Section 6: Convex uniform prismatic polychora: polychora #48–64 and infinite sets

Section 7: Uniform polychora derived from hyperspherical tetrahedron B4: all duplicates of prior polychora