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]
Alternative 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 Horton Conway)
- Ico (Jonathan Bowers: for icositetrachoron)
- Ambohexadecachoron (Neil Sloane & John Horton Conway)
- Four-dimensional rhombic-dodecahedron-analogue
Schläfli symbols: {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]
Alternative 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
- Rico (Jonathan Bowers: for rectified
icositetrachoron)
- Amboicositetrachoron (Neil Sloane & John Horton
Conway)
Schläfli symbols: 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]
Alternative names:
- Truncated 24-cell
- Truncated polyoctahedron
- Tico (Jonathan Bowers: for truncated
icositetrachoron)
Schläfli symbols: 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]
Alternative names:
- Cantellated 24-cell (Norman W. Johnson)
- Cantellated icositetrachoron
- Cantellated polyoctahedron
- Srico (Jonathan Bowers: for small rhombated
icositetrachoron)
Schläfli symbols:
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 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]
Alternative names:
- Runcinated 24-cell (Norman W. Johnson)
- Runcinated icositetrachoron
- Runcinated polyoctahedron
- Spic (Jonathan Bowers: for small
prismatotetracontaoctachoron)
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]
Alternative names:
- Bitruncated 24-cell (Norman W. Johnson)
- Bitruncated icositetrachoron
- Bitruncated polyoctahedron
- Cont (Jonathan Bowers: for
tetracontaoctachoron)
Symmetry group: [[3,4,3]], the extended
icositetrachoric group, of order 2304
Schläfli symbols: 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:
- Tetragonal 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]
Alternative names:
- Cantitruncated 24-cell (Norman W. Johnson)
- Cantitruncated icositetrachoron
- Cantitruncated polyoctahedron
- Grico (Jonathan Bowers: for great rhombated
icositetrachoron)
Schläfli symbols:
t0,1,2{3,4,3} or
t1,2,3{3,4,3}
Elements:
- Cells: 24 truncated cuboctahedra, 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:
- Sphenoid (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]
Alternative names:
- Runcitruncated 24-cell (Norman W. Johnson)
- Runcitruncated icositetrachoron
- Runcitruncated polyoctahedron
- Prico (Jonathan Bowers: for prismatorhombated
icositetrachoron)
Schläfli symbols:
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]
Alternative names:
- Omnitruncated 24-cell (Norman W. Johnson)
- Omnitruncated icositetrachoron
- Omnitruncated polyoctahedron
- Gippic (Jonathan Bowers: for great
prismatotetracontaoctachoron)
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:
- Phyllic disphenoid (chiral tetrahedron with two kinds
of faces): 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]
Alternative names:
- Snub 24-cell
- Snub polyoctahedron
- Sadi (Jonathan Bowers: for snub
disicositetrachoron)
Symmetry group: [3+,4,3],
the ionic diminished icositetrachoric group, of order
576
Schläfli symbols: 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
Multidimensional Glossary: Explanations of some geometrical terms and
concepts
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 glomeric tetrahedron
B4: all duplicates of prior
polychora