4. Convex uniform polychora based on the hecatonicosachoron
(120cell) and hexacosichoron (600cell)
Symmetry group of all polychora in this section: [5,3,3]
or [3,3,5], the diploid hexacosichoric group, of order
14400

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5
 Hecatonicosachoron [32]
Alternative names:
 120cell (most frequently used)
 Hecatonicosahedroid or hecatoncaiicosahedroid
 Hekatonikosahedroid or 120hedroid (Henry Parker
Manning)
 Polydodecahedron (John Horton Conway)
 Hi (Jonathan Bowers: for hecatonicosachoron)
Schläfli symbols: {5,3,3}, also
t_{0}{5,3,3} or
t_{3}{3,3,5}
Elements:
 Cells: 120 dodecahedra
 Faces: 720 pentagons
 Edges: 1200
 Vertices: 600
Vertex figure:
 Regular tetrahedron, edge length tau
(=[sqrt(5)+1]/2)

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5
 Icosidodecahedral hexacosihecatonicosachoron [33]
Alternative names:
 Rectified 120cell (Norman W. Johnson)
 Rectified hecatonicosichoron
 Rectified polydodecahedron
 Rahi (Jonathan Bowers: for rectified
hecatonicosachoron)
 Ambohecatonicosachoron (Neil Sloane & John Horton
Conway)
Schläfli symbols: r{5,3,3}, also
t_{1}{5,3,3} or
t_{2}{3,3,5}
Elements:
 Cells: 120 icosidodecahedra, 600 tetrahedra
 Faces: 2400 triangles (all joining icosidodecahedra to
tetrahedra), 720 pentagons (all joining icosidodecahedra to
icosidodecahedra)
 Edges: 3600
 Vertices: 1200 (located at the midpoints of the
edges of a hecatonicosachoron, or at the centroids of the faces
of a hexacosichoron)
Vertex figure:
 Right equilateraltriangular prism: bases 2
equilateral triangles, edge length 1; lateral faces 3 rectangles,
edge lengths 1, tau

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5
 Icosahedral hexacosihecatonicosachoron [34]
Alternative names:
 Rectified 600cell (Norman W. Johnson)
 Rectified hexacosichoron
 Rectified polytetrahedron
 Rox (Jonathan Bowers: for rectified
hexacosichoron)
 Ambohexacosichoron (Neil Sloane & John Horton Conway)
Schläfli symbols: r{3,3,5}, also
t_{1}{3,3,5} or
t_{2}{5,3,3}
Elements:
 Cells: 120 icosahedra, 600 octahedra
 Faces: 3600 triangles (1200 joining octahedra
to octahedra, 2400 joining icosahedra to octahedra)
 Edges: 3600
 Vertices: 720 (located
at the midpoints of the edges of a hexacosichoron, or at the
centroids of the faces of a hecatonicosachoron)
Vertex figure:
 Uniform pentagonal prism, edge length 1
ooo(o)
5
 Hexacosichoron [35]
Alternative names:
 600cell (most frequently used)
 Hexacosihedroid
 Hexakosioihedroid or 600hedroid (Henry
Parker Manning)
 Polytetrahedron (John Horton Conway)
 Ex (Jonathan Bowers: for hexacosichoron)
Schläfli symbols: {3,3,5}, also
t_{0}{3,3,5} or
t_{3}{5,3,3}
Elements:
 Cells: 600 tetrahedra
 Faces: 1200 triangles (joining tetrahedra to
tetrahedra)
 Edges: 720
 Vertices: 120
Vertex figure:
 Regular icosahedron, edge length 1

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5
 Truncated hecatonicosachoron [36]
Alternative names:
 Truncated 120cell
 Truncated polydodecahedron
 Thi (Jonathan Bowers: for truncated
hecatonicosachoron)
Schläfli symbols: t{5,3,3}, also
t_{0,1}{5,3,3} or
t_{2,3}{3,3,5}
Elements:
 Cells: 120 truncated dodecahedra, 600 tetrahedra
 Faces: 2400 triangles (all joining truncated
dodecahedra to tetrahedra), 720 decagons (all joining truncated
dodecahedra to truncated dodecahedra)
 Edges: 4800
 Vertices: 2400 (located 1/(2+tau) from the
ends of each edge of a unit hecatonicosichoron)
Vertex figure:
 Equilateraltriangular pyramid (or
“triangular spike”): base an equilateral triangle, edge
length 1; all 3 lateral triangles isosceles, edge lengths 1,
sqrt(2+tau), sqrt(2+tau)

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5
 [Small] prismatohexacosihecatonicosachoron [37]
Alternative names:
 Cantellated 120cell (Norman W. Johnson)
 Cantellated hecatonicosachoron
 Cantellated polydodecahedron
 Srahi (Jonathan Bowers: for small rhombated
hecatonicosachoron)
Schläfli symbols:
t_{0,2}{5,3,3} or
t_{1,3}{3,3,5}
Elements:
 Cells: 120 rhombicosidodecahedra, 600 octahedra, 1200
triangular prisms
 Faces: 4800 triangles (2400 joining
rhombicosidodecahedra to octahedra, 2400 joining octahedra to
triangular prisms), 3600 squares (all joining
rhombicosidodecahedra to triangular prisms), 720 pentagons (all
joining rhombicosidodecahedra to rhombicosidodecahedra)
 Edges: 10800
 Vertices: 3600
Vertex figure:
 Square wedge: pentahedron with square base, edge
length 1, and wedge edge, length tau, symmetrically
located above plane of base and parallel to 2 opposite square
edges; lateral faces joining wedge edge to square are 2 isosceles
triangles, edge lengths 1, sqrt(2), sqrt(2),
alternating with 2 trapezoids, edge lengths 1, sqrt(2),
tau, sqrt(2)

(o)oo(o)
5
 [Small] diprismatohexacosihecatonicosachoron [38]
Alternative names:
 Runcinated 120cell (Norman W. Johnson)
 Runcinated hecatonicosachoron
 Runcinated polydodecahedron
 Runcinated 600cell (Norman W. Johnson)
 Runcinated hexacosichoron
 Runcinated polytetrahedron
 Sidpixhi (Jonathan Bowers: for small
diprismatohexacosihecatonicosachoron)
Schläfli symbols:
t_{0,3}{5,3,3} or
t_{0,3}{3,3,5}
Elements:
 Cells: 120 dodecahedra, 600 tetrahedra, 1200
triangular prisms, 720 pentagonal prisms
 Faces: 2400 triangles (all joining tetrahedra to
triangular prisms), 3600 squares (all joining triangular prisms
to pentagonal prisms), 1440 pentagons (all joining pentagonal
prisms to dodecahedra)
 Edges: 7200
 Vertices: 2400
Vertex figure:
 Equilateraltriangular antipodium
(equilateraltriangular antiprism with unequal bases): one base
an equilateral triangle, edge length 1, the other base an
equilateral triangle, edge length tau; 6 lateral faces are
3 isosceles triangles, edge lengths 1, sqrt(2),
sqrt(2), alternating with 3 other isosceles triangles,
edge lengths tau, sqrt(2), sqrt(2)

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5
 Truncatedicosahedral hexacosihecatonicosachoron [39]
Alternative names:
 Bitruncated 120cell (Norman W. Johnson)
 Bitruncated hecatonicosachoron
 Bitruncated polydodecahedron
 Bitruncated 600cell (Norman W. Johnson)
 Bitruncated hexacosichoron
 Bitruncated polytetrahedron
 Xhi (Jonathan Bowers: for
hexacosihecatonicosachoron)
Schläfli symbols: 2t{5,3,3} or 2t{3,3,5}, also
t_{1,2}{5,3,3} or
t_{1,2}{3,3,5}
Elements:
 Cells: 120 truncated icosahedra, 600 truncated
tetrahedra
 Faces: 1200 triangles (all joining truncated
tetrahedra to truncated tetrahedra), 720 pentagons (all joining
truncated icosahedra to truncated icosahedra), 2400 hexagons (all
joining truncated icosahedra to truncated tetrahedra)
 Edges: 7200
 Vertices: 3600
Vertex figure:
 Digonal disphenoid: tetrahedron with 2 opposite edges
lengths 1, tau; all 4 lateral edges length sqrt(3)

o(o)o(o)
5
 Icosidodecahedral prismatohexacosihecatonicosachoron
[40]
Alternative names:
 Rectified icosahedral hexacosihecatonicosachoron
 Cantellated 600cell (Norman W. Johnson)
 Cantellated hexacosichoron
 Cantellated polytetrahedron
 Srix (Jonathan Bowers: for small rhombated
hexacosichoron)
Schläfli symbols:
t_{0,2}{3,3,5} or
t_{1,3}{5,3,3}
Elements:
 Cells: 120 icosidodecahedra, 600 cuboctahedra, 720
pentagonal prisms
 Faces: 3600 triangles (1200 joining cuboctahedra to
cuboctahedra, 2400 joining cuboctahedra to icosidodecahedra),
3600 squares (all joining cuboctahedra to pentagonal prisms),
1440 pentagons (all joining icosidodecahedra to pentagonal
prisms)
 Edges: 10800
 Vertices: 3600 (located at the midpoints of the
edges of an icosahedral hexacosihecatonicosachoron)
Vertex figure:
 Right isoscelestriangular prism: base an isosceles
triangle, edge lengths sqrt(2), sqrt(2),
tau; height 1

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5
 Truncated hexacosichoron [41]
Alternative names:
 Truncated 600cell
 Truncated polytetrahedron
 Tex (Jonathan Bowers: for truncated
hexacosichoron)
 “Fourdimensional soccer ball”
Schläfli symbols: t{3,3,5}, also
t_{0,1}{3,3,5} or
t_{2,3}{5,3,3}
Elements:
 Cells: 120 icosahedra, 600 truncated tetrahedra
 Faces: 2400 triangles (all joining icosahedra to
truncated tetrahedra), 1200 hexagons (all joining truncated
tetrahedra to truncated tetrahedra)
 Edges: 4320
 Vertices: 1440 (located 1/3 and 2/3 the way along
each edge of a hexacosichoron)
Vertex figure:
 Regularpentagonal pyramid (or “pentagonal
spike”): base a regular pentagon, edge length 1; all 5
lateral edges length sqrt(3)

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5
 Great prismatohexacosihecatonicosachoron [42]
Alternative names:
 Cantitruncated 120cell (Norman W. Johnson)
 Cantitruncated hecatonicosachoron
 Cantitruncated polydodecahedron
 Grahi (Jonathan Bowers: for great rhombated
hecatonicosachoron)
Schläfli symbols:
t_{0,1,2}{5,3,3} or
t_{1,2,3}{3,3,5}
Elements:
 Cells: 120 truncated icosidodecahedra, 600 truncated
tetrahedra, 1200 triangular prisms
 Faces: 2400 triangles (all joining truncated
tetrahedra to triangular prisms), 3600 squares (all joining
truncated icosidodecahedra to triangular prisms), 2400 hexagons
(all joining truncated icosidodecahedra to truncated tetrahedra),
720 decagons (all joining truncated icosidodecahedra to truncated
icosidodecahedra)
 Edges: 14400
 Vertices: 7200
Vertex figure:
 Sphenoid (isoscelestriangular pyramid or bilaterally
symmetric tetrahedron): one face an isosceles triangle, edge
lengths 1, sqrt(2), sqrt(2), joined to another
isosceles triangle, edge lengths 1, sqrt(3),
sqrt(3); other 2 faces congruent scalene triangles, edge
lengths sqrt(2), sqrt(3),
sqrt(2+tau), joined so that edges length 1 and
sqrt (2+tau) are opposite

(o)(o)o(o)
5
 Truncateddodecahedral
diprismatohexacosihecatonicosachoron [43]
Alternative names:
 Runcitruncated 120cell (Norman W. Johnson)
 Runcitruncated hecatonicosachoron
 Runcitruncated polydodecahedron
 Prix (Jonathan Bowers: for prismatorhombated
hexacosichoron)
Schläfli symbols:
t_{0,1,3}{5,3,3} or
t_{0,2,3}{3,3,5}
Elements:
 Cells: 120 truncated dodecahedra, 600 cuboctahedra,
1200 triangular prisms, 720 decagonal prisms
 Faces: 4800 triangles (2400 joining truncated
dodecahedra to cuboctahedra, 2400 joining triangular prisms to
cuboctahedra), 7200 squares (3600 joining cuboctahedra to
decagonal prisms, 3600 joining triangular prisms to decagonal
prisms), 1440 decagons (all joining truncated dodecahedra to
decagonal prisms)
 Edges: 18000
 Vertices: 7200
Vertex figure:
 Rectangular pyramid: base a rectangle, edge lengths 1,
sqrt(2); lateral faces (1) isosceles triangle, edge
lengths 1, sqrt(2), sqrt(2), and (2) isosceles
triangle, edge lengths 1, sqrt(2+tau),
sqrt(2+tau), alternating with (3 and 4) isosceles
triangles, edge lengths sqrt(2), sqrt(2),
sqrt(2+tau)

(o)o(o)(o)
5
 Rhombicosidodecahedral
diprismatohexacosihecatonicosachoron [44]
Alternative names:
 Runcitruncated 600cell (Norman W. Johnson)
 Runcitruncated hexacosichoron
 Runcitruncated polytetrahedron
 Prahi (Jonathan Bowers: for prismatorhombated
hecatonicosachoron)
Schläfli symbols:
t_{0,2,3}{5,3,3} or
t_{0,1,3}{3,3,5}
Elements:
 Cells: 120 rhombicosidodecahedra, 600 truncated
tetrahedra, 1200 hexagonal prisms, 720 pentagonal prisms
 Faces: 2400 triangles (all joining
rhombicosidodecahedra to truncated tetrahedra), 7200 squares
(3600 joining rhombicosidodecahedra to hexagonal prisms, 3600
joining pentagonal prisms to hexagonal prisms), 1440 pentagons
(all joining rhombicosidodecahedra to pentagonal prisms), 2400
hexagons (all joining truncated tetrahedra to hexagonal prisms)
 Edges: 18000
 Vertices: 7200
Vertex figure:
 Trapezoidal pyramid: base a trapezoid with edge
lengths 1, sqrt(2), tau, sqrt(2);
lateral triangles are (1) isosceles triangle, edge lengths
sqrt(2), sqrt(2), tau, (2) isosceles
triangle, edge lengths 1, sqrt(3), sqrt(3),
alternating with (3 and 4) congruent isosceles triangles, edge
lengths sqrt(2), sqrt(2), sqrt(3)

o(o)(o)(o)
5
 Truncatedicosahedral prismatohexacosihecatonicosachoron
[45]
Alternative names:
 Truncated icosahedral hexacosihecatonicosachoron
 Cantitruncated 600cell (Norman W. Johnson)
 Cantitruncated hexacosichoron
 Cantitruncated polytetrahedron
 Grix (Jonathan Bowers: for great rhombated
hexacosichoron)
Schläfli symbols:
t_{1,2,3}{5,3,3} or
t_{0,1,2}{3,3,5}
Elements:
 Cells: 120 truncated icosahedra, 600 truncated
octahedra, 720 pentagonal prisms
 Faces: 3600 squares (all joining truncated octahedra
to pentagonal prisms), 1440 pentagons (all joining truncated
icosahedra to pentagonal prisms), 3600 hexagons (2400 joining
truncated icosahedra to truncated octahedra, 1200 joining
truncated octahedra to truncated octahedra)
 Edges: 14400
 Vertices: 7200 (located 1/3 and 2/3 the way along
each edge of an icosahedral hexacosihecatonicosachoron)
Vertex figure:
 Sphenoid (isoscelestriangular pyramid or bilaterally
symmetric tetrahedron): base an isosceles triangle, edge
lengths sqrt(3), sqrt(3), tau; lateral
faces (1) isosceles triangle, edge lengths sqrt(2),
sqrt(2), tau, and (2 and 3) congruent isosceles
triangles, edge lengths sqrt(2), sqrt(3),
sqrt(3)

(o)(o)(o)(o)
5
 Great diprismatohexacosihecatonicosachoron
[46]
Alternative names:
 Omnitruncated 120cell (Norman W. Johnson)
 Omnitruncated hecatonicosachoron
 Omnitruncated polydodecahedron
 Omnitruncated 600cell (Norman W. Johnson)
 Omnitruncated hexacosichoron
 Omnitruncated polytetrahedron
 Gidpixhi (Jonathan Bowers: for great
diprismatohexacosihecatonicosachoron)
Schläfli symbols:
t_{0,1,2,3}{5,3,3} or
t_{0,1,2,3}{3,3,5}
Elements:
 Cells: 120 truncated icosidodecahedra, 600 truncated
octahedra, 1200 hexagonal prisms, 720 decagonal prisms
 Faces: 10800 squares (3600 joining truncated
icosidodecahedra to hexagonal prisms, 3600 joining truncated
octahedra to decagonal prisms, 3600 joining hexagonal prisms to
decagonal prisms), 4800 hexagons (2400 joining truncated
icosidodecahedra to truncated octahedra, 2400 joining truncated
octahedra to hexagonal prisms), 1440 decagons (all joining
truncated icosidodecahedra to decagonal prisms)
 Edges: 28800
 Vertices: 14400
Vertex figure:
 Chiral scalene tetrahedron: 3 edges length
sqrt(2) form chain through all 4 vertices; other edges
length sqrt(3), sqrt(3), sqrt(2+tau)
in that order form complementary chain through the 4 vertices;
dextro and laevo versions each occur at 7200
vertices
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
(5cell): polychora #1–9
Section
2: Convex uniform polychora based on the tesseract
(hypercube) and hexadecachoron (16cell): polychora
#10–21
Section
3: Convex uniform polychora based on the icositetrachoron
(24cell): polychora #22–31
Section
5: The anomalous nonWythoffian convex uniform polychoron:
polychoron #47
Section
6: Convex uniform prismatic polychora: polychora #48–64
and infinite sets
Section
7: Uniform polychora derived from glomeric tetrahedron
B_{4}: all duplicates of prior
polychora