4. Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell)


Symmetry group of all polychora in this section: [5,3,3] or [3,3,5], the diploid hexacosichoric group, of order 14400


(o)----o-----o-----o
5
Hecatonicosachoron [32]
Alternative names:
120-cell (most frequently used)
Hecatonicosahedroid or hecatoncaiicosahedroid
Hekatonikosahedroid or 120-hedroid (Henry Parker Manning)
Polydodecahedron (John Horton Conway)
Hi (Jonathan Bowers: for hecatonicosachoron)

Schläfli symbols: {5,3,3}, also t0{5,3,3} or t3{3,3,5}

Elements:
Cells: 120 dodecahedra
Faces: 720 pentagons
Edges: 1200
Vertices: 600

Vertex figure:Regular Tetrahedron
Regular tetrahedron, edge length tau (=[sqrt(5)+1]/2)


 o----(o)----o-----o
5
Icosidodecahedral hexacosihecatonicosachoron [33]
Alternative names:
Rectified 120-cell (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 t1{5,3,3} or t2{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:Triangular Prism
Right equilateral-triangular prism: bases 2 equilateral triangles, edge length 1; lateral faces 3 rectangles, edge lengths 1, tau


 o-----o----(o)----o
5
Icosahedral hexacosihecatonicosachoron [34]
Alternative names:
Rectified 600-cell (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 t1{3,3,5} or t2{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:Pentagonal Prism
Uniform pentagonal prism, edge length 1


 o-----o-----o----(o)
5
Hexacosichoron [35]
Alternative names:
600-cell (most frequently used)
Hexacosihedroid
Hexakosioihedroid or 600-hedroid (Henry Parker Manning)
Polytetrahedron (John Horton Conway)
Ex (Jonathan Bowers: for hexacosichoron)

Schläfli symbols: {3,3,5}, also t0{3,3,5} or t3{5,3,3}

Elements:
Cells: 600 tetrahedra
Faces: 1200 triangles (joining tetrahedra to tetrahedra)
Edges: 720
Vertices: 120

Vertex figure:
Regular Icosahedron
Regular icosahedron, edge length 1


(o)---(o)----o-----o
5
Truncated hecatonicosachoron [36]
Alternative names:
Truncated 120-cell
Truncated polydodecahedron
Thi (Jonathan Bowers: for truncated hecatonicosachoron)

Schläfli symbols: t{5,3,3}, also t0,1{5,3,3} or t2,3{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:Equilateral-triangular Pyramid
Equilateral-triangular 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)


(o)----o----(o)----o
5
[Small] prismatohexacosihecatonicosachoron [37]
Alternative names:
Cantellated 120-cell (Norman W. Johnson)
Cantellated hecatonicosachoron
Cantellated polydodecahedron
Srahi (Jonathan Bowers: for small rhombated hecatonicosachoron)

Schläfli symbols: t0,2{5,3,3} or t1,3{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
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)----o-----o----(o)
5
[Small] diprismatohexacosihecatonicosachoron [38]
Alternative names:
Runcinated 120-cell (Norman W. Johnson)
Runcinated hecatonicosachoron
Runcinated polydodecahedron
Runcinated 600-cell (Norman W. Johnson)
Runcinated hexacosichoron
Runcinated polytetrahedron
Sidpixhi (Jonathan Bowers: for small diprismatohexacosihecatonicosachoron)

Schläfli symbols: t0,3{5,3,3} or t0,3{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:Equilateral-triangular Antipodium
Equilateral-triangular antipodium (equilateral-triangular 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)


 o----(o)---(o)----o
5
Truncated-icosahedral hexacosihecatonicosachoron [39]
Alternative names:
Bitruncated 120-cell (Norman W. Johnson)
Bitruncated hecatonicosachoron
Bitruncated polydodecahedron
Bitruncated 600-cell (Norman W. Johnson)
Bitruncated hexacosichoron
Bitruncated polytetrahedron
Xhi (Jonathan Bowers: for hexacosihecatonicosachoron)

Schläfli symbols: 2t{5,3,3} or 2t{3,3,5}, also t1,2{5,3,3} or t1,2{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
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 600-cell (Norman W. Johnson)
Cantellated hexacosichoron
Cantellated polytetrahedron
Srix (Jonathan Bowers: for small rhombated hexacosichoron)

Schläfli symbols: t0,2{3,3,5} or t1,3{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 Isosceles-triangular Prism
Right isosceles-triangular prism: base an isosceles triangle, edge lengths sqrt(2), sqrt(2), tau; height 1

 o-----o----(o)---(o)
5
Truncated hexacosichoron [41]
Alternative names:
Truncated 600-cell
Truncated polytetrahedron
Tex (Jonathan Bowers: for truncated hexacosichoron)
“Four-dimensional soccer ball”

Schläfli symbols: t{3,3,5}, also t0,1{3,3,5} or t2,3{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:Regular-pentagonal Pyramid
Regular-pentagonal pyramid (or “pentagonal spike”): base a regular pentagon, edge length 1; all 5 lateral edges length sqrt(3)


(o)---(o)---(o)----o
5
Great prismatohexacosihecatonicosachoron [42]
Alternative names:
Cantitruncated 120-cell (Norman W. Johnson)
Cantitruncated hecatonicosachoron
Cantitruncated polydodecahedron
Grahi (Jonathan Bowers: for great rhombated hecatonicosachoron)

Schläfli symbols: t0,1,2{5,3,3} or t1,2,3{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
Sphenoid (isosceles-triangular 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
Truncated-dodecahedral diprismatohexacosihecatonicosachoron [43]
Alternative names:
Runcitruncated 120-cell (Norman W. Johnson)
Runcitruncated hecatonicosachoron
Runcitruncated polydodecahedron
Prix (Jonathan Bowers: for prismatorhombated hexacosichoron)

Schläfli symbols: t0,1,3{5,3,3} or t0,2,3{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
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 600-cell (Norman W. Johnson)
Runcitruncated hexacosichoron
Runcitruncated polytetrahedron
Prahi (Jonathan Bowers: for prismatorhombated hecatonicosachoron)

Schläfli symbols: t0,2,3{5,3,3} or t0,1,3{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
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
Truncated-icosahedral prismatohexacosihecatonicosachoron [45]
Alternative names:
Truncated icosahedral hexacosihecatonicosachoron
Cantitruncated 600-cell (Norman W. Johnson)
Cantitruncated hexacosichoron
Cantitruncated polytetrahedron
Grix (Jonathan Bowers: for great rhombated hexacosichoron)

Schläfli symbols: t1,2,3{5,3,3} or t0,1,2{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
Sphenoid (isosceles-triangular 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 120-cell (Norman W. Johnson)
Omnitruncated hecatonicosachoron
Omnitruncated polydodecahedron
Omnitruncated 600-cell (Norman W. Johnson)
Omnitruncated hexacosichoron
Omnitruncated polytetrahedron
Gidpixhi (Jonathan Bowers: for great diprismatohexacosihecatonicosachoron)

Schläfli symbols: t0,1,2,3{5,3,3} or t0,1,2,3{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
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 (5-cell): polychora #1–9

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

Section 3: Convex uniform polychora based on the icositetrachoron (24-cell): polychora #22–31

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 glomeric tetrahedron B4: all duplicates of prior polychora