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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, edge length tau (=[sqrt(5)+1]/2)

 o----(o)----o-----o
    5

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:

Right equilateral-triangular prism: bases 2 equilateral triangles, edge length 1; lateral faces 3 rectangles, edge lengths 1, tau

 o-----o----(o)----o
    5

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:

Uniform pentagonal prism, edge length 1

 o-----o-----o----(o)
    5

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, edge length 1

(o)---(o)----o-----o
    5

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:

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

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)----o-----o----(o)
    5

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:

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

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

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 isosceles-triangular prism: base an isosceles triangle, edge lengths sqrt(2), sqrt(2), tau; height 1

 o-----o----(o)---(o)
    5

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:

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

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 (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

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

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

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 (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

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 (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