1. Convex uniform polychora based on the pentachoron (5-cell)


Symmetry group of all polychora in this section except #5, 6, and 9: [3,3,3], the diploid pentachoric group, of order 120


(o)----o-----o-----o
Pentachoron [1]
Alternative names:
5-cell (most frequently used)
Pentatope
Pentahedroid (Henry Parker Manning)
[Four-dimensional] simplex
Pen (Jonathan Bowers: for pentachoron)

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

Elements:
Cells: 5 tetrahedra
Faces: 10 triangles (all joining tetrahedra to tetrahedra)
Edges: 10
Vertices: 5

Vertex figure:Regular Tetrahedron
Regular tetrahedron, edge length 1


 o----(o)----o-----o
Dispentachoron [2]
Alternative names:
Rectified 5-cell (Norman W. Johnson)
Rectified pentachoron
Rectified pentatope
Rectified [four-dimensional] simplex
Rap (Jonathan Bowers: for rectified pentachoron)
Ambopentachoron (Neil Sloane & John Horton Conway)

Schläfli symbols: r{3,3,3}, also t1{3,3,3} or t2{3,3,3}

Elements:
Cells: 5 octahedra, 5 tetrahedra
Faces: 30 triangles (10 joining octahedra to octahedra, 20 joining octahedra to tetrahedra)
Edges: 30
Vertices: 10 (located at the midpoints of the edges of a pentachoron, or at the centroids of its faces)

Vertex figure:Triangular Prism
Uniform triangular prism, edge length 1


(o)---(o)----o-----o
Truncated pentachoron [3]
Alternative names:
Truncated 5-cell
Truncated pentatope
Truncated [four-dimensional] simplex
Tip (Jonathan Bowers: for truncated pentachoron)

Schläfli symbols: t{3,3,3}, also t0,1{3,3,3} or t2,3{3,3,3}

Elements:
Cells: 5 truncated tetrahedra, 5 tetrahedra
Faces: 20 triangles (all joining tetrahedra to truncated tetrahedra), 10 hexagons (all joining truncated tetrahedra to truncated tetrahedra)
Edges: 40
Vertices: 20 (located 1/3 and 2/3 the way along each edge of a pentachoron)

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(3), sqrt(3)


(o)----o----(o)----o
[Small] prismatodispentachoron [4]
Alternative names:
Rectified dispentachoron
Cantellated 5-cell (Norman W. Johnson)
Cantellated pentachoron
Cantellated pentatope
Cantellated [four-dimensional] simplex
Srip (Jonathan Bowers: for small rhombated pentachoron)

Schläfli symbols: t0,2{3,3,3} or t1,3{3,3,3}

Elements:
Cells: 5 cuboctahedra, 5 octahedra, 10 triangular prisms
Faces: 50 triangles (10 joining cuboctahedra to cuboctahedra, 20 joining cuboctahedra to octahedra, 20 joining octahedra to triangular prisms), 30 squares (all joining cuboctahedra to triangular prisms)
Edges: 90
Vertices: 30 (located at the midpoints of the edges of a dispentachoron)

Vertex figure:Right Isosceles-triangular Prism
Right isosceles-triangular prism: base an isosceles triangle, edge lengths 1, sqrt(2), sqrt(2); height 1


(o)----o-----o----(o)
[Small] prismatodecachoron [5]
Alternative names:
Runcinated 5-cell (Norman W. Johnson)
Runcinated pentachoron
Runcinated pentatope
Runcinated [four-dimensional] simplex
Spid (Jonathan Bowers: for small prismatodecachoron)

Symmetry group: [[3,3,3]], the extended pentachoric group, of order 240

Schläfli symbol: t0,3{3,3,3}

Elements:
Cells: 10 tetrahedra, 20 triangular prisms
Faces: 40 triangles (all joining tetrahedra to triangular prisms), 30 squares (all joining triangular prisms to triangular prisms)
Edges: 60
Vertices: 20

Vertex figure:Elongate Equilateral-triangular Antiprism
Elongate equilateral-triangular antiprism: both bases equilateral triangles, edge length 1; all 6 lateral faces isosceles triangles edge lengths 1, sqrt(2), sqrt(2)


 o----(o)---(o)----o
[Truncated-tetrahedral] decachoron [6]
Alternative names:
Bitruncated 5-cell (Norman W. Johnson)
Bitruncated pentachoron
Bitruncated pentatope
Bitruncated [four-dimensional] simplex
Deca (Jonathan Bowers: for decachoron)

Symmetry group: [[3,3,3]], the extended pentachoric group, of order 240

Schläfli symbols: 2t{3,3,3}, also t1,2{3,3,3}

Elements:
Cells: 10 truncated tetrahedra
Faces: 20 triangles, 20 hexagons (all 40 faces joining truncated tetrahedra to truncated tetrahedra)
Edges: 60
Vertices: 30

Vertex figure:Tetragonal Disphenoid
Tetragonal disphenoid: tetrahedron with 2 opposite edges length 1; all 4 lateral edges length sqrt(3)


(o)---(o)---(o)----o
Great prismatodispentachoron [7]
Alternative names:
Truncated dispentachoron
Cantitruncated 5-cell (Norman W. Johnson)
Cantitruncated pentachoron
Cantitruncated pentatope
Cantitruncated [four-dimensional] simplex
Grip (Jonathan Bowers: for great rhombated pentachoron)

Schläfli symbols: t0,1,2{3,3,3} or t1,2,3{3,3,3}

Elements:
Cells: 5 truncated octahedra, 5 truncated tetrahedra, 10 triangular prisms
Faces: 20 triangles (all joining truncated tetrahedra to triangular prisms), 30 squares (all joining truncated octahedra to triangular prisms), 30 hexagons (10 joining truncated octahedra to truncated octahedra, 20 joining truncated octahedra to truncated tetrahedra)
Edges: 120
Vertices: 60 (located 1/3 and 2/3 the way along each edge of a dispentachoron)

Vertex figure:Sphenoid
Sphenoid (isosceles-triangular pyramid): base an isosceles triangle with edges length 1, sqrt(2), sqrt(2); all lateral edges length sqrt(3)


(o)---(o)----o----(o)
Diprismatodispentachoron [8]
Alternative names:
Runcitruncated 5-cell (Norman W. Johnson)
Runcitruncated pentachoron
Runcitruncated pentatope
Runcitruncated [four-dimensional] simplex
Prip (Jonathan Bowers: for prismatorhombated pentachoron)

Schläfli symbols: t0,1,3{3,3,3} or t0,2,3{3,3,3}

Elements:
Cells: 5 truncated tetrahedra, 5 cuboctahedra, 10 triangular prisms, 10 hexagonal prisms
Faces: 40 triangles (20 joining truncated tetrahedra to cuboctahedra, 20 joining cuboctahedra to triangular prisms), 60 squares (30 joining cuboctahedra to hexagonal prisms, 30 joining triangular prisms to hexagonal prisms), 20 hexagons (all joining truncated tetrahedra to hexagonal prisms)
Edges: 150
Vertices: 60

Vertex figure:Rectangular Pyramid
Rectangular pyramid: base a rectangle with edges length 1, sqrt(2); lateral faces an isosceles triangle with edges length 1, sqrt(2), sqrt(2) and an isosceles triangle with edges length 1, sqrt(3), sqrt(3), alternating with 2 congruent isosceles triangles with edges sqrt(2), sqrt(2), sqrt(3)


(o)---(o)---(o)---(o)
Great prismatodecachoron [9]
Alternative names:
Omnitruncated 5-cell (Norman W. Johnson)
Omnitruncated pentachoron
Omnitruncated pentatope
Omnitruncated [four-dimensional] simplex
Gippid (Jonathan Bowers: for great prismatodecachoron)

Symmetry group: [[3,3,3]], the extended pentachoric group, of order 240

Schläfli symbol: t0,1,2,3{3,3,3}

Elements:
Cells: 10 truncated octahedra, 20 hexagonal prisms
Faces: 90 squares (60 joining truncated octahedra to hexagonal prisms, 30 joining hexagonal prisms to hexagonal prisms), 60 hexagons (40 joining truncated octahedra to hexagonal prisms, 20 joining truncated octahedra to truncated octahedra)
Edges: 240
Vertices: 120

Vertex figure:Phyllic Disphenoid
Phyllic disphenoid with two kinds of isosceles-triangular faces (chiral tetrahedron): two faces with edges sqrt(2), sqrt(3), sqrt(3), other two with edges sqrt(2), sqrt(2), sqrt(3), arranged so the sqrt(2) edges form a chain and the sqrt(3) edges form the complementary chain passing through all 4 vertices; dextro and laevo versions each occur at 60 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 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 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 glomeric tetrahedron B4: all duplicates of prior polychora