1. Convex uniform polychora based on the pentachoron
(5cell)
Symmetry group of all polychora in this section except
#5, 6, and 9: [3,3,3], the diploid pentachoric group, of
order 120

(o)ooo
 Pentachoron [1]
Alternative names:
 5cell (most frequently used)
 Pentatope
 Pentahedroid (Henry Parker Manning)
 [Fourdimensional] simplex
 Pen (Jonathan Bowers: for pentachoron)
Schläfli symbols: {3,3,3}, also
t_{0}{3,3,3} or
t_{3}{3,3,3}
Elements: Cells: 5 tetrahedra
 Faces: 10 triangles (all joining tetrahedra to
tetrahedra)
 Edges: 10
 Vertices: 5
Vertex figure:
 Regular tetrahedron, edge length 1

o(o)oo
 Dispentachoron [2]
Alternative names:
 Rectified 5cell (Norman W. Johnson)
 Rectified pentachoron
 Rectified pentatope
 Rectified [fourdimensional] simplex
 Rap (Jonathan Bowers: for rectified
pentachoron)
 Ambopentachoron (Neil Sloane & John Horton Conway)
Schläfli symbols: r{3,3,3}, also
t_{1}{3,3,3} or
t_{2}{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: Uniform triangular prism, edge
length 1

(o)(o)oo
 Truncated pentachoron [3]
Alternative names:
 Truncated 5cell
 Truncated pentatope
 Truncated [fourdimensional] simplex
 Tip (Jonathan Bowers: for truncated
pentachoron)
Schläfli symbols: t{3,3,3}, also
t_{0,1}{3,3,3} or
t_{2,3}{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:
 Equilateraltriangular 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 5cell (Norman W. Johnson)
 Cantellated pentachoron
 Cantellated pentatope
 Cantellated [fourdimensional] simplex
 Srip (Jonathan Bowers: for small rhombated
pentachoron)
Schläfli symbols:
t_{0,2}{3,3,3} or
t_{1,3}{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 isoscelestriangular prism: base an isosceles
triangle, edge lengths 1, sqrt(2), sqrt(2); height
1

(o)oo(o)
 [Small] prismatodecachoron [5]
Alternative names:
 Runcinated 5cell (Norman W. Johnson)
 Runcinated pentachoron
 Runcinated pentatope
 Runcinated [fourdimensional] simplex
 Spid (Jonathan Bowers: for small
prismatodecachoron)
Symmetry group: [[3,3,3]], the extended pentachoric
group, of order 240
Schläfli symbol:
t_{0,3}{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 equilateraltriangular 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
 [Truncatedtetrahedral] decachoron [6]
Alternative names: Bitruncated 5cell (Norman W.
Johnson)
 Bitruncated pentachoron
 Bitruncated pentatope
 Bitruncated [fourdimensional] 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
t_{1,2}{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: 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 5cell (Norman W. Johnson)
 Cantitruncated pentachoron
 Cantitruncated pentatope
 Cantitruncated [fourdimensional] simplex
 Grip (Jonathan Bowers: for great rhombated
pentachoron)
Schläfli symbols:
t_{0,1,2}{3,3,3} or
t_{1,2,3}{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 (isoscelestriangular 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 5cell (Norman W. Johnson)
 Runcitruncated pentachoron
 Runcitruncated pentatope
 Runcitruncated [fourdimensional] simplex
 Prip (Jonathan Bowers: for prismatorhombated
pentachoron)
Schläfli symbols:
t_{0,1,3}{3,3,3} or
t_{0,2,3}{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: 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 5cell (Norman W. Johnson)
 Omnitruncated pentachoron
 Omnitruncated pentatope
 Omnitruncated [fourdimensional] simplex
 Gippid (Jonathan Bowers: for great
prismatodecachoron)
Symmetry group: [[3,3,3]], the extended pentachoric
group, of order 240
Schläfli symbol:
t_{0,1,2,3}{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 with two
kinds of isoscelestriangular 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 (16cell): polychora #10–21
Section
3: Convex uniform polychora based on the icositetrachoron
(24cell): polychora #22–31
Section
4: Convex uniform polychora based on the hecatonicosachoron
(120cell) and hexacosichoron (600cell): polychora #32–46
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