2. Convex uniform polychora based on the tesseract (8cell)
and hexadecachoron (16cell)
Symmetry group of all numbered polychora in this section:
[4,3,3] or [3,3,4], the diploid hexadecachoric group, of
order 384

(o)ooo
4
 Tesseract [10]
Alternative names:
 Tessaract
 [Fourdimensional] hypercube
 8cell
 Octachoron
 Octahedroid (Henry Parker Manning)
 Tes (Jonathan Bowers: for tesseract)
 [Fourdimensional] measure polytope
 [Fourdimensional regular] orthotope
 Cubic prism
 Cubic hyperprism
 Square duoprism
 Square double prism
 Square hyperprism
Schläfli symbols: {4,3,3}, also
t_{0}{4,3,3} or
t_{3}{3,3,4}, also {4,3}x{ }, t{2,4}x{ },
{4}x{4}, {4}x{ }x{ }, t{2}x{4}, t{2}xt{2}, t{2}x{ }x{ }, { }x{
}x{ }x{ }, or t_{0,1,2,3}{2,2,2}
Elements:
 Cells: 8 cubes
 Faces: 24 squares (all joining cubes to cubes)
 Edges: 32
 Vertices: 16
Vertex figure:
 Regular tetrahedron, edge length sqrt(2)

o(o)oo
4
 Tesseractihexadecachoron [11]
Alternative names:
 Rectified tesseract (Norman W. Johnson)
 Rit (Jonathan Bowers: for rectified tesseract)
 Rectified [fourdimensional] hypercube
 Rectified 8cell
 Rectified octachoron
 Rectified [fourdimensional] measure polytope
 Rectified [fourdimensional regular] orthotope
 Runcic tesseract (Norman W. Johnson)
 Runcic [fourdimensional] hypercube
 Runcic 8cell
 Runcic octachoron
 Runcic [fourdimensional] measure polytope
 Runcic [fourdimensional regular] orthotope
 Ambotesseract (Neil Sloane & John Horton Conway)
Schläfli symbols: r{4,3,3}, also
t_{1}{4,3,3},
t_{2}{3,3,4},
h_{3}{4,3,3}, or
t_{2}{3,3^{1,1}}
Elements: Cells: 8 cuboctahedra, 16
tetrahedra
 Faces: 64 triangles (all joining cuboctahedra to
tetrahedra), 24 squares (all joining cuboctahedra to
cuboctahedra)
 Edges: 96
 Vertices: 32 (located at the midpoints of the edges of
a tesseract, or at the centroids of the faces of a
hexadecachoron)
Vertex figure:
 Right equilateraltriangular prism: bases 2
equilateral triangles, edge length 1; lateral faces 3 rectangles,
edge lengths 1, sqrt(2)

oo(o)o
4
 Icositetrachoron [as “rectified
16cell”; not counted, duplicate of 22]

ooo(o)
4
 Hexadecachoron [12]
Alternative names:
 16cell (most frequently used)
 Hexadecahedroid or hexacaidecahedroid
 Hexadekahedroid or 16hedroid (Henry Parker Manning)
 Hex (Jonathan Bowers: for hexadecachoron)
 [Fourdimensional] cross polytope
 [Fourdimensional regular] orthoplex
 [Fourdimensional] hemicube
 Half tesseract or hemitesseract or alternated tesseract or
demitesseract
 Half [fourdimensional] hypercube or hemihypercube or
alternated hypercube or demihypercube
 Half 8cell or half octachoron
 Half [fourdimensional] measure polytope
 Half [fourdimensional regular] orthotope
 Tetrahedral antiprism
Schläfli symbols: {3,3,4}, also h{4,3,3},
t_{0}{3,3,4},
t_{3}{4,3,3}, or
{3,3^{1,1}}
Elements: Cells: 16 tetrahedra
 Faces: 32 triangles (all joining tetrahedra to
tetrahedra)
 Edges: 24
 Vertices: 8
Vertex figure: Regular octahedron, edge
length 1

(o)(o)oo
4
 Truncated tesseract [13]
Alternative names:
 Truncated [fourdimensional] hypercube
 Truncated 8cell
 Tat (Jonathan Bowers: for truncated tesseract)
 Truncated octachoron
 Truncated [fourdimensional] measure polytope
 Truncated [fourdimensional regular] orthotope
Schläfli symbols: t{4,3,3}, also
t_{0,1}{4,3,3} or
t_{2,3}{3,3,4}
Elements: Cells: 8 truncated cubes, 16
tetrahedra
 Faces: 64 triangles (all joining truncated cubes to
tetrahedra), 24 octagons (all joining truncated cubes to
truncated cubes)
 Edges: 128
 Vertices: 64 (located 1–sqrt(2)/2 from the
ends of each edge of a unit tesseract)
Vertex figure:
 Equilateraltriangular pyramid (or “triangular
spike”): base an equilateral triangle, edge
length 1; all 3 lateral triangles isosceles, edge lengths 1,
sqrt(2+sqrt(2)),sqrt(2+sqrt(2))

(o)o(o)o
4
 [Small] prismatotesseractihexadecachoron [14]
Alternative names:
 Cantellated tesseract (Norman W. Johnson)
 Cantellated [fourdimensional] hypercube
 Cantellated 8cell
 Cantellated octachoron
 Srit (Jonathan Bowers: for small rhombated
tesseract)
 Cantellated [fourdimensional] measure polytope
 Cantellated [fourdimensional regular] orthotope
Schläfli symbols:
t_{0,2}{4,3,3} or
t_{1,3}{3,3,4}
Elements:
 Cells: 8 rhombicuboctahedra, 16 octahedra, 32
triangular prisms
 Faces: 128 triangles (64 joining rhombicuboctahedra to
octahedra, 64 joining octahedra to triangular prisms), 120
squares (24 joining rhombicuboctahedra to rhombicuboctahedra, 96
joining rhombicuboctahedra to triangular prisms)
 Edges: 288
 Vertices: 96
Vertex figure:
 Square wedge: pentahedron with square base, edge
length 1, and wedge edge, length sqrt(2), 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),
sqrt(2), sqrt(2)

(o)oo(o)
4
 [Small] diprismatotesseractihexadecachoron [15]
Alternative names:
 Runcinated tesseract (Norman W. Johnson)
 Runcinated [fourdimensional] hypercube
 Runcinated 8cell
 Runcinated octachoron
 Sidpith (Jonathan Bowers: for small
diprismatotesseractihexadecachoron)
 Runcinated [fourdimensional] measure polytope
 Runcinated [fourdimensional regular] orthotope
 Runcinated 16cell (Norman W. Johnson)
 Runcinated hexadecachoron
 Runcinated [fourdimensional] cross polytope
 Runcinated [fourdimensional regular] orthoplex
Schläfli symbols:
t_{0,3}{4,3,3} or
t_{0,3}{3,3,4}
Elements:
 Cells: 32 cubes, 16 tetrahedra, 32 triangular prisms
 Faces: 64 triangles (all joining tetrahedra to
triangular prisms), 144 squares (48 joining cubes to cubes, 96
joining cubes to triangular prisms)
 Edges: 192
 Vertices: 64
Vertex figure:
 Equilateraltriangular antipodium (antiprism with
unequal bases): one base an equilateral triangle, edge length 1,
the other base an equilateral triangle, edge length
sqrt(2); 6 lateral faces are 3 isosceles triangles, edge
lengths 1, sqrt(2), sqrt(2), alternating with 3
equilateral triangles, edge length sqrt(2)

o(o)(o)o
4
 Truncatedoctahedral tesseractihexadecachoron [16]
Alternative names:
 Bitruncated tesseract (Norman W. Johnson)
 Bitruncated [fourdimensional] hypercube
 Bitruncated 8cell
 Bitruncated octachoron
 Bitruncated [fourdimensional] measure polytope
 Bitruncated [fourdimensional regular] orthotope
 Bitruncated 16cell (Norman W. Johnson)
 Bitruncated hexadecachoron
 Tah (Jonathan Bowers: for
tesseractihexadecachoron)
 Bitruncated [fourdimensional] cross polytope
 Bitruncated [fourdimensional regular] orthoplex
 Runcicantic tesseract (Norman W. Johnson)
 Runcicantic [fourdimensional] hypercube
 Runcicantic 8cell
 Runcicantic octachoron
 Runcicantic [fourdimensional] measure polytope
 Runcicantic [fourdimensional regular] orthotope
 Runcicantic 16cell (Norman W. Johnson)
 Runcicantic hexadecachoron
 Runcicantic [fourdimensional] cross polytope
 Runcicantic [fourdimensional regular] orthoplex
Schläfli symbols: 2t{4,3,3} or 2t{3,3,4}, also
t_{1,2}{4,3,3},
t_{1,2}{3,3,4},
h_{2,3}{4,3,3}, or
t_{1,2}{3,3^{1,1}}
Elements:
 Cells: 8 truncated octahedra, 16 truncated tetrahedra
 Faces: 32 triangles (all joining truncated tetrahedra
to truncated tetrahedra), 24 squares (all joining truncated
octahedra to truncated octahedra), 64 hexagons (all joining
truncated octahedra to truncated tetrahedra)
 Edges: 192
 Vertices: 96
Vertex figure:
 Digonal disphenoid: tetrahedron with 2 opposite edges
length 1, sqrt(2); all 4 lateral edges length
sqrt(3)

o(o)o(o)
4
 Disicositetrachoron [as “cantellated
16cell”; not counted, duplicate of 23]

oo(o)(o)
4
 Truncated hexadecachoron [17]
Alternative names:
 Truncated 16cell
 Thex (Jonathan Bowers: for truncated
hexadecachoron)
 Truncated [fourdimensional] cross polytope
 Truncated [fourdimensional regular] orthoplex
 Cantic tesseract (Norman W. Johnson)
 Cantic [fourdimensional] hypercube
 Cantic 8cell
 Cantic octachoron
 Cantic [fourdimensional] measure polytope
 Cantic [fourdimensional regular] orthotope
Schläfli symbols: t{3,3,4}, also
t_{2,3}{4,3,3},
t_{0,1}{3,3,4},
h_{2}{4,3,3}, or
t_{0,1}{3,3^{1,1}}
Elements:
 Cells: 8 octahedra, 16 truncated tetrahedra
 Faces: 64 triangles (all joining truncated tetrahedra
to octahedra), 32 hexagons (all joining truncated tetrahedra to
truncated tetrahedra)
 Edges: 120
 Vertices: 48
Vertex figure:
 Square pyramid (or “square spike”): square
base, edge length 1; all 4 lateral faces isosceles triangles,
edge lengths 1, sqrt(3), sqrt(3)

(o)(o)(o)o
4
 Great prismatotesseractihexadecachoron [18]
Alternative names:
 Cantitruncated tesseract (Norman W. Johnson)
 Cantitruncated [fourdimensional] hypercube
 Cantitruncated 8cell
 Cantitruncated octachoron
 Grit (Jonathan Bowers: for great rhombated
tesseract)
 Cantitruncated [fourdimensional] measure polytope
 Cantitruncated [fourdimensional regular] orthotope
Schläfli symbols:
t_{0,1,2}{4,3,3} or
t_{1,2,3}{3,3,4}
Elements:
 Cells: 8 truncated cuboctahedra, 16 truncated
tetrahedra, 32 triangular prisms
 Faces: 64 triangles (all joining truncated tetrahedra
to triangular prisms), 96 squares (all joining truncated
cuboctahedra to triangular prisms), 64 hexagons (all joining
truncated cuboctahedra to truncated tetrahedra), 24 octagons (all
joining truncated cuboctahedra to truncated cuboctahedra)
 Edges: 384
 Vertices: 192
Vertex figure:
 Sphenoid (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+sqrt(2)), joined so that
edges length 1 and sqrt(2+sqrt(2)) are opposite

(o)(o)o(o)
4
 Truncatedcubic diprismatotesseractihexadecachoron
[19]
Alternative names:
 Runcitruncated tesseract (Norman W. Johnson)
 Runcitruncated [fourdimensional] hypercube
 Runcitruncated 8cell
 Runcitruncated octachoron
 Proh (Jonathan Bowers: for prismatorhombated
hexadecachoron)
 Runcitruncated [fourdimensional] measure polytope
 Runcitruncated [fourdimensional regular] orthotope
Schläfli symbols:
t_{0,1,3}{4,3,3} or
t_{0,2,3}{3,3,4}
Elements:
 Cells: 8 truncated cubes, 16 cuboctahedra, 32
triangular prisms, 24 octagonal prisms
 Faces: 128 triangles (64 joining truncated cubes to
cuboctahedra, 64 joining triangular prisms to cuboctahedra), 192
squares (96 joining cuboctahedra to octagonal prisms, 96 joining
triangular prisms to octagonal prisms), 48 octagons (all joining
truncated cubes to octagonal prisms)
 Edges: 480
 Vertices: 192
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+sqrt(2)),
sqrt(2+sqrt(2)), alternating with (3 and 4)
isosceles triangles, edge lengths sqrt(2), sqrt(2),
sqrt(2+sqrt(2))

(o)o(o)(o)
4
 Rhombicuboctahedral diprismatotesseractihexadecachoron
[20]
Alternative names:
 Runcitruncated 16cell (Norman W. Johnson)
 Runcitruncated hexadecachoron
 Prit (Jonathan Bowers: for prismatorhombated
tesseract)
 Runcitruncated [fourdimensional] cross polytope
 Runcitruncated [fourdimensional regular] orthoplex
Schläfli symbols:
t_{0,2,3}{4,3,3} or
t_{0,1,3}{3,3,4}
Elements:
 Cells: 8 rhombicuboctahedra, 16 truncated tetrahedra,
32 hexagonal prisms, 24 cubes
 Faces: 64 triangles (all joining rhombicuboctahedra to
truncated tetrahedra), 240 squares (48 joining rhombicuboctahedra
to cubes, 96 joining rhombicuboctahedra to hexagonal prisms, 96
joining cubes to hexagonal prisms), 64 hexagons (all joining
truncated tetrahedra to hexagonal prisms)
 Edges: 480
 Vertices: 192
Vertex figure:
 Trapezoidal pyramid: base a trapezoid with edge
lengths 1, sqrt(2), sqrt(2), sqrt(2); 4
lateral triangles are (1) equilateral triangle, edge length
sqrt(2), (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)
4
 Truncated icositetrachoron [as “cantitruncated
16cell”; not counted, duplicate of 24]

(o)(o)(o)(o)
4
 Great diprismatotesseractihexadecachoron [21]
Alternative names:
 Omnitruncated tesseract (Norman W. Johnson)
 Gidpith (Jonathan Bowers: for great
diprismatotesseractihexadecachoron)
 Omnitruncated [fourdimensional] hypercube
 Omnitruncated 8cell
 Omnitruncated octachoron
 Omnitruncated [fourdimensional] measure polytope
 Omnitruncated [fourdimensional regular] orthotope
 Omnitruncated 16cell (Norman W. Johnson)
 Omnitruncated hexadecachoron
 Omnitruncated [fourdimensional] cross polytope
 Omnitruncated [fourdimensional regular] orthoplex
Schläfli symbols:
t_{0,1,2,3}{4,3,3} or
t_{0,1,2,3}{3,3,4}
Elements: Cells: 8 truncated cuboctahedra,
16 truncated octahedra, 32 hexagonal prisms, 24 octagonal prisms
 Faces: 288 squares (96 joining truncated cuboctahedra
to hexagonal prisms, 96 joining truncated octahedra to octagonal
prisms, 96 joining hexagonal prisms to octagonal prisms), 128
hexagons (64 joining truncated cuboctahedra to truncated
octahedra, 64 joining truncated octahedra to hexagonal prisms),
48 octagons (all joining truncated cuboctahedra to octagonal
prisms)
 Edges: 768
 Vertices: 384
Vertex figure:
 Chiral scalene tetrahedron with 4 different faces: 3
edges length sqrt(2) form chain through all 4 vertices;
other edges length sqrt(3), sqrt(3),
sqrt(2+sqrt(2)) in that order form the
complementary chain through the 4 vertices; dextro and
laevo versions each occur at 192 vertices

( )ooo
4
 Hexadecachoron [as “half tesseract”
h{4,3,3}; not counted, duplicate of 12]

( )o(o)o
4
 Truncated hexadecachoron [as “cantic
tesseract” h_{2}{4,3,3}; not counted,
duplicate of 17]

( )oo(o)
4
 Tesseractihexadecachoron [as “runcic
tesseract” h_{3}{4,3,3}; not counted,
duplicate of 11]

( )o(o)(o)
4
 Truncatedoctahedral tesseractihexadecachoron [as
“runcicantic tesseract”
h_{2,3}{4,3,3}; not counted, duplicate of
16]
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
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