2. Convex uniform polychora based on the tesseract (8-cell)
and hexadecachoron (16-cell)
Symmetry group of all numbered polychora in this section:
[4,3,3] or [3,3,4], the diploid hexadecachoric group, of
order 384
-
(o)----o-----o-----o
4
- Tesseract [10]
Alternative names:
- Tessaract
- [Four-dimensional] hypercube
- 8-cell
- Octachoron
- Octahedroid (Henry Parker Manning)
- Tes (Jonathan Bowers: for tesseract)
- [Four-dimensional] measure polytope
- [Four-dimensional regular] orthotope
- Cubic prism
- Cubic hyperprism
- Square duoprism
- Square double prism
- Square hyperprism
Schläfli symbols: {4,3,3}, also
t0{4,3,3} or
t3{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 t0,1,2,3{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)----o-----o
4
- Tesseractihexadecachoron [11]
Alternative names:
- Rectified tesseract (Norman W. Johnson)
- Rit (Jonathan Bowers: for rectified tesseract)
- Rectified [four-dimensional] hypercube
- Rectified 8-cell
- Rectified octachoron
- Rectified [four-dimensional] measure polytope
- Rectified [four-dimensional regular] orthotope
- Runcic tesseract (Norman W. Johnson)
- Runcic [four-dimensional] hypercube
- Runcic 8-cell
- Runcic octachoron
- Runcic [four-dimensional] measure polytope
- Runcic [four-dimensional regular] orthotope
- Ambotesseract (Neil Sloane & John Horton Conway)
Schläfli symbols: r{4,3,3}, also
t1{4,3,3},
t2{3,3,4},
h3{4,3,3}, or
t2{3,31,1}
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 equilateral-triangular prism: bases 2
equilateral triangles, edge length 1; lateral faces 3 rectangles,
edge lengths 1, sqrt(2)
-
o-----o----(o)----o
4
- Icositetrachoron [as “rectified
16-cell”; not counted, duplicate of 22]
-
o-----o-----o----(o)
4
- Hexadecachoron [12]
Alternative names:
- 16-cell (most frequently used)
- Hexadecahedroid or hexacaidecahedroid
- Hexadekahedroid or 16-hedroid (Henry Parker Manning)
- Hex (Jonathan Bowers: for hexadecachoron)
- [Four-dimensional] cross polytope
- [Four-dimensional regular] orthoplex
- [Four-dimensional] hemicube
- Half tesseract or hemitesseract or alternated tesseract or
demitesseract
- Half [four-dimensional] hypercube or hemihypercube or
alternated hypercube or demihypercube
- Half 8-cell or half octachoron
- Half [four-dimensional] measure polytope
- Half [four-dimensional regular] orthotope
- Tetrahedral antiprism
Schläfli symbols: {3,3,4}, also h{4,3,3},
t0{3,3,4},
t3{4,3,3}, or
{3,31,1}
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)----o-----o
4
- Truncated tesseract [13]
Alternative names:
- Truncated [four-dimensional] hypercube
- Truncated 8-cell
- Tat (Jonathan Bowers: for truncated tesseract)
- Truncated octachoron
- Truncated [four-dimensional] measure polytope
- Truncated [four-dimensional regular] orthotope
Schläfli symbols: t{4,3,3}, also
t0,1{4,3,3} or
t2,3{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:
- Equilateral-triangular 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 [four-dimensional] hypercube
- Cantellated 8-cell
- Cantellated octachoron
- Srit (Jonathan Bowers: for small rhombated
tesseract)
- Cantellated [four-dimensional] measure polytope
- Cantellated [four-dimensional regular] orthotope
Schläfli symbols:
t0,2{4,3,3} or
t1,3{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)----o-----o----(o)
4
- [Small] diprismatotesseractihexadecachoron [15]
Alternative names:
- Runcinated tesseract (Norman W. Johnson)
- Runcinated [four-dimensional] hypercube
- Runcinated 8-cell
- Runcinated octachoron
- Sidpith (Jonathan Bowers: for small
diprismatotesseractihexadecachoron)
- Runcinated [four-dimensional] measure polytope
- Runcinated [four-dimensional regular] orthotope
- Runcinated 16-cell (Norman W. Johnson)
- Runcinated hexadecachoron
- Runcinated [four-dimensional] cross polytope
- Runcinated [four-dimensional regular] orthoplex
Schläfli symbols:
t0,3{4,3,3} or
t0,3{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:
- Equilateral-triangular 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
- Truncated-octahedral tesseractihexadecachoron [16]
Alternative names:
- Bitruncated tesseract (Norman W. Johnson)
- Bitruncated [four-dimensional] hypercube
- Bitruncated 8-cell
- Bitruncated octachoron
- Bitruncated [four-dimensional] measure polytope
- Bitruncated [four-dimensional regular] orthotope
- Bitruncated 16-cell (Norman W. Johnson)
- Bitruncated hexadecachoron
- Tah (Jonathan Bowers: for
tesseractihexadecachoron)
- Bitruncated [four-dimensional] cross polytope
- Bitruncated [four-dimensional regular] orthoplex
- Runcicantic tesseract (Norman W. Johnson)
- Runcicantic [four-dimensional] hypercube
- Runcicantic 8-cell
- Runcicantic octachoron
- Runcicantic [four-dimensional] measure polytope
- Runcicantic [four-dimensional regular] orthotope
- Runcicantic 16-cell (Norman W. Johnson)
- Runcicantic hexadecachoron
- Runcicantic [four-dimensional] cross polytope
- Runcicantic [four-dimensional regular] orthoplex
Schläfli symbols: 2t{4,3,3} or 2t{3,3,4}, also
t1,2{4,3,3},
t1,2{3,3,4},
h2,3{4,3,3}, or
t1,2{3,31,1}
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
16-cell”; not counted, duplicate of 23]
-
o-----o----(o)---(o)
4
- Truncated hexadecachoron [17]
Alternative names:
- Truncated 16-cell
- Thex (Jonathan Bowers: for truncated
hexadecachoron)
- Truncated [four-dimensional] cross polytope
- Truncated [four-dimensional regular] orthoplex
- Cantic tesseract (Norman W. Johnson)
- Cantic [four-dimensional] hypercube
- Cantic 8-cell
- Cantic octachoron
- Cantic [four-dimensional] measure polytope
- Cantic [four-dimensional regular] orthotope
Schläfli symbols: t{3,3,4}, also
t2,3{4,3,3},
t0,1{3,3,4},
h2{4,3,3}, or
t0,1{3,31,1}
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 [four-dimensional] hypercube
- Cantitruncated 8-cell
- Cantitruncated octachoron
- Grit (Jonathan Bowers: for great rhombated
tesseract)
- Cantitruncated [four-dimensional] measure polytope
- Cantitruncated [four-dimensional regular] orthotope
Schläfli symbols:
t0,1,2{4,3,3} or
t1,2,3{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
- Truncated-cubic diprismatotesseractihexadecachoron
[19]
Alternative names:
- Runcitruncated tesseract (Norman W. Johnson)
- Runcitruncated [four-dimensional] hypercube
- Runcitruncated 8-cell
- Runcitruncated octachoron
- Proh (Jonathan Bowers: for prismatorhombated
hexadecachoron)
- Runcitruncated [four-dimensional] measure polytope
- Runcitruncated [four-dimensional regular] orthotope
Schläfli symbols:
t0,1,3{4,3,3} or
t0,2,3{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 16-cell (Norman W. Johnson)
- Runcitruncated hexadecachoron
- Prit (Jonathan Bowers: for prismatorhombated
tesseract)
- Runcitruncated [four-dimensional] cross polytope
- Runcitruncated [four-dimensional regular] orthoplex
Schläfli symbols:
t0,2,3{4,3,3} or
t0,1,3{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
16-cell”; 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 [four-dimensional] hypercube
- Omnitruncated 8-cell
- Omnitruncated octachoron
- Omnitruncated [four-dimensional] measure polytope
- Omnitruncated [four-dimensional regular] orthotope
- Omnitruncated 16-cell (Norman W. Johnson)
- Omnitruncated hexadecachoron
- Omnitruncated [four-dimensional] cross polytope
- Omnitruncated [four-dimensional regular] orthoplex
Schläfli symbols:
t0,1,2,3{4,3,3} or
t0,1,2,3{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
-
( )----o-----o-----o
4
- Hexadecachoron [as “half tesseract”
h{4,3,3}; not counted, duplicate of 12]
-
( )----o----(o)----o
4
- Truncated hexadecachoron [as “cantic
tesseract” h2{4,3,3}; not counted,
duplicate of 17]
-
( )----o-----o----(o)
4
- Tesseractihexadecachoron [as “runcic
tesseract” h3{4,3,3}; not counted,
duplicate of 11]
-
( )----o----(o)---(o)
4
- Truncated-octahedral tesseractihexadecachoron [as
“runcicantic tesseract”
h2,3{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
(5-cell): polychora #1–9
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