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
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:Triangular Prism
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
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
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
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
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
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
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
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
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
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
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