6. Convex uniform prismatic polychora


(o)   (o)   (o)   (o)
Tesseract [as “omnitruncated digonal-dihedral dichoron” t0,1,2,3{2,2,2}; not counted, duplicate of 10]

( )   ( )   ( )   ( )
Hexadecachoron [as “snub digonal-dihedral dichoron” s{2,2,2}; not counted, duplicate of 12]

( )   ( )   ( )   (o)
Tetrahedral prism [as “digonal antiprismatic prism” s{2}h{ }x{ } or sr{2,2}x{ }; not counted, duplicate of 48]

( )---( )   ( )   (o)
Octahedral prism [as “triangular antiprismatic prism” s{3}h{ }x{ }, sr{2,3}x{ }, or sr{3,2}x{ }; not counted, duplicate of 51]

( )---( )   ( )   (o)
p
p-gonal antiprismatic prism [not counted, infinite family (p>3)]
Alternative names:
p-gonal antiprismatic hyperprism

Symmetry group: [2p,2+]x[ ], the augmented dyadic skew 2p-gonal group, of order 8p

Schläfli symbols: s{p}h{ }x{ }, also sr{2,p}x{ } or sr{p,2}x{ }

Elements:
Cells: 2 p-gonal antiprisms, 2 p-gonal prisms, 2p triangular prisms
Faces: 4p triangles (all joining p-gonal antiprisms to triangular prisms), 4p squares (2p joining p-gonal prisms to triangular prisms, 2p joining triangular prisms to triangular prisms), 4 p-gons (joining p-gonal antiprisms to p-gonal prisms)
Edges: 10p
Vertices: 4p

Vertex figure:Trapezoidal Pyramid
Trapezoidal pyramid: base a trapezoid, edge lengths 2cos(pi/p), 1, 1, 1; all 4 lateral edges sqrt(2)


 o----( )   ( )   (o)
2p
p-gonal antiprismatic prism [not counted, infinite family (p>1, tetrahedral prism for p=2, octahedral prism for p=3): see above]

( )---( )   (o)   (o)
Triangular-square duoprism [not counted, member of infinite duoprism family: see below]

( )---( )   (o)   (o)
p
Square-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3, square duoprism = tesseract for p=4): see below]

(o)----o    (o)   (o)
Triangular-square duoprism [not counted, member of infinite duoprism family]
Alternative names:
Triangular-square prism
Triangular-square double prism
Triangular-square hyperprism

Symmetry group: [3]x[4], the triangular-square duoprismatic group, of order 48 (direct product of triangular and square dihedral groups)

Schläfli symbols: {3}x{4}, also t{2,3}x{ }, {3}xt{2}, or {3}x{ }x{ }

Elements:
Cells: 4 triangular prisms, 3 cubes
Faces: 4 triangles (all joining triangular prisms to triangular prisms), 15 squares (3 joining cubes to cubes, 12 joining triangular prisms to cubes)
Edges: 24
Vertices: 12

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 1, sqrt(2), all 4 lateral edges length sqrt(2), making 2 triangles equilateral, the other 2 isosceles


(o)---(o)   (o)   (o)
Square-hexagonal duoprism [not counted, member of infinite duoprism family]
Alternative names:
Square-hexagonal prism
Square-hexagonal double prism
Square-hexagonal hyperprism

Symmetry group: [4]x[6], the square-hexagonal duoprismatic group, of order 96 (direct product of square and hexagonal dihedral groups)

Schläfli symbols: {4}x{6}, also t{2,6}x{ }, {4}xt{3}, {4}xt0,1{3}, t{3}x{ }x{ }, t0,1{3}x{ }x{ }, {6}xt{2}, t{3}xt{2}, or {6}x{ }x{ }

Elements:
Cells: 4 hexagonal prisms, 6 cubes
Faces: 30 squares (6 joining cubes to cubes, 24 joining cubes to hexagonal prisms), 4 hexagons (all joining hexagonal prisms to hexagonal prisms)
Edges: 48
Vertices: 24

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths sqrt(2), sqrt(3), all 4 lateral edges length sqrt(2), making 2 triangles equilateral, the other 2 isosceles


(o)----o    (o)   (o)
p
Square-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3, square duoprism = tesseract for p=4)]
Alternative names:
Square-p-gonal prism
Square-p-gonal double prism
Square-p-gonal hyperprism

Symmetry group: [4,3,3] or [3,3,4], the dyadic hexadecachoric group, of order 384 (if p=4); [4]x[p], the square-p-gonal duoprismatic group, of order 16p (direct product of square and p-gonal dihedral groups, if p>4)

Schläfli symbols: {4}x{p}, also t{2,p}x{ }, {p}x{ }x{ }, or t{2}x{p}

Elements:
Cells: 4 p-gonal prisms (cubes if p=4), p cubes
Faces: 5p squares (p joining cubes to cubes, 4p joining cubes to p-gonal prisms), 4 p-gons (all joining p-gonal prisms to p-gonal prisms: extra squares if p=4)
Edges: 8p
Vertices: 4p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths sqrt(2), 2cos(pi/p); all 4 lateral edges length sqrt(2), making 2 triangles equilateral, the other 2 isosceles; regular tetrahedron, edge length sqrt(2), if p=4


(o)---(o)   (o)   (o)
p
Square-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
Square-2p-gonal prism
Square-2p-gonal double prism
Square-2p-gonal hyperprism

Symmetry group: [4]x[2p], the square-2p-gonal duoprismatic group, of order 32p (direct product of square and 2p-gonal dihedral groups)

Schläfli symbols: {4}x{2p}, also t{2,2p}x{ }, {4}xt{p}, {4}xt0,1{p}, {2p}x{ }x{ }, t{p}x{ }x{ }, t0,1{p}x{ }x{ }, t{2}xt{2p}, or t{2}xt{p}

Elements:
Cells: 4 2p-gonal prisms, 2p cubes
Faces: 10p squares (2p joining cubes to cubes, 8p joining cubes to 2p-gonal prisms), 4 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 16p
Vertices: 8p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths sqrt(2), 2cos(pi/2p), all 4 lateral edges length sqrt(2), making 2 triangles equilateral, the other 2 isosceles


(o)----o    (o)----o
Triangular duoprism [not counted, member of infinite duoprism family]
Alternative names:
Triangular double prism
Triangular hyperprism

Symmetry group: [[3]x[3]], the triangular duoprismatic group, of order 72 (direct product of two triangular dihedral groups and an inversion)

Schläfli symbol: {3}x{3}

Elements:
Cells: 6 triangular prisms
Faces: 6 triangles (all joining triangular prisms to triangular prisms), 9 squares (all joining triangular prisms to triangular prisms)
Edges: 18
Vertices: 9

Vertex figure:Tetragonal Disphenoid
Tetragonal disphenoid: tetrahedron with 2 opposite edges length 1, all 4 lateral edges length sqrt(2), making all 4 faces congruent isosceles triangles


(o)---(o)   (o)----o
Triangular-hexagonal duoprism [not counted, member of infinite duoprism family]
Alternative names:
Triangular-hexagonal prism
Triangular-hexagonal double prism
Triangular-hexagonal hyperprism

Symmetry group: [3]x[6], the triangular-hexagonal duoprismatic group, of order 72 (direct product of triangular and hexagonal dihedral groups)

Schläfli symbols: {3}x{6}, also {3}xt{3} or {3}xt0,1{3}

Elements:
Cells: 3 hexagonal prisms, 6 triangular prisms
Faces: 6 triangles (all joining triangular prisms to triangular prisms), 18 squares (all joining triangular prisms to hexagonal prisms), 3 hexagons (all joining hexagonal prisms to hexagonal prisms)
Edges: 36
Vertices: 18

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 1, sqrt(3), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles


(o)---(o)   (o)---(o)
Hexagonal duoprism [not counted, member of infinite duoprism family]
Alternative names:
Hexagonal double prism
Hexagonal hyperprism

Symmetry group: [[6]x[6]], the hexagonal duoprismatic group, of order 288 (direct product of two hexagonal dihedral groups and an inversion)

Schläfli symbols: {6}x{6}, also {6}xt{3}, {6}xt0,1{3}, t{3}xt{3}, t{3}xt0,1{3}, or t0,1{3}xt0,1{3}

Elements:
Cells: 12 hexagonal prisms
Faces: 36 squares (all joining hexagonal prisms to hexagonal prisms), 12 hexagons (all joining hexagonal prisms to hexagonal prisms)
Edges: 72
Vertices: 36

Vertex figure:Tetragonal Disphenoid
Tetragonal disphenoid: tetrahedron with 2 opposite edges length sqrt(3), all 4 lateral edges length sqrt(2), making all 4 faces congruent isosceles triangles


(o)----o    (o)----o
p
Triangular-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
Triangular-p-gonal prism
Triangular-p-gonal double prism
Triangular-p-gonal hyperprism

Symmetry group: [p]x[3], the triangular-p-gonal duoprismatic group, of order 12p (direct product of triangular and p-gonal dihedral groups)

Schläfli symbol: {p}x{3}

Elements:
Cells: 3 p-gonal prisms (cubes if p=4), p triangular prisms
Faces: p triangles (all joining triangular prisms to triangular prisms), 3p squares (all joining triangular prisms to p-gonal prisms), 3 p-gons (all joining p-gonal prisms to p-gonal prisms: extra squares if p=4)
Edges: 6p
Vertices: 3p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 1, 2cos(pi/p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles; 2 triangles equilateral if p=4


(o)----o    (o)---(o)
p
Hexagonal-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
Hexagonal-p-gonal prism
Hexagonal-p-gonal double prism
Hexagonal-p-gonal hyperprism

Symmetry group: [[6]x[6]], the hexagonal duoprismatic group, of order 288 (direct product of two hexagonal dihedral groups and an inversion, if p=6); [6]x[p], the hexagonal-2p-gonal duoprismatic group, of order 24p (direct product of hexagonal and 2p-gonal dihedral groups, if p~=6)

Schläfli symbols: {6}x{p}, also t{3}x{p} or t0,1{3}x{p}

Elements:
Cells: 6 p-gonal prisms (cubes if p=4), p hexagonal prisms
Faces: 6p squares (all joining p-gonal prisms to hexagonal prisms), p hexagons (all joining hexagonal prisms to hexagonal prisms), 6 p-gons (all joining p-gonal prisms to p-gonal prisms: extra squares if p=4)
Edges: 12p
Vertices: 6p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths sqrt(3), 2cos(pi/p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles; not scalene, and all 4 faces congruent isosceles triangles (tetragonal disphenoid), when p=6


(o)---(o)   (o)----o
p
Triangular-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
Triangular-2p-gonal prism
Triangular-2p-gonal double prism
Triangular-2p-gonal hyperprism

Symmetry group: [3]x[2p], the square-2p-gonal duoprismatic group, of order 24p (direct product of triangular and 2p-gonal dihedral groups)

Schläfli symbols: {3}x{2p}, also {3}xt{p} or {3}xt0,1{p}

Elements:
Cells: 3 2p-gonal prisms, 2p triangular prisms
Faces: 2p triangles (all joining triangular prisms to triangular prisms), 6p squares (all joining triangular prisms to 2p-gonal prisms), 3 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 12p
Vertices: 6p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 1, 2cos(pi/2p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles


(o)---(o)   (o)---(o)
p
Hexagonal-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
Hexagonal-2p-gonal prism
Hexagonal-2p-gonal double prism
Hexagonal-2p-gonal hyperprism

Symmetry group: [2p]x[6], the hexagonal-2p-gonal duoprismatic group, of order 48p (direct product of hexagonal and 2p-gonal dihedral groups)

Schläfli symbols: {2p}x{6}, also {2p}xt{3}, {2p}xt0,1{3}, t{p}x{6}, t{p}xt{3}, t{p}xt0,1{3}, t0,1{p}x{6}, t0,1{p}xt{3}, or t0,1{p}xt0,1{3}

Elements:
Cells: 6 2p-gonal prisms, 2p hexagonal prisms
Faces: 12p squares (all joining hexagonal prisms to 2p-gonal prisms), 2p hexagons (all joining hexagonal prisms to hexagonal prisms), 6 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 24p
Vertices: 12p

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths sqrt(3), 2cos(pi/2p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles


(o)----o    (o)----o
    p           p
p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3, square duoprism = tesseract if p=4)]
Alternative names:
p-gonal double prism
p-gonal hyperprism

Symmetry group: [4,3,3] or [3,3,4], the dyadic hexadecachoric group, of order 384 (if p=4); [[p]x[p]], the p-gonal duoprismatic group, of order 8p2 (direct product of two p-gonal dihedral groups and an inversion, if p>4)

Schläfli symbol: {p}x{p}

Elements:
Cells: 2p p-gonal prisms (cubes if p=4)
Faces: p2 squares (all joining p-gonal prisms to p-gonal prisms), 2p p-gons (all joining p-gonal prisms to p-gonal prisms: extra squares if p=4)
Edges: 2p2
Vertices: p2

Vertex figure:Tetragonal Disphenoid
Tetragonal disphenoid: tetrahedron with 2 opposite edges length 2cos(pi/p), all 4 lateral edges length sqrt(2), making all 4 faces congruent isosceles triangles; regular tetrahedron, edge length sqrt(2), if p=4


(o)---(o)   (o)----o
    p           p
p-gonal-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3, square-octagonal duoprism if p=4)]
Alternative names:
p-gonal-2p-gonal prism
p-gonal-2p-gonal double prism
p-gonal-2p-gonal hyperprism

Symmetry group: [2p]x[p], the p-gonal-2p-gonal duoprismatic group, of order 8p2 (direct product of p-gonal and 2p-gonal dihedral groups)

Schläfli symbols: {2p}x{p}, also t{p}x{p} or t0,1{p}x{p}

Elements:
Cells: 2p p-gonal prisms (cubes if p=4), p 2p-gonal prisms
Faces: 2p2 squares (all joining p-gonal prisms to 2p-gonal prisms), 2p p-gons (all joining p-gonal prisms to p-gonal prisms: extra squares if p=4), p 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 4p2
Vertices: 2p2

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 2cos(pi/p), 2cos(pi/2p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles; 2 triangles equilateral if p=4


(o)---(o)   (o)---(o)
    p           p
2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
2p-gonal double prism
2p-gonal hyperprism

Symmetry group: [[2p]x[2p]], the 2p-gonal duoprismatic group, of order 32p2 (direct product of two 2p-gonal dihedral groups and an inversion)

Schläfli symbols: {2p}x{2p}, also {2p}xt{p}, {2p}xt0,1{p}, t{p}x{2p}, t{p}xt{p}, t{p}xt0,1{p}, t0,1{p}x{2p}, t0,1{p}xt{p}, or t0,1{p}xt0,1{p}

Elements:
Cells: 4p 2p-gonal prisms
Faces: 4p2 squares (all joining 2p-gonal prisms to 2p-gonal prisms), 4p 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 8p2
Vertices: 4p2

Vertex figure:Tetragonal Disphenoid
Tetragonal disphenoid: tetrahedron with 2 opposite edges length 2cos(pi/2p), all 4 lateral edges length sqrt(2), making all 4 faces congruent isosceles triangles


(o)----o    (o)----o
    p           q
q-gonal-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>q>3)]
Alternative names:
q-gonal-p-gonal prism
q-gonal-p-gonal double prism
q-gonal-p-gonal hyperprism

Symmetry group: [p]x[q], the p-gonal-q-gonal duoprismatic group, of order 4pq (direct product of p-gonal and q-gonal dihedral groups)

Schläfli symbols: {p}x{q}, also t0{p}x{q}, {p}xt0{q}, or t0{p}xt0{q}

Elements:
Cells: p q-gonal prisms (cubes if q=4), q p-gonal prisms
Faces: pq squares (all joining q-gonal prisms to p-gonal prisms), p q-gons (all joining q-gonal prisms to q-gonal prisms: extra squares if q=4), q p-gons (all joining p-gonal prisms to p-gonal prisms)
Edges: 2pq
Vertices: pq

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 2cos(pi/p), 2cos(pi/q), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles; 2 triangles equilateral if p=4


(o)---(o)   (o)----o
    p           q
q-gonal-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>q>3)]
Alternative names:
q-gonal-2p-gonal prism
q-gonal-2p-gonal double prism
q-gonal-2p-gonal hyperprism

Symmetry group: [2p]x[q], the 2p-gonal-q-gonal duoprismatic group, of order 8pq (direct product of 2p-gonal and q-gonal dihedral groups)

Schläfli symbols: {2p}x{q}, also t{p}x{q} or t0,1{p}x{q}

Elements:
Cells: 2p q-gonal prisms (cubes if q=4), q 2p-gonal prisms
Faces: 2pq squares (all joining q-gonal prisms to 2p-gonal prisms), 2p q-gons (all joining q-gonal prisms to q-gonal prisms: extra squares if q=4), q 2p-gons (all joining 2p-gonal prisms to 2p-gonal prisms)
Edges: 4pq
Vertices: 2pq

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 2cos(pi/q), 2cos(pi/2p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles


(o)----o    (o)---(o)
    p           q
2q-gonal-p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>q>3)]
Alternative names:
2q-gonal-p-gonal prism
2q-gonal-p-gonal double prism
2q-gonal-p-gonal hyperprism

Symmetry group: [[p]x[p]], the p-gonal duoprismatic group, of order 8p2 (direct product of 2 p-gonal dihedral groups and an inversion, if p=2q); [p]x[2q], the p-gonal-q-gonal duoprismatic group, of order 8pq (direct product of p-gonal and q-gonal dihedral groups, if p~=2q)

Schläfli symbols: {p}x{2q}, also {p}xt{q} or {p}xt0,1{q}

Elements:
Cells: p 2q-gonal prisms, 2q p-gonal prisms
Faces: 2pq squares (all joining p-gonal prisms to 2q-gonal prisms), p 2q-gons (all joining 2q-gonal prisms to 2q-gonal prisms), 2q p-gons (all joining p-gonal prisms to p-gonal prisms)
Edges: 4pq
Vertices: 2pq

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 2cos(pi/2q), 2cos(pi/p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles; tetragonal disphenoid if p=2q (all 4 triangles are congruent isosceles)


(o)---(o)   (o)---(o)
    p           q
2q-gonal-2p-gonal duoprism [not counted, infinite subfamily of infinite duoprism family (p>q>3)]
Alternative names:
2q-gonal-2p-gonal prism
2q-gonal-2p-gonal double prism
2q-gonal-2p-gonal hyperprism

Symmetry group: [2p]x[2q], the 2q-gonal-2p-gonal duoprismatic group, of order 16pq (direct product of 2p-gonal and 2q-gonal dihedral groups)

Schläfli symbols: {2p}x{2q}, also {2p}xt{q}, {2p}xt0,1{q}, t{p}x{2q}, t{p}xt{q}, t{p}xt0,1{q}, t0,1{p}x{2q}, t0,1{p}xt{q}, or t0,1{p}xt0,1{q}

Elements:
Cells: 2p 2q-gonal prisms, 2q 2p-gonal prisms
Faces: 4pq squares (all joining 2q-gonal prisms to 2p-gonal prisms), 2p 2q-gons (all joining 2q-gonal prisms), 2q 2p-gons (all joining 2p-gonal prisms)
Edges: 8pq
Vertices: 4pq

Vertex figure:Digonal Disphenoid
Digonal disphenoid: tetrahedron with 2 opposite edges lengths 2cos(pi/2q), 2cos(pi/2p), all 4 lateral edges length sqrt(2), making the faces 2 kinds of isosceles triangles


(o)----o-----o    (o)
Tetrahedral prism [48]
Alternative names:
Tetrahedral dyadic prism (Norman W. Johnson)
Tepe (Jonathan Bowers: for tetrahedral prism)
Tetrahedral hyperprism
Digonal antiprismatic prism
Digonal antiprismatic hyperprism

Symmetry group: [3,3]x[ ], the dyadic tetrahedral-prismatic group, of order 48

Schläfli symbols: {3,3}x{ }, also s{2,2}x{ } or s{2}h{ }x{ }

Elements:
Cells: 2 tetrahedra, 4 triangular prisms
Faces: 8 triangles (all joining tetrahedra to triangular prisms), 6 squares (all joining triangular prisms to triangular prisms)
Edges: 16
Vertices: 8

Vertex figure:Equilateral-triangular Pyramid
Equilateral-triangular pyramid (or “triangular spike”): base an equilateral triangle, edge length 1; all 3 lateral edges length sqrt(2)


 o----(o)----o    (o)
Octahedral prism [as “rectified tetrahedral prism” r{3,3}x{ }; not counted, duplicate of 51]

(o)----o----(o)   (o)
Cuboctahedral prism [as “rhombioctahedral prism” rr{3,3}x{ }; not counted, duplicate of 50]

(o)---(o)----o    (o)
Truncated-tetrahedral prism [49]
Alternative names:
Truncated-tetrahedral dyadic prism (Norman W. Johnson)
Tuttip (Jonathan Bowers: for truncated-tetrahedral prism)
Truncated-tetrahedral hyperprism

Symmetry group: [3,3]x[ ], the dyadic tetrahedral-prismatic group, of order 48

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

Elements:
Cells: 2 truncated tetrahedra, 4 triangular prisms, 4 hexagonal prisms
Faces: 8 triangles (all joining truncated tetrahedra to triangular prisms), 18 squares (6 joining hexagonal prisms to hexagonal prisms, 12 joining triangular prisms to hexagonal prisms), 8 hexagons (all joining truncated tetrahedra to hexagonal prisms)
Edges: 48
Vertices: 24

Vertex figure:Isosceles-triangular Pyramid
Isosceles-triangular pyramid: base an isosceles triangle, edge lengths 1, sqrt(3), sqrt(3); all 3 lateral edges length sqrt(2)


(o)---(o)---(o)   (o)
Truncated-octahedral prism [as “great-octahedral prism” tr{3,3}x{ }; not counted, duplicate of 54]

( )---( )---( )   (o)
Icosahedral prism [as “snub-tetrahedral prism” sr{3,3}x{ }; not counted, duplicate of 59]

(o)----o-----o    (o)
4
Tesseract [as “cubic prism” {4,3}x{ }, {4}x{ }x{ }, or { }x{ }x{ }x{ }; not counted, duplicate of 10]

 o----(o)----o    (o)
4
Cuboctahedral prism [50]
Alternative names:
Cuboctahedral dyadic prism (Norman W. Johnson)
Cope (Jonathan Bowers: for cuboctahedral prism)
Cuboctahedral hyperprism
Rhombioctahedral prism
Rhombioctahedral hyperprism

Symmetry group: The dyadic octahedral-prismatic group [3,4]x[ ] or [4,3]x[ ], of order 96

Schläfli symbols: r{3,4}x{ } or r{4,3}x{ }, also rr{3,3}x{ }

Elements:
Cells: 2 cuboctahedra, 8 triangular prisms, 6 cubes
Faces: 16 triangles (all joining cuboctahedra to triangular prisms), 36 squares (12 joining cuboctahedra to cubes, 24 joining cubes to triangular prisms)
Edges: 60
Vertices: 24

Vertex figure:Rectangular Pyramid
Rectangular pyramid: base a rectangle, edges length 1, sqrt(2); all 4 lateral edges length sqrt(2), making 2 triangles equilateral, the other 2 isosceles


 o-----o----(o)   (o)
4
Octahedral prism [51]
Alternative names:
Octahedral dyadic prism (Norman W. Johnson)
Ope (Jonathan Bowers: for octahedral prism)
Octahedral hyperprism
Triangular antiprismatic prism

Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic octahedral-prismatic group, of order 96

Schläfli symbols: {3,4}x{ }, also t0{3,4}x{ }, t2{4,3}x{ }, r{3,3}x{ }, sr{2,3}x{ }, sr{3,2}x{ }, or s{3}h{ }x{ }

Elements:
Cells: 2 octahedra, 8 triangular prisms
Faces: 16 triangles (all joining octahedra to triangular prisms), 12 squares (all joining triangular prisms to triangular prisms)
Edges: 30
Vertices: 12

Vertex figure:Square Pyramid
Square pyramid: base a square, edge length 1; all 4 lateral edges length sqrt(2)


(o)---(o)----o    (o)
4
Truncated-cubic prism [52]
Alternative names:
Truncated-cubic dyadic prism (Norman W. Johnson)
Ticcup (Jonathan Bowers: for truncated-cubic prism)
Truncated-cubic hyperprism

Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic octahedral-prismatic group, of order 96

Schläfli symbols: t{4,3}x{ }, also t0,1{4,3}x{ }, t1,2{3,4}x{ }

Elements:
Cells: 2 truncated cubes, 6 octagonal prisms, 8 triangular prisms
Faces: 16 triangles (all joining truncated cubes to triangular prisms), 36 squares (12 joining octagonal prisms to octagonal prisms, 24 joining triangular prisms to octagonal prisms), 12 octagons (all joining truncated cubes to octagonal prisms)
Edges: 96
Vertices: 48

Vertex figure:Isosceles-triangular Pyramid
Isosceles-triangular pyramid: base an isosceles triangle, edge lengths 1, sqrt(2+sqrt(2)), sqrt(2+sqrt(2)); all 3 lateral edges length sqrt(2)


(o)----o----(o)   (o)
4
[Small-]rhombicuboctahedral prism [53]
Alternative names:
[Small-]rhombicuboctahedral dyadic prism (Norman W. Johnson)
Sircope (Jonathan Bowers: for small-rhombicuboctahedral prism)
[Small-]rhombicuboctahedral hyperprism

Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic octahedral-prismatic group, of order 96

Schläfli symbols: rr{3,4}x{ } or rr{4,3}x{ }

Elements:
Cells: 2 rhombicuboctahedra, 8 triangular prisms, 18 cubes
Faces: 16 triangles (all joining rhombicuboctahedra to triangular prisms), 84 squares (36 joining rhombicuboctahedra to cubes, 24 joining cubes to cubes, 24 joining triangular prisms to cubes)
Edges: 120
Vertices: 48

Vertex figure:Trapezoidal Pyramid
Trapezoidal pyramid: base a trapezoid with edges length 1, sqrt(2), sqrt(2), sqrt(2); all 4 lateral edges length sqrt(2), making 3 of the lateral triangles equilateral, the other isosceles


 o----(o)---(o)   (o)
4
Truncated-octahedral prism [54]
Alternative names:
Truncated-octahedral dyadic prism (Norman W. Johnson)
Tope (Jonathan Bowers: for truncated-octahedral prism)
Truncated-octahedral hyperprism
Great-octahedral prism
Great-octahedral hyperprism

Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic octahedral-prismatic group, of order 96

Schläfli symbols: t{3,4}x{ }, also t0,1{3,4}x{ }, t1,2{4,3}x{ }, tr{3,3}x{ }, and t0,1,2{3,3}x{ }

Elements:
Cells: 2 truncated octahedra, 6 cubes, 8 hexagonal prisms
Faces: 48 squares (12 joining truncated octahedra to cubes, 12 joining hexagonal prisms to hexagonal prisms, 24 joining cubes to hexagonal prisms), 16 hexagons (all joining truncated octahedra to hexagonal prisms)
Edges: 96
Vertices: 48

Vertex figure:Isosceles-triangular Pyramid
Isosceles-triangular pyramid: base an isosceles triangle, edge lengths sqrt(2), sqrt(3), sqrt(3); all 3 lateral edges length sqrt(2), making one lateral face an equilateral triangle, the other two isosceles


(o)---(o)---(o)   (o)
4
Truncated-cuboctahedral prism [55]
Alternative names:
Truncated-cuboctahedral dyadic prism (Norman W. Johnson)
Gircope (Jonathan Bowers: for great-rhombicuboctahedral prism)
Truncated-cuboctahedral hyperprism
Great-rhombicuboctahedral prism
Great-rhombicuboctahedral hyperprism

Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic octahedral-prismatic group, of order 96

Schläfli symbols: tr{3,4}x{ } or tr{4,3}x{ }, also t0,1,2{4,3}x{ } or t0,1,2{3,4}x{ }

Elements:
Cells: 2 truncated cuboctahedra, 6 octagonal prisms, 8 hexagonal prisms, 12 cubes
Faces: 96 squares (24 joining truncated cuboctahedra to cubes, 24 joining cubes to hexagonal prisms, 24 joining cubes to octagonal prisms), 16 hexagons (all joining truncated cuboctahedra to hexagonal prisms), 12 octagons (all joining truncated cuboctahedra to octagonal prisms)
Edges: 192
Vertices: 96

Vertex figure:Scalene-triangular Pyramid
Chiral scalene-triangular pyramid: base a scalene triangle, edge lengths sqrt(2), sqrt(3), sqrt(2+sqrt(2)); all 3 lateral edges length sqrt(2), making one lateral face an equilateral triangle and the other two isosceles; dextro and laevo versions each occur at 48 vertices


( )---( )---( )   (o)
4
Snub-cuboctahedral prism [56]
Alternative names:
Snub-cuboctahedral dyadic prism (Norman W. Johnson)
Sniccup (Jonathan Bowers: for snub-cubic prism)
Snub-cuboctahedral hyperprism
Snub-cubic prism
Snub-cubic hyperprism

Symmetry group: [3,4]+x[ ] or [4,3]+x[ ], the direct octahedral-prismatic group, of order 48

Schläfli symbols: sr{4,3}x{ } or sr{3,4}x{ }

Elements:
Cells: 2 snub cuboctahedra, 6 cubes, 32 triangular prisms
Faces: 64 triangles (all joining snub cuboctahedra to triangular prisms), 72 squares (12 joining snub cuboctahedra to cubes, 24 joining cubes to triangular prisms, 36 joining triangular prisms to triangular prisms)
Edges: 144
Vertices: 48

Vertex figure:Pentagonal Pyramid
Pentagonal pyramid: base a slightly irregular pentagon, 4 edge lengths all 1, 5th edge length sqrt(2); all 5 lateral edges length sqrt(2), making one lateral triangle equilateral, the other 4 isosceles


(o)----o-----o    (o)
5
Dodecahedral prism [57]
Alternative names:
Dodecahedral dyadic prism (Norman W. Johnson)
Dope (Jonathan Bowers: for dodecahedral prism)
Dodecahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: {5,3}x{ }, also t0{5,3}x{ } or t2{3,5}x{ }

Elements:
Cells: 2 dodecahedra, 12 pentagonal prisms
Faces: 30 squares (all joining pentagonal prisms to pentagonal prisms), 24 pentagons (all joining dodecahedra to pentagonal prisms)
Edges: 80
Vertices: 40

Vertex figure:Equilateral-triangular
Pyramid
Equilateral-triangular pyramid: base an equilateral triangle, edge length tau; all 5 lateral edges length sqrt(2)


 o----(o)----o    (o)
5
Icosidodecahedral prism [58]
Alternative names:
Icosidodecahedral dyadic prism (Norman W. Johnson)
Iddip (Jonathan Bowers: for icosidodecahedral prism)
Icosidodecahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: r{3,5}x{ } or r{5,3}x{ }

Elements:
Cells: 2 icosidodecahedra, 20 triangular prisms, 12 pentagonal prisms
Faces: 40 triangles (all joining icosidodecahedra to triangular prisms), 60 squares (all joining pentagonal prisms to triangular prisms), 24 pentagons (all joining icosidodecahedra to pentagonal prisms)
Edges: 150
Vertices: 60

Vertex figure:Rectangular Pyramid
Rectangular pyramid: base a rectangle, edges length 1, tau; all 4 lateral edges length sqrt(2)


 o-----o----(o)   (o)
5
Icosahedral prism [59]
Alternative names:
Icosahedral dyadic prism (Norman W. Johnson)
Ipe (Jonathan Bowers: for icosahedral prism)
Icosahedral hyperprism
Snub-octahedral prism
Snub-octahedral hyperprism
Snub-tetrahedral prism
Snub-tetrahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: {3,5}x{ }, also t0{3,5}x{ }, t2{5,3}x{ }, or sr{3,3}x{ }

Elements:
Cells: 2 icosahedra, 20 triangular prisms
Faces: 40 triangles (all joining icosahedra to triangular prisms), 30 squares (all joining triangular prisms to triangular prisms)
Edges: 72
Vertices: 24

Vertex figure:Regular-pentagonal Pyramid
Regular-pentagonal pyramid: base a regular pentagon, edge length 1; all 5 lateral edges length sqrt(2)


(o)---(o)----o    (o)
5
Truncated-dodecahedral prism [60]
Alternative names:
Truncated-dodecahedral dyadic prism (Norman W. Johnson)
Tiddip (Jonathan Bowers: for truncated-dodecahedral prism)
Truncated-dodecahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: t{5,3}x{ }, also t0,1{5,3}x{ } or t1,2{3,5}x{ }

Elements:
Cells: 2 truncated dodecahedra, 12 decagonal prisms, 20 triangular prisms
Faces: 40 triangles (all joining truncated dodecahedra to triangular prisms), 90 squares (60 joining triangular prisms to decagonal prisms, 30 joining decagonal prisms to decagonal prisms), 24 decagons (all joining truncated dodecahedra to decagonal prisms)
Edges: 240
Vertices: 120

Vertex figure:Isosceles-triangular Pyramid
Isosceles-triangular pyramid: base an isosceles triangle, edge lengths 1, sqrt(2+tau), sqrt(2+tau); all 3 lateral edges length sqrt(2)


(o)----o----(o)   (o)
5
[Small-]rhombicosidodecahedral prism [61]
Alternative names:
[Small-]rhombicosidodecahedral dyadic prism (Norman W. Johnson)
Sriddip (Jonathan Bowers: for small-rhombicosidodecahedral prism)
[Small-]rhombicosidodecahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: rr{5,3}x{ } or rr{3,5}x{ }, also t1{5,3}x{ } or t1{3,5}x{ }

Elements:
Cells: 2 rhombicosidodecahedra, 12 pentagonal prisms, 20 triangular prisms, 30 cubes
Faces: 40 triangles (all joining rhombicosidodecahedra to triangular prisms), 180 squares (60 joining rhombicosidodecahedra to cubes, 60 joining triangular prisms to cubes, 60 joining pentagonal prisms to cubes), 24 pentagons (all joining rhombicosidodecahedra to pentagonal prisms)
Edges: 300
Vertices: 120

Vertex figure:Trapezoidal Pyramid
Trapezoidal pyramid: base a trapezoid with edges length 1, sqrt(2), tau, sqrt(2); all 4 lateral edges length sqrt(2), making 2 of the lateral triangles equilateral, the other two isosceles


 o----(o)---(o)   (o)
5
Truncated-icosahedral prism [62]
Alternative names:
Truncated-icosahedral dyadic prism (Norman W. Johnson)
Tipe (Jonathan Bowers: for truncated-icosahedral prism)
Truncated-icosahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: t{3,5}x{ }, also t0,1{3,5}x{ } or t1,2{5,3}x{ }

Elements:
Cells: 2 truncated icosahedra, 12 pentagonal prisms, 20 hexagonal prisms
Faces: 90 squares (60 joining pentagonal prisms to hexagonal prisms, 30 joining hexagonal prisms to hexagonal prisms), 24 pentagons (all joining truncated icosahedra to pentagonal prisms), 40 hexagons (all joining truncated icosahedra to hexagonal prisms)
Edges: 240
Vertices: 120

Vertex figure:Isosceles-triangular Pyramid
Isosceles-triangular pyramid: base an isosceles triangle, edge lengths tau, sqrt(3), sqrt(3); all 3 lateral edges length sqrt(2)


(o)---(o)---(o)   (o)
5
Truncated-icosidodecahedral prism [63]
Alternative names:
Truncated-icosidodecahedral dyadic prism (Norman W. Johnson)
Griddip (Jonathan Bowers: for great-rhombicosidodecahedral prism)
Truncated-icosidodecahedral hyperprism
Great-rhombicosidodecahedral prism
Great-rhombicosidodecahedral hyperprism

Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic icosahedral-prismatic group, of order 240

Schläfli symbols: tr{3,5}x{ } or tr{5,3}x{ }, also t0,1,2{5,3}x{ } or t0,1,2{3,5}x{ }

Elements:
Cells: 2 truncated icosidodecahedra, 12 decagonal prisms, 20 hexagonal prisms, 30 cubes
Faces: 240 squares (60 joining truncated icosidodecahedra to cubes, 60 joining cubes to hexagonal prisms, 60 joining cubes to decagonal prisms, 60 joining hexagonal prisms to decagonal prisms), 40 hexagons (all joining truncated icosidodecahedra to hexagonal prisms), 24 decagons (all joining truncated icosidodecahedra to decagonal prisms)
Edges: 480
Vertices: 240

Vertex figure:Scalene-triangular Pyramid

Chiral scalene-triangular pyramid: base a scalene triangle, edge lengths sqrt(2), sqrt(3), sqrt(2+tau); all 3 lateral edges length sqrt(2), making one lateral face an equilateral triangle and the other two isosceles; dextro and laevo versions each occur at 120 vertices

( )---( )---( )   (o)
5
Snub-icosidodecahedral prism [64]
Alternative names:
Snub-icosidodecahedral dyadic prism (Norman W. Johnson)
Sniddip (Jonathan Bowers: for snub-dodecahedral prism)
Snub-icosidodecahedral hyperprism
Snub-dodecahedral prism
Snub-dodecahedral hyperprism

Symmetry group: [3,5]+x[ ] or [5,3]+x[ ], the direct icosahedral-prismatic group, of order 120

Schläfli symbols: sr{3,5}x{ } or sr{5,3}x{ }

Elements:
Cells: 2 snub icosidodecahedra, 12 pentagonal prisms, 80 triangular prisms
Faces: 160 triangles (all joining snub icosidodecahedra to triangular prisms), 150 squares (60 joining pentagonal prisms to triangular prisms, 90 joining triangular prisms to triangular prisms), 24 pentagons (all joining snub icosidodecahedra to pentagonal prisms)
Edges: 360
Vertices: 120

Vertex figure:Pentagonal Pyramid
Pentagonal pyramid: base a slightly irregular pentagon, 4 edge lengths all 1, 5th edge length tau; all 5 lateral edges length sqrt(2)


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 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 7: Uniform polychora derived from glomeric tetrahedron B4: all duplicates of prior polychora