6. Convex uniform prismatic polychora
(o) (o) (o) (o)
 Tesseract [as “omnitruncated digonaldihedral
dichoron” t_{0,1,2,3}{2,2,2}; not
counted, duplicate of 10]

( ) ( ) ( ) ( )
 Hexadecachoron [as “snub digonaldihedral
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
 pgonal antiprismatic prism [not counted,
infinite family (p>3)]
Alternative names:
 pgonal antiprismatic hyperprism
Symmetry group:
[2p,2^{+}]x[ ], the augmented
dyadic skew 2pgonal group, of order 8p
Schläfli symbols: s{p}h{ }x{ }, also
sr{2,p}x{ } or sr{p,2}x{ }
Elements: Cells: 2 pgonal
antiprisms, 2 pgonal prisms, 2p triangular
prisms
 Faces: 4p triangles (all joining pgonal
antiprisms to triangular prisms), 4p squares (2p
joining pgonal prisms to triangular prisms, 2p
joining triangular prisms to triangular prisms), 4 pgons
(joining pgonal antiprisms to pgonal prisms)
 Edges: 10p
 Vertices: 4p
Vertex figure: Trapezoidal pyramid: base a
trapezoid, edge lengths 2cos(pi/p), 1, 1, 1;
all 4
lateral edges sqrt(2)

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

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

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

(o)o (o) (o)
 Triangularsquare
duoprism [not counted, member of infinite duoprism family]
Alternative names:
 Triangularsquare prism
 Triangularsquare double prism
 Triangularsquare hyperprism
Symmetry group: [3]x[4], the triangularsquare
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: 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)
 Squarehexagonal duoprism [not counted, member of
infinite duoprism family]
Alternative names:
 Squarehexagonal prism
 Squarehexagonal double prism
 Squarehexagonal hyperprism
Symmetry group: [4]x[6], the squarehexagonal
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}xt_{0,1}{3}, t{3}x{ }x{ },
t_{0,1}{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: 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
 Squarepgonal duoprism [not counted, infinite
subfamily of infinite duoprism family (p>3, square
duoprism = tesseract for p=4)]
Alternative names:
 Squarepgonal prism
 Squarepgonal double prism
 Squarepgonal hyperprism
Symmetry group: [4,3,3] or [3,3,4], the dyadic
hexadecachoric group, of order 384 (if p=4);
[4]x[p], the squarepgonal duoprismatic
group, of order 16p (direct product of square and
pgonal 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 pgonal prisms (cubes if p=4),
p cubes
 Faces: 5p squares (p joining cubes to
cubes, 4p joining cubes to pgonal prisms), 4
pgons (all joining pgonal prisms to
pgonal prisms: extra squares if p=4)
 Edges: 8p
 Vertices: 4p
Vertex figure:
 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
 Square2pgonal duoprism [not counted, infinite
subfamily of infinite duoprism family (p>3)]
Alternative names:
 Square2pgonal prism
 Square2pgonal double prism
 Square2pgonal hyperprism
Symmetry group: [4]x[2p], the
square2pgonal duoprismatic group, of order
32p (direct product of square and 2pgonal dihedral
groups)
Schläfli symbols: {4}x{2p}, also
t{2,2p}x{ }, {4}xt{p},
{4}xt_{0,1}{p}, {2p}x{ }x{
}, t{p}x{ }x{ },
t_{0,1}{p}x{ }x{ },
t{2}xt{2p}, or t{2}xt{p}
Elements:
 Cells: 4 2pgonal prisms, 2p cubes
 Faces: 10p squares (2p joining cubes to
cubes, 8p joining cubes to 2pgonal prisms), 4
2pgons (all joining 2pgonal prisms to
2pgonal prisms)
 Edges: 16p
 Vertices: 8p
Vertex figure:
 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: 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
 Triangularhexagonal duoprism [not counted, member of
infinite duoprism family]
Alternative names:
 Triangularhexagonal prism
 Triangularhexagonal double prism
 Triangularhexagonal hyperprism
Symmetry group: [3]x[6], the triangularhexagonal
duoprismatic group, of order 72 (direct product of triangular
and hexagonal dihedral groups)
Schläfli symbols: {3}x{6}, also {3}xt{3} or
{3}xt_{0,1}{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: 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}xt_{0,1}{3}, t{3}xt{3},
t{3}xt_{0,1}{3}, or
t_{0,1}{3}xt_{0,1}{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: 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
 Triangularpgonal duoprism [not counted,
infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
 Triangularpgonal prism
 Triangularpgonal double prism
 Triangularpgonal hyperprism
Symmetry group: [p]x[3], the
triangularpgonal duoprismatic group, of order
12p (direct product of triangular and pgonal
dihedral groups)
Schläfli symbol: {p}x{3}
Elements:
 Cells: 3 pgonal 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 pgonal prisms), 3 pgons (all
joining pgonal prisms to pgonal prisms: extra
squares if p=4)
 Edges: 6p
 Vertices: 3p
Vertex figure:
 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
 Hexagonalpgonal duoprism [not counted,
infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
 Hexagonalpgonal prism
 Hexagonalpgonal double prism
 Hexagonalpgonal 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 hexagonal2pgonal duoprismatic
group, of order 24p (direct product of hexagonal and
2pgonal dihedral groups, if p~=6)
Schläfli symbols: {6}x{p}, also
t{3}x{p} or t_{0,1}{3}x{p}
Elements:
 Cells: 6 pgonal prisms (cubes if p=4),
p hexagonal prisms
 Faces: 6p squares (all joining pgonal
prisms to hexagonal prisms), p hexagons (all joining
hexagonal prisms to hexagonal prisms), 6 pgons (all
joining pgonal prisms to pgonal prisms: extra
squares if p=4)
 Edges: 12p
 Vertices: 6p
Vertex figure:
 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
 Triangular2pgonal duoprism [not counted,
infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
 Triangular2pgonal prism
 Triangular2pgonal double prism
 Triangular2pgonal hyperprism
Symmetry group: [3]x[2p], the
square2pgonal duoprismatic group, of order
24p (direct product of triangular and 2pgonal
dihedral groups)
Schläfli symbols: {3}x{2p}, also
{3}xt{p} or {3}xt_{0,1}{p}
Elements:
 Cells: 3 2pgonal prisms, 2p triangular
prisms
 Faces: 2p triangles (all joining triangular
prisms to triangular prisms), 6p squares (all joining
triangular prisms to 2pgonal prisms), 3 2pgons
(all joining 2pgonal prisms to 2pgonal prisms)
 Edges: 12p
 Vertices: 6p
Vertex figure:
 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
 Hexagonal2pgonal duoprism [not counted,
infinite subfamily of infinite duoprism family (p>3)]
Alternative names:
 Hexagonal2pgonal prism
 Hexagonal2pgonal double prism
 Hexagonal2pgonal hyperprism
Symmetry group: [2p]x[6], the
hexagonal2pgonal duoprismatic group, of order
48p (direct product of hexagonal and 2pgonal
dihedral groups)
Schläfli symbols: {2p}x{6}, also
{2p}xt{3}, {2p}xt_{0,1}{3},
t{p}x{6}, t{p}xt{3},
t{p}xt_{0,1}{3},
t_{0,1}{p}x{6},
t_{0,1}{p}xt{3}, or
t_{0,1}{p}xt0,1{3}
Elements:
 Cells: 6 2pgonal prisms, 2p hexagonal
prisms
 Faces: 12p squares (all joining hexagonal
prisms to 2pgonal prisms), 2p hexagons (all
joining hexagonal prisms to hexagonal prisms), 6 2pgons
(all joining 2pgonal prisms to 2pgonal prisms)
 Edges: 24p
 Vertices: 12p
Vertex figure:
 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
 pgonal duoprism [not counted, infinite
subfamily of infinite duoprism family (p>3, square
duoprism = tesseract if p=4)]
Alternative names:
 pgonal double prism
 pgonal hyperprism
Symmetry group: [4,3,3] or [3,3,4], the dyadic
hexadecachoric group, of order 384 (if p=4);
[[p]x[p]], the pgonal duoprismatic group,
of order 8p^{2} (direct product of
two pgonal dihedral groups and an inversion, if
p>4)
Schläfli symbol: {p}x{p}
Elements:
 Cells: 2p pgonal prisms (cubes if p=4)
 Faces: p^{2} squares
(all joining pgonal prisms to pgonal prisms),
2p pgons (all joining pgonal prisms to
pgonal prisms: extra squares if p=4)
 Edges: 2p^{2}
 Vertices: p^{2}
Vertex figure:
 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
 pgonal2pgonal duoprism [not counted,
infinite subfamily of infinite duoprism family (p>3,
squareoctagonal duoprism if p=4)]
Alternative names:
 pgonal2pgonal prism
 pgonal2pgonal double prism
 pgonal2pgonal hyperprism
Symmetry group: [2p]x[p], the
pgonal2pgonal duoprismatic group, of order
8p^{2} (direct product of
pgonal and 2pgonal dihedral groups)
Schläfli symbols: {2p}x{p}, also
t{p}x{p} or
t_{0,1}{p}x{p}
Elements:
 Cells: 2p pgonal prisms (cubes if p=4),
p 2pgonal prisms
 Faces: 2p^{2} squares
(all joining pgonal prisms to 2pgonal prisms),
2p pgons (all joining pgonal prisms to
pgonal prisms: extra squares if p=4), p
2pgons (all joining 2pgonal prisms to
2pgonal prisms)
 Edges: 4p^{2}
 Vertices: 2p^{2}
Vertex figure:
 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
 2pgonal duoprism [not counted, infinite
subfamily of infinite duoprism family (p>3)]
Alternative names:
 2pgonal double prism
 2pgonal hyperprism
Symmetry group: [[2p]x[2p]], the
2pgonal duoprismatic group, of order
32p^{2} (direct product of two
2pgonal dihedral groups and an inversion)
Schläfli symbols: {2p}x{2p}, also
{2p}xt{p},
{2p}xt_{0,1}{p},
t{p}x{2p}, t{p}xt{p},
t{p}xt_{0,1}{p},
t_{0,1}{p}x{2p},
t_{0,1}{p}xt{p}, or
t_{0,1}{p}xt_{0,1}{p}
Elements:
 Cells: 4p 2pgonal prisms
 Faces: 4p^{2} squares
(all joining 2pgonal prisms to 2pgonal prisms),
4p 2pgons (all joining 2pgonal prisms to
2pgonal prisms)
 Edges: 8p^{2}
 Vertices: 4p^{2}
Vertex figure:
 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
 qgonalpgonal duoprism [not counted,
infinite subfamily of infinite duoprism family
(p>q>3)]
Alternative names:
 qgonalpgonal prism
 qgonalpgonal double prism
 qgonalpgonal hyperprism
Symmetry group: [p]x[q], the
pgonalqgonal duoprismatic group, of order
4pq (direct product of pgonal and qgonal
dihedral groups)
Schläfli symbols: {p}x{q}, also
t_{0}{p}x{q},
{p}xt_{0}{q}, or
t_{0}{p}xt_{0}{q}
Elements:
 Cells: p qgonal prisms (cubes if q=4),
q pgonal prisms
 Faces: pq squares (all joining qgonal
prisms to pgonal prisms), p qgons (all joining
qgonal prisms to qgonal prisms: extra squares if
q=4), q pgons (all joining pgonal prisms
to pgonal prisms)
 Edges: 2pq
 Vertices: pq
Vertex figure:
 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
 qgonal2pgonal duoprism [not counted,
infinite subfamily of infinite duoprism family
(p>q>3)]
Alternative names:
 qgonal2pgonal prism
 qgonal2pgonal double prism
 qgonal2pgonal hyperprism
Symmetry group: [2p]x[q], the
2pgonalqgonal duoprismatic group, of
order 8pq (direct product of 2pgonal and
qgonal dihedral groups)
Schläfli symbols: {2p}x{q}, also
t{p}x{q} or
t_{0,1}{p}x{q}
Elements:
 Cells: 2p qgonal prisms (cubes if q=4),
q 2pgonal prisms
 Faces: 2pq squares (all joining qgonal
prisms to 2pgonal prisms), 2p qgons (all joining
qgonal prisms to qgonal prisms: extra squares if
q=4), q 2pgons (all joining 2pgonal
prisms to 2pgonal prisms)
 Edges: 4pq
 Vertices: 2pq
Vertex figure:
 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
 2qgonalpgonal duoprism [not counted,
infinite subfamily of infinite duoprism family
(p>q>3)]
Alternative names:
 2qgonalpgonal prism
 2qgonalpgonal double prism
 2qgonalpgonal hyperprism
Symmetry group: [[p]x[p]], the
pgonal duoprismatic group, of order
8p^{2} (direct product of 2
pgonal dihedral groups and an inversion, if
p=2q); [p]x[2q], the
pgonalqgonal duoprismatic group, of order
8pq (direct product of pgonal and qgonal
dihedral groups, if p~=2q)
Schläfli symbols: {p}x{2q}, also
{p}xt{q} or
{p}xt_{0,1}{q}
Elements:
 Cells: p 2qgonal prisms, 2q
pgonal prisms
 Faces: 2pq squares (all joining pgonal
prisms to 2qgonal prisms), p 2qgons (all
joining 2qgonal prisms to 2qgonal prisms),
2q pgons (all joining pgonal prisms to
pgonal prisms)
 Edges: 4pq
 Vertices: 2pq
Vertex figure:
 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
 2qgonal2pgonal duoprism [not counted,
infinite subfamily of infinite duoprism family
(p>q>3)]
Alternative names:
 2qgonal2pgonal prism
 2qgonal2pgonal double prism
 2qgonal2pgonal hyperprism
Symmetry group: [2p]x[2q], the
2qgonal2pgonal duoprismatic group, of
order 16pq (direct product of 2pgonal and
2qgonal dihedral groups)
Schläfli symbols: {2p}x{2q}, also
{2p}xt{q},
{2p}xt_{0,1}{q},
t{p}x{2q}, t{p}xt{q},
t{p}xt_{0,1}{q},
t_{0,1}{p}x{2q},
t_{0,1}{p}xt{q}, or
t_{0,1}{p}xt_{0,1}{q}
Elements:
 Cells: 2p 2qgonal prisms, 2q
2pgonal prisms
 Faces: 4pq squares (all joining 2qgonal
prisms to 2pgonal prisms), 2p 2qgons (all
joining 2qgonal prisms), 2q 2pgons (all
joining 2pgonal prisms)
 Edges: 8pq
 Vertices: 4pq
Vertex figure:
 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)oo (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
tetrahedralprismatic 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:
 Equilateraltriangular 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)
 Truncatedtetrahedral prism [49]
Alternative names:
 Truncatedtetrahedral dyadic prism (Norman W. Johnson)
 Tuttip (Jonathan Bowers: for truncatedtetrahedral
prism)
 Truncatedtetrahedral hyperprism
Symmetry group: [3,3]x[ ], the dyadic
tetrahedralprismatic group, of order 48
Schläfli symbols: t{3,3}x{ }, also
t_{0,1}{3,3}x{ } or
t_{1,2}{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:
 Isoscelestriangular pyramid: base an isosceles
triangle, edge lengths 1, sqrt(3), sqrt(3); all 3
lateral edges length sqrt(2)

(o)(o)(o) (o)
 Truncatedoctahedral prism [as “greatoctahedral
prism” tr{3,3}x{ }; not counted, duplicate of 54]

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

(o)oo (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 octahedralprismatic
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: base a rectangle, edges length 1,
sqrt(2); all 4 lateral edges length sqrt(2), making
2 triangles equilateral, the other 2 isosceles

oo(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
octahedralprismatic group, of order 96
Schläfli symbols: {3,4}x{ }, also
t_{0}{3,4}x{ },
t_{2}{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: base a square, edge length 1; all 4
lateral edges length sqrt(2)

(o)(o)o (o)
4
 Truncatedcubic prism [52]
Alternative names:
 Truncatedcubic dyadic prism (Norman W. Johnson)
 Ticcup (Jonathan Bowers: for truncatedcubic
prism)
 Truncatedcubic hyperprism
Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic
octahedralprismatic group, of order 96
Schläfli symbols: t{4,3}x{ }, also
t_{0,1}{4,3}x{ },
t_{1,2}{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:
 Isoscelestriangular 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
smallrhombicuboctahedral prism)
 [Small]rhombicuboctahedral hyperprism
Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic
octahedralprismatic 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: 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
 Truncatedoctahedral prism [54]
Alternative names:
 Truncatedoctahedral dyadic prism (Norman W. Johnson)
 Tope (Jonathan Bowers: for truncatedoctahedral
prism)
 Truncatedoctahedral hyperprism
 Greatoctahedral prism
 Greatoctahedral hyperprism
Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic
octahedralprismatic group, of order 96
Schläfli symbols: t{3,4}x{ }, also
t_{0,1}{3,4}x{ },
t_{1,2}{4,3}x{ }, tr{3,3}x{ }, and
t_{0,1,2}{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:
 Isoscelestriangular 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
 Truncatedcuboctahedral prism
[55]
Alternative names:
 Truncatedcuboctahedral dyadic prism (Norman W.
Johnson)
 Gircope (Jonathan Bowers: for
greatrhombicuboctahedral prism)
 Truncatedcuboctahedral hyperprism
 Greatrhombicuboctahedral prism
 Greatrhombicuboctahedral hyperprism
Symmetry group: [3,4]x[ ] or [4,3]x[ ], the dyadic
octahedralprismatic group, of order 96
Schläfli symbols: tr{3,4}x{ } or tr{4,3}x{ },
also t_{0,1,2}{4,3}x{ } or
t_{0,1,2}{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:
 Chiral scalenetriangular 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
 Snubcuboctahedral prism [56]
Alternative names:
 Snubcuboctahedral dyadic prism (Norman W. Johnson)
 Sniccup (Jonathan Bowers: for snubcubic prism)
 Snubcuboctahedral hyperprism
 Snubcubic prism
 Snubcubic hyperprism
Symmetry group: [3,4]^{+}x[ ]
or [4,3]^{+}x[ ], the direct
octahedralprismatic 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: 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)oo (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
icosahedralprismatic group, of order 240
Schläfli symbols: {5,3}x{ }, also
t_{0}{5,3}x{ } or
t_{2}{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:
 Equilateraltriangular 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
icosahedralprismatic 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: base a rectangle, edges length 1,
tau; all 4 lateral edges length sqrt(2)

oo(o) (o)
5
 Icosahedral prism [59]
Alternative names:
 Icosahedral dyadic prism (Norman W. Johnson)
 Ipe (Jonathan Bowers: for icosahedral prism)
 Icosahedral hyperprism
 Snuboctahedral prism
 Snuboctahedral hyperprism
 Snubtetrahedral prism
 Snubtetrahedral hyperprism
Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic
icosahedralprismatic group, of order 240
Schläfli symbols: {3,5}x{ }, also
t_{0}{3,5}x{ },
t_{2}{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:
 Regularpentagonal pyramid: base a regular pentagon,
edge length 1; all 5 lateral edges length sqrt(2)

(o)(o)o (o)
5
 Truncateddodecahedral prism [60]
Alternative names:
 Truncateddodecahedral dyadic prism (Norman W.
Johnson)
 Tiddip (Jonathan Bowers: for truncateddodecahedral
prism)
 Truncateddodecahedral hyperprism
Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic
icosahedralprismatic group, of order 240
Schläfli symbols: t{5,3}x{ }, also
t_{0,1}{5,3}x{ } or
t_{1,2}{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:
 Isoscelestriangular 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
smallrhombicosidodecahedral prism)
 [Small]rhombicosidodecahedral hyperprism
Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic
icosahedralprismatic group, of order 240
Schläfli symbols: rr{5,3}x{ } or rr{3,5}x{ },
also t_{1}{5,3}x{ } or
t_{1}{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: 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
 Truncatedicosahedral prism [62]
Alternative names:
 Truncatedicosahedral dyadic prism (Norman W. Johnson)
 Tipe (Jonathan Bowers: for truncatedicosahedral
prism)
 Truncatedicosahedral hyperprism
Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic
icosahedralprismatic group, of order 240
Schläfli symbols: t{3,5}x{ }, also
t_{0,1}{3,5}x{ } or
t_{1,2}{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:
 Isoscelestriangular pyramid: base an isosceles
triangle, edge lengths tau, sqrt(3),
sqrt(3); all 3 lateral edges length sqrt(2)

(o)(o)(o) (o)
5
 Truncatedicosidodecahedral prism [63]
Alternative names:
 Truncatedicosidodecahedral dyadic prism (Norman W.
Johnson)
 Griddip (Jonathan Bowers: for
greatrhombicosidodecahedral prism)
 Truncatedicosidodecahedral hyperprism
 Greatrhombicosidodecahedral prism
 Greatrhombicosidodecahedral hyperprism
Symmetry group: [3,5]x[ ] or [5,3]x[ ], the dyadic
icosahedralprismatic group, of order 240
Schläfli symbols: tr{3,5}x{ } or tr{5,3}x{ },
also t_{0,1,2}{5,3}x{ } or
t_{0,1,2}{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:
Chiral scalenetriangular 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
 Snubicosidodecahedral prism [64]
Alternative names:
 Snubicosidodecahedral dyadic prism (Norman W.
Johnson)
 Sniddip (Jonathan Bowers: for snubdodecahedral
prism)
 Snubicosidodecahedral hyperprism
 Snubdodecahedral prism
 Snubdodecahedral hyperprism
Symmetry group: [3,5]^{+}x[ ]
or [5,3]^{+}x[ ], the direct
icosahedralprismatic 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: 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
(5cell): polychora #1–9
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
7: Uniform polychora derived from glomeric tetrahedron
B_{4}: all duplicates of prior polychora