List of uniform polyhedra
Encyclopedia
Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols.
This list includes:
Not included are:
s from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.
(*1) : The Great disnub dirhombidodecahedron has 120 edges shared by four faces. If counted as two pairs, then there are a total 360 edges. Because of this edge-degeneracy, it is not always considered a uniform polyhedron.
This list includes:
- all 75 nonprismatic uniformUniform polyhedronA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
polyhedraPolyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
; - a few representatives of the infinite sets of prismPrism (geometry)In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
s and antiprismAntiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s; - one special case polyhedron, Skilling's figure with overlapping edges.
Not included are:
- 40 potential uniform polyhedra with degenerate vertex figureVertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
s which have overlapping edges (not counted by Coxeter); - 11 uniform tessellations with convex faces;
- 14 uniform tilings with nonconvex faces;
- the infinite set of Uniform tilings in hyperbolic planeUniform tilings in hyperbolic planeThere are an infinite number of uniform tilings on the hyperbolic plane based on the where 1/p + 1/q + 1/r ...
.
Indexing
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:- [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
- [W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
- [K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetryDihedral symmetry in three dimensionsThis article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...
, 6-9 with tetrahedral symmetryTetrahedral symmetry150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
, 10-26 with Octahedral symmetryOctahedral symmetry150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
, 46-80 with icosahedral symmetryIcosahedral symmetryA regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
. - [U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.
Table of polyhedra
The convex forms are listed in order of degree of vertex configurationVertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...
s from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.
Convex forms (3 faces/vertex)
Name | Picture | Solid class |
Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... |
R | 3|2 3 | 3.3.3 |
Tet | T_{d} | W001 | U01 | K06 | 4 | 6 | 4 | 2 | 4{3} | |
Triangular prism Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.... |
P | 2 3|2 | 3.4.4 |
Trip | D_{3h} | -- | -- | -- | 6 | 9 | 5 | 2 | 2{3}+3{4} | |
Truncated tetrahedron Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :... |
A | 2 3|3 | 3.6.6 |
Tut | T_{d} | W006 | U02 | K07 | 12 | 18 | 8 | 2 | 4{3}+4{6} | |
Truncated cube Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... |
A | 2 3|4 | 3.8.8 |
Tic | O_{h} | W008 | U09 | K14 | 24 | 36 | 14 | 2 | 8{3}+6{8} | |
Truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :... |
A | 2 3|5 | 3.10.10 |
Tid | I_{h} | W010 | U26 | K31 | 60 | 90 | 32 | 2 | 20{3}+12{10} | |
Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... |
R | 3|2 4 | 4.4.4 |
Cube | O_{h} | W003 | U06 | K11 | 8 | 12 | 6 | 2 | 6{4} | |
Pentagonal prism Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :... |
P | 2 5|2 | 4.4.5 |
Pip | D_{5h} | -- | U76 | K01 | 10 | 15 | 7 | 2 | 5{4}+2{5} | |
Hexagonal prism Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces... |
P | 2 6|2 | 4.4.6 |
Hip | D_{6h} | -- | -- | -- | 12 | 18 | 8 | 2 | 6{4}+2{6} | |
Octagonal prism Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :... |
P | 2 8|2 | 4.4.8 |
Op | D_{8h} | -- | -- | -- | 16 | 24 | 10 | 2 | 8{4}+2{8} | |
Decagonal prism Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron.... |
P | 2 10|2 | 4.4.10 |
Dip | D_{10h} | -- | -- | -- | 20 | 30 | 12 | 2 | 10{4}+2{10} | |
Dodecagonal prism Dodecagonal prism In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :... |
P | 2 12|2 | 4.4.12 |
Twip | D_{12h} | -- | -- | -- | 24 | 36 | 14 | 2 | 12{4}+2{12} | |
Truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... |
A | 2 4|3 | 4.6.6 |
Toe | O_{h} | W007 | U08 | K13 | 24 | 36 | 14 | 2 | 6{4}+8{6} | |
Great rhombicuboctahedron Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges... |
A | 2 3 4| | 4.6.8 |
Girco | O_{h} | W015 | U11 | K16 | 48 | 72 | 26 | 2 | 12{4}+8{6}+6{8} | |
Great rhombicosidodecahedron Truncated icosidodecahedron In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... |
A | 2 3 5| | 4.6.10 |
Grid | I_{h} | W016 | U28 | K33 | 120 | 180 | 62 | 2 | 30{4}+20{6}+12{10} | |
Dodecahedron | R | 3|2 5 | 5.5.5 |
Doe | I_{h} | W005 | U23 | K28 | 20 | 30 | 12 | 2 | 12{5} | |
Truncated icosahedron Truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.... |
A | 2 5|3 | 5.6.6 |
Ti | I_{h} | W009 | U25 | K30 | 60 | 90 | 32 | 2 | 12{5}+20{6} |
Convex forms (4 faces/vertex)
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... |
R | 4|2 3 | 3.3.3.3 |
Oct | O_{h} | W002 | U05 | K10 | 6 | 12 | 8 | 2 | 8{3} | |
Square antiprism Square antiprism In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps... |
P | |2 2 4 | 3.3.3.4 |
Squap | D_{4d} | -- | -- | -- | 8 | 16 | 10 | 2 | 8{3}+2{4} | |
Pentagonal antiprism Pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces... |
P | |2 2 5 | 3.3.3.5 |
Pap | D_{5d} | -- | U77 | K02 | 10 | 20 | 12 | 2 | 10{3}+2{5} | |
Hexagonal antiprism Hexagonal antiprism In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms... |
P | |2 2 6 | 3.3.3.6 |
Hap | D_{6d} | -- | -- | -- | 12 | 24 | 14 | 2 | 12{3}+2{6} | |
Octagonal antiprism Octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms... |
P | |2 2 8 | 3.3.3.8 |
Oap | D_{8d} | -- | -- | -- | 16 | 32 | 18 | 2 | 16{3}+2{8} | |
Decagonal antiprism Decagonal antiprism In geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- External links :*... |
P | |2 2 10 | 3.3.3.10 |
Dap | D_{10d} | -- | -- | -- | 20 | 40 | 22 | 2 | 20{3}+2{10} | |
Dodecagonal antiprism Dodecagonal antiprism In geometry, the dodecagonal antiprism is the tenth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.... |
P | |2 2 12 | 3.3.3.12 |
Twap | D_{12d} | -- | -- | -- | 24 | 48 | 26 | 2 | 24{3}+2{12} | |
Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... |
A | 2|3 4 | 3.4.3.4 |
Co | O_{h} | W011 | U07 | K12 | 12 | 24 | 14 | 2 | 8{3}+6{4} | |
Small rhombicuboctahedron | A | 3 4|2 | 3.4.4.4 |
Sirco | O_{h} | W013 | U10 | K15 | 24 | 48 | 26 | 2 | 8{3}+(6+12){4} | |
Small rhombicosidodecahedron | A | 3 5|2 | 3.4.5.4 |
Srid | I_{h} | W014 | U27 | K32 | 60 | 120 | 62 | 2 | 20{3}+30{4}+12{5} | |
Icosidodecahedron Icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon... |
A | 2|3 5 | 3.5.3.5 |
Id | I_{h} | W012 | U24 | K29 | 30 | 60 | 32 | 2 | 20{3}+12{5} |
Convex forms (5 faces/vertex)
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... |
R | 5|2 3 | 3.3.3.3.3 |
Ike | I_{h} | W004 | U22 | K27 | 12 | 30 | 20 | 2 | 20{3} | |
Snub cube Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each... |
A | |2 3 4 | 3.3.3.3.4 |
Snic | O | W017 | U12 | K17 | 24 | 60 | 38 | 2 | (8+24){3}+6{4} | |
Snub dodecahedron Snub dodecahedron In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.... |
A | |2 3 5 | 3.3.3.3.5 |
Snid | I | W018 | U29 | K34 | 60 | 150 | 92 | 2 | (20+60){3}+12{5} |
Nonconvex forms with convex faces
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Octahemioctahedron Octahemioctahedron In geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U3. Its vertex figure is a crossed quadrilateral.It is one of nine hemipolyhedra with 4 hexagonal faces passing through the model center.- Related polyhedra :... |
C+ | ^{3}/_{2} 3|3 | 6.^{3}/_{2}.6.3 |
Oho | O_{h} | W068 | U03 | K08 | 12 | 24 | 12 | 0 | 8{3}+4{6} | |
Tetrahemihexahedron Tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of .... |
C+ | ^{3}/_{2} 3|2 | 4.^{3}/_{2}.4.3 |
Thah | T_{d} | W067 | U04 | K09 | 6 | 12 | 7 | 1 | 4{3}+3{4} | |
Cubohemioctahedron Cubohemioctahedron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces... |
C+ | ^{4}/_{3} 4|3 | 6.^{4}/_{3}.6.4 |
Cho | O_{h} | W078 | U15 | K20 | 12 | 24 | 10 | -2 | 6{4}+4{6} | |
Great dodecahedron | R+ | ^{5}/_{2}|2 5 | (5.5.5.5.5)/_{2} |
Gad | I_{h} | W021 | U35 | K40 | 12 | 30 | 12 | -6 | 12{5} | |
Great icosahedron | R+ | ^{5}/_{2}|2 3 | (3.3.3.3.3)/_{2} |
Gike | I_{h} | W041 | U53 | K58 | 12 | 30 | 20 | 2 | 20{3} | |
Great ditrigonal icosidodecahedron | C+ | ^{3}/_{2}|3 5 | (5.3.5.3.5.3)/_{2} |
Gidtid | I_{h} | W087 | U47 | K52 | 20 | 60 | 32 | -8 | 20{3}+12{5} | |
Small rhombihexahedron Small rhombihexahedron In geometry, the small rhombihexahedron is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces , 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.-Related polyhedra:... |
C+ | ^{3}/_{2} 2 4| | 4.8.^{4}/_{3}.8 |
Sroh | O_{h} | W086 | U18 | K23 | 24 | 48 | 18 | -6 | 12{4}+6{8} | |
Small cubicuboctahedron Small cubicuboctahedron In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces , 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
C+ | ^{3}/_{2} 4|4 | 8.^{3}/_{2}.8.4 |
Socco | O_{h} | W069 | U13 | K18 | 24 | 48 | 20 | -4 | 8{3}+6{4}+6{8} | |
Nonconvex great rhombicuboctahedron | C+ | ^{3}/_{2} 4|2 | 4.^{3}/_{2}.4.4 |
Querco | O_{h} | W085 | U17 | K22 | 24 | 48 | 26 | 2 | 8{3}+(6+12){4} | |
Small dodecahemidodecahedron Small dodecahemidodecahedron In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U51. Its vertex figure alternates two regular pentagons and decagons as a crossed quadrilateral.... |
C+ | ^{5}/_{4} 5|5 | 10.^{5}/_{4}.10.5 |
Sidhid | I_{h} | W091 | U51 | K56 | 30 | 60 | 18 | -12 | 12{5}+6{10} | |
Great dodecahemicosahedron Great dodecahemicosahedron In geometry, the great dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U65. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :... |
C+ | ^{5}/_{4} 5|3 | 6.^{5}/_{4}.6.5 |
Gidhei | I_{h} | W102 | U65 | K70 | 30 | 60 | 22 | -8 | 12{5}+10{6} | |
Small icosihemidodecahedron Small icosihemidodecahedron In geometry, the small icosihemidodecahedron is a uniform star polyhedron, indexed as U49. Its vertex figure alternates two regular triangles and decagons as a crossed quadrilateral.... |
C+ | ^{3}/_{2} 3|5 | 10.^{3}/_{2}.10.3 |
Seihid | I_{h} | W089 | U49 | K54 | 30 | 60 | 26 | -4 | 20{3}+6{10} | |
Small dodecicosahedron Small dodecicosahedron In geometry, the small dodecicosahedron is a nonconvex uniform polyhedron, indexed as U50. Its vertex figure is a crossed quadrilateral.-Related polyhedra:It shares its vertex arrangement with the great stellated truncated dodecahedron... |
C+ | ^{3}/_{2} 3 5| | 10.6.^{10}/_{9}.^{6}/_{5} |
Siddy | I_{h} | W090 | U50 | K55 | 60 | 120 | 32 | -28 | 20{6}+12{10} | |
Small rhombidodecahedron Small rhombidodecahedron In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
C+ | 2 ^{5}/_{2} 5| | 10.4.^{10}/_{9}.^{4}/_{3} |
Sird | I_{h} | W074 | U39 | K44 | 60 | 120 | 42 | -18 | 30{4}+12{10} | |
Small dodecicosidodecahedron Small dodecicosidodecahedron In geometry, the small dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U33. Its vertex figure is a crossed quadrilateral.-Related polyhedra:... |
C+ | ^{3}/_{2} 5|5 | 10.^{3}/_{2}.10.5 |
Saddid | I_{h} | W072 | U33 | K38 | 60 | 120 | 44 | -16 | 20{3}+12{5}+12{10} | |
Rhombicosahedron Rhombicosahedron In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :... |
C+ | 2 ^{5}/_{2} 3| | 6.4.^{6}/_{5}.^{4}/_{3} |
Ri | I_{h} | W096 | U56 | K61 | 60 | 120 | 50 | -10 | 30{4}+20{6} | |
Great icosicosidodecahedron Great icosicosidodecahedron In geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron... |
C+ | ^{3}/_{2} 5|3 | 6.^{3}/_{2}.6.5 |
Giid | I_{h} | W088 | U48 | K53 | 60 | 120 | 52 | -8 | 20{3}+12{5}+20{6} |
Nonconvex prismatic forms
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Pentagrammic prism Pentagrammic prism In geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U78 as a uniform polyhedron.... |
P+ | 2 ^{5}/_{2}|2 | ^{5}/_{2}.4.4 |
Stip | D_{5h} | -- | U78 | K03 | 10 | 15 | 7 | 2 | 5{4}+2{^{5}/_{2}} | |
Heptagrammic prism (7/3) | P+ | 2 ^{7}/_{3}|2 | ^{7}/_{3}.4.4 |
Giship | D_{7h} | -- | -- | -- | 14 | 21 | 9 | 2 | 7{4}+2{^{7}/_{3}} | |
Heptagrammic prism (7/2) | P+ | 2 ^{7}/_{2}|2 | ^{7}/_{2}.4.4 |
Ship | D_{7h} | -- | -- | -- | 14 | 21 | 9 | 2 | 7{4}+2{^{7}/_{2}} | |
Pentagrammic antiprism Pentagrammic antiprism In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U79 as a uniform polyhedron.... |
P+ | |2 2 ^{5}/_{2} | ^{5}/_{2}.3.3.3 |
Stap | D_{5h} | -- | U79 | K04 | 10 | 20 | 12 | 2 | 10{3}+2{^{5}/_{2}} | |
Pentagrammic crossed-antiprism Pentagrammic crossed-antiprism In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.... |
P+ | |2 2 ^{5}/_{3} | ^{5}/_{3}.3.3.3 |
Starp | D_{5d} | -- | U80 | K05 | 10 | 20 | 12 | 2 | 10{3}+2{^{5}/_{2}} |
Other nonconvex forms with nonconvex faces
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Small stellated dodecahedron | R+ | 5|2 ^{5}/_{2} | (^{5}/_{2})^{5} |
Sissid | I_{h} | W020 | U34 | K39 | 12 | 30 | 12 | -6 | 12{^{5}/_{2}} | |
Great stellated dodecahedron | R+ | 3|2 ^{5}/_{2} | (^{5}/_{2})^{3} |
Gissid | I_{h} | W022 | U52 | K57 | 20 | 30 | 12 | 2 | 12{^{5}/_{2}} | |
Ditrigonal dodecadodecahedron Ditrigonal dodecadodecahedron In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.- Related polyhedra :Its convex hull is a regular dodecahedron... |
S+ | 3|^{5}/_{3} 5 | (^{5}/_{3}.5)^{3} |
Ditdid | I_{h} | W080 | U41 | K46 | 20 | 60 | 24 | -16 | 12{5}+12{^{5}/_{2}} | |
Small ditrigonal icosidodecahedron Small ditrigonal icosidodecahedron In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron... |
S+ | 3|^{5}/_{2} 3 | (^{5}/_{2}.3)^{3} |
Sidtid | I_{h} | W070 | U30 | K35 | 20 | 60 | 32 | -8 | 20{3}+12{^{5}/_{2}} | |
Stellated truncated hexahedron | S+ | 2 3|^{4}/_{3} | ^{8}/_{3}.^{8}/_{3}.3 |
Quith | O_{h} | W092 | U19 | K24 | 24 | 36 | 14 | 2 | 8{3}+6{^{8}/_{3}} | |
Great rhombihexahedron Great rhombihexahedron In geometry, the great rhombihexahedron is a nonconvex uniform polyhedron, indexed as U21. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
S+ | ^{4}/_{3}^{3}/_{2} 2| | 4.^{8}/_{3}.^{4}/_{3}.^{8}/_{5} |
Groh | O_{h} | W103 | U21 | K26 | 24 | 48 | 18 | -6 | 12{4}+6{^{8}/_{3}} | |
Great cubicuboctahedron Great cubicuboctahedron In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14.- Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra... |
S+ | 3 4|^{4}/_{3} | ^{8}/_{3}.3.^{8}/_{3}.4 |
Gocco | O_{h} | W077 | U14 | K19 | 24 | 48 | 20 | -4 | 8{3}+6{4}+6{^{8}/_{3}} | |
Great dodecahemidodecahedron Great dodecahemidodecahedron In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. Its vertex figure is a crossed quadrilateral.... |
S+ | ^{5}/_{3}^{5}/_{2}|^{5}/_{3} | ^{10}/_{3}.^{5}/_{3}.^{10}/_{3}.^{5}/_{2} |
Gidhid | I_{h} | W107 | U70 | K75 | 30 | 60 | 18 | -12 | 12{^{5}/_{2}}+6{^{10}/_{3}} | |
Small dodecahemicosahedron Small dodecahemicosahedron In geometry, the small dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U62. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :... |
S+ | ^{5}/_{3}^{5}/_{2}|3 | 6.^{5}/_{3}.6.^{5}/_{2} |
Sidhei | I_{h} | W100 | U62 | K67 | 30 | 60 | 22 | -8 | 12{^{5}/_{2}}+10{6} | |
Dodecadodecahedron | S+ | 2|^{5}/_{2} 5 | (^{5}/_{2}.5)^{2} |
Did | I_{h} | W073 | U36 | K41 | 30 | 60 | 24 | -6 | 12{5}+12{^{5}/_{2}} | |
Great icosihemidodecahedron Great icosihemidodecahedron In geometry, the great icosihemidodecahedron is a nonconvex uniform polyhedron, indexed as U71. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with 6 decagrammic faces passing through the model center.- Related polyhedra :... |
S+ | ^{3}/_{2} 3|^{5}/_{3} | ^{10}/_{3}.^{3}/_{2}.^{10}/_{3}.3 |
Geihid | I_{h} | W106 | U71 | K76 | 30 | 60 | 26 | -4 | 20{3}+6{^{10}/_{3}} | |
Great icosidodecahedron | S+ | 2|^{5}/_{2} 3 | (^{5}/_{2}.3)^{2} |
Gid | I_{h} | W094 | U54 | K59 | 30 | 60 | 32 | 2 | 20{3}+12{^{5}/_{2}} | |
Cubitruncated cuboctahedron Cubitruncated cuboctahedron In geometry, the cubitruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16.- Convex hull :Its convex hull is a nonuniform truncated cuboctahedron.- Cartesian coordinates :... |
S+ | ^{4}/_{3} 3 4| | ^{8}/_{3}.6.8 |
Cotco | O_{h} | W079 | U16 | K21 | 48 | 72 | 20 | -4 | 8{6}+6{8}+6{^{8}/_{3}} | |
Great truncated cuboctahedron | S+ | ^{4}/_{3} 2 3| | ^{8}/_{3}.4.6 |
Quitco | O_{h} | W093 | U20 | K25 | 48 | 72 | 26 | 2 | 12{4}+8{6}+6{^{8}/_{3}} | |
Truncated great dodecahedron | S+ | 2 ^{5}/_{2}|5 | 10.10.^{5}/_{2} |
Tigid | I_{h} | W075 | U37 | K42 | 60 | 90 | 24 | -6 | 12{^{5}/_{2}}+12{10} | |
Small stellated truncated dodecahedron | S+ | 2 5|^{5}/_{3} | ^{10}/_{3}.^{10}/_{3}.5 |
Quitsissid | I_{h} | W097 | U58 | K63 | 60 | 90 | 24 | -6 | 12{5}+12{^{10}/_{3}} | |
Great stellated truncated dodecahedron | S+ | 2 3|^{5}/_{3} | ^{10}/_{3}.^{10}/_{3}.3 |
Quitgissid | I_{h} | W104 | U66 | K71 | 60 | 90 | 32 | 2 | 20{3}+12{^{10}/_{3}} | |
Truncated great icosahedron | S+ | 2 ^{5}/_{2}|3 | 6.6.^{5}/_{2} |
Tiggy | I_{h} | W095 | U55 | K60 | 60 | 90 | 32 | 2 | 12{^{5}/_{2}}+20{6} | |
Great dodecicosahedron Great dodecicosahedron In geometry, the great dodecicosahedron is a nonconvex uniform polyhedron, indexed as U63. Its vertex figure is a crossed quadrilateral.It has a composite Wythoff symbol, 3 5/3 |, requiring two different Schwarz triangles to generate it: and .Its vertex figure 6.10/3.6/5.10/7 is also ambiguous,... |
S+ | ^{5}/_{3}^{5}/_{2} 3| | 6.^{10}/_{3}.^{6}/_{5}.^{10}/_{7} |
Giddy | I_{h} | W101 | U63 | K68 | 60 | 120 | 32 | -28 | 20{6}+12{^{10}/_{3}} | |
Great rhombidodecahedron Great rhombidodecahedron In geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
S+ | ^{3}/_{2}^{5}/_{3} 2| | 4.^{10}/_{3}.^{4}/_{3}.^{10}/_{7} |
Gird | I_{h} | W109 | U73 | K78 | 60 | 120 | 42 | -18 | 30{4}+12{^{10}/_{3}} | |
Icosidodecadodecahedron Icosidodecadodecahedron In geometry, the icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U44. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms... |
S+ | ^{5}/_{3} 5|3 | 6.^{5}/_{3}.6.5 |
Ided | I_{h} | W083 | U44 | K49 | 60 | 120 | 44 | -16 | 12{5}+12{^{5}/_{2}}+20{6} | |
Small ditrigonal dodecicosidodecahedron Small ditrigonal dodecicosidodecahedron In geometry, the small ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U43. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron... |
S+ | ^{5}/_{3} 3|5 | 10.^{5}/_{3}.10.3 |
Sidditdid | I_{h} | W082 | U43 | K48 | 60 | 120 | 44 | -16 | 20{3}+12{^{;5}/_{2}}+12{10} | |
Great ditrigonal dodecicosidodecahedron Great ditrigonal dodecicosidodecahedron In geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron... |
S+ | 3 5|^{5}/_{3} | ^{10}/_{3}.3.^{10}/_{3}.5 |
Gidditdid | I_{h} | W081 | U42 | K47 | 60 | 120 | 44 | -16 | 20{3}+12{5}+12{^{10}/_{3}} | |
Great dodecicosidodecahedron Great dodecicosidodecahedron In geometry, the great dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U61.- Related polyhedra :It shares its vertex arrangement with the truncated great dodecahedron and the uniform compounds of 6 or 12 pentagonal prisms... |
S+ | ^{5}/_{2} 3|^{5}/_{3} | ^{10}/_{3}.^{5}/_{2}.^{10}/_{3}.3 |
Gaddid | I_{h} | W099 | U61 | K66 | 60 | 120 | 44 | -16 | 20{3}+12{^{5}/_{2}}+12{^{10}/_{3}} | |
Small icosicosidodecahedron Small icosicosidodecahedron In geometry, the small icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U31.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron... |
S+ | ^{5}/_{2} 3|3 | 6.^{5}/_{2}.6.3 |
Siid | I_{h} | W071 | U31 | K36 | 60 | 120 | 52 | -8 | 20{3}+12{^{5}/_{2}}+20{6} | |
Rhombidodecadodecahedron | S+ | ^{5}/_{2} 5|2 | 4.^{5}/_{2}.4.5 |
Raded | I_{h} | W076 | U38 | K43 | 60 | 120 | 54 | -6 | 30{4}+12{5}+12{^{5}/_{2}} | |
Nonconvex great rhombicosidodecahedron | S+ | ^{5}/_{3} 3|2 | 4.^{5}/_{3}.4.3 |
Qrid | I_{h} | W105 | U67 | K72 | 60 | 120 | 62 | 2 | 20{3}+30{4}+12{^{5}/_{2}} | |
Snub dodecadodecahedron | S+ | |2 ^{5}/_{2} 5 | 3.3.^{5}/_{2}.3.5 |
Siddid | I | W111 | U40 | K45 | 60 | 150 | 84 | -6 | 60{3}+12{5}+12{^{5}/_{2}} | |
Inverted snub dodecadodecahedron | S+ | |^{5}/_{3} 2 5 | 3.^{5}/_{3}.3.3.5 |
Isdid | I | W114 | U60 | K65 | 60 | 150 | 84 | -6 | 60{3}+12{5}+12{^{5}/_{2}} | |
Great snub icosidodecahedron | S+ | |2 ^{5}/_{2} 3 | 3.^{4}.^{5}/_{2} |
Gosid | I | W116 | U57 | K62 | 60 | 150 | 92 | 2 | (20+60){3}+12{^{5}/_{2}} | |
Great inverted snub icosidodecahedron | S+ | |^{5}/_{3} 2 3 | 3.^{3}.^{5}/_{3} |
Gisid | I | W113 | U69 | K74 | 60 | 150 | 92 | 2 | (20+60){3}+12{^{5}/_{2}} | |
Great retrosnub icosidodecahedron | S+ | |^{3}/_{2}^{5}/_{3} 2 | (3^{4}.^{5}/_{2})/_{2} |
Girsid | I | W117 | U74 | K79 | 60 | 150 | 92 | 2 | (20+60){3}+12{^{5}/_{2}} | |
Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,... |
S+ | |^{5}/_{3}^{5}/_{2} 3 | 3^{3}.^{5}/_{3}.3.^{5}/_{2} |
Gisdid | I | W115 | U64 | K69 | 60 | 180 | 104 | -16 | (20+60){3}+(12+12){^{5}/_{2}} | |
Snub icosidodecadodecahedron Snub icosidodecadodecahedron In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46.- Cartesian coordinates :Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of... |
S+ | |^{5}/_{3} 3 5 | 3.^{3}.5.^{5}/_{3} |
Sided | I | W112 | U46 | K51 | 60 | 180 | 104 | -16 | (20+60){3}+12{5}+12{^{5}/_{2}} | |
Small snub icosicosidodecahedron Small snub icosicosidodecahedron In geometry, the small snub icosicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces , 180 edges, and 60 vertices.- Convex hull :Its convex hull is a nonuniform truncated icosahedron.... |
S+ | |^{5}/_{2} 3 3 | 3^{5}.^{5}/_{2} |
Seside | I_{h} | W110 | U32 | K37 | 60 | 180 | 112 | -8 | (40+60){3}+12{^{5}/_{2}} | |
Small retrosnub icosicosidodecahedron Small retrosnub icosicosidodecahedron In geometry, the small retrosnub icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U72.- Convex hull :Its convex hull is a nonuniform truncated dodecahedron.- Cartesian coordinates :... |
S+ | |^{3}/_{2}^{3}/_{2}^{5}/_{2} | (3^{5}.^{5}/_{3})/_{2} |
Sirsid | I_{h} | W118 | U72 | K77 | 60 | 180 | 112 | -8 | (40+60){3}+12{^{5}/_{2}} | |
Great dirhombicosidodecahedron Great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex... |
S+ | |^{3}/_{2}^{5}/_{3} 3 ^{5}/_{2} |
(4.^{5}/_{3}.4.3. 4.^{5}/_{2}.4.^{3}/_{2})/_{2} |
Gidrid | I_{h} | W119 | U75 | K80 | 60 | 240 | 124 | -56 | 40{3}+60{4}+24{^{5}/_{2}} | |
Icositruncated dodecadodecahedron Icositruncated dodecadodecahedron In geometry, the icositruncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U45.- Convex hull :Its convex hull is a nonuniform great rhombicosidodecahedron.- Cartesian coordinates :... |
S+ | ^{5}/_{3} 3 5| | ^{10}/_{3}.6.10 |
Idtid | I_{h} | W084 | U45 | K50 | 120 | 180 | 44 | -16 | 20{6}+12{10}+12{^{10}/_{3}} | |
Truncated dodecadodecahedron | S+ | ^{5}/_{3} 2 5| | ^{10}/_{3}.4.10 |
Quitdid | I_{h} | W098 | U59 | K64 | 120 | 180 | 54 | -6 | 30{4}+12{10}+12{^{10}/_{3}} | |
Great truncated icosidodecahedron | S+ | ^{5}/_{3} 2 3| | ^{10}/_{3}.4.6 |
Gaquatid | I_{h} | W108 | U68 | K73 | 120 | 180 | 62 | 2 | 30{4}+20{6}+12{^{10}/_{3}} |
Special case
Name | Picture | Solid class |
Wythoff symbol |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Bowers-style acronym |
Symmetry group |
W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Great disnub dirhombidodecahedron Great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge... Skilling's figure |
S++ | | (3/2) 5/3 (3) 5/2 | (^{5}/_{2}.4.3.3.3.4. ^{5}/_{3}.4.^{3}/_{2}.^{3}/_{2}.^{3}/_{2}.4)/_{2} |
Gidisdrid | I_{h} | -- | -- | -- | 60 | 240 (*1) | 204 | 24 | 120{3}+60{4}+24{^{5}/_{2}} |
(*1) : The Great disnub dirhombidodecahedron has 120 edges shared by four faces. If counted as two pairs, then there are a total 360 edges. Because of this edge-degeneracy, it is not always considered a uniform polyhedron.
Column key
- Solid classes
- R = 5 Platonic solidPlatonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s - R+= 4 Kepler-Poinsot polyhedra
- A = 13 Archimedean solidArchimedean solidIn geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
s - C+= 14 Non-convex polyhedra with only convexConvex polygonIn geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...
faces (all of these uniform polyhedra have faces which intersect each other) - S+= 39 Non-convex polyhedra with complexComplex polygonThe term complex polygon can mean two different things:*In computer graphics, as a polygon which is neither convex nor concave.*In geometry, as a polygon in the unitary plane, which has two complex dimensions.-Computer graphics:...
(star) faces - P = Infinite series of Convex Regular Prisms and Antiprisms
- P+= Infinite series of Non-convex uniform prismsPrism (geometry)In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
and antiprismAntiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s (these all contain complex (star) faces) - T = 11 PlanarPlane (mathematics)In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
tessellationTessellationA tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s
- R = 5 Platonic solid
- Bowers style acronym - A unique pronounceable abbreviated name created by mathematician Jonathan Bowers
- Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
- Kaleido software indexing: K01-K80
(prisms 1-5, Tetrahedron 6+) - Magnus Wenninger Polyhedron Models: W001-W119
- 1-18 - 5 convex regular and 13 convex semiregular
- 20-22, 41 - 4 non-convex regular
- 19-66 Special 48 stellations/compounds (Nonregulars not given on this list)
- 67-109 - 43 non-convex non-snub uniform
- 110-119 - 10 non-convex snub uniform
- Chi: the Euler characteristicEuler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero. - For the plane tilings, the numbers given of vertices, edges and faces show the ratio of such elements in one period of the pattern, which in each case is a rhombusRhombusIn Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
(sometimes a right-angled rhombus, i.e. a squareSquare (geometry)In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
). - Note on Vertex figure images:
- The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images help see their relations. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front.
External links
- Stella: Polyhedron Navigator - Software able to generate and print nets for all uniform polyhedra. Used to create most images on this page.
- Paper models
- Uniform indexing: U1-U80, (Tetrahedron first)
- Uniform Polyhedra (80), Paul Bourke
- http://mathworld.wolfram.com/UniformPolyhedron.html
- http://www.mathconsult.ch/showroom/unipoly
- http://gratrix.net/polyhedra/uniform/summary
- http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm
- http://www.buddenbooks.com/jb/uniform/
- Kaleido Indexing: K1-K80 (Pentagonal prism first)
- http://www.math.technion.ac.il/~rl/kaleido
- http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra
- http://bulatov.org/polyhedra/uniform
- http://www.orchidpalms.com/polyhedra/uniform/uniform.html
- http://www.math.technion.ac.il/~rl/kaleido
- Also
- http://www.polyedergarten.de/polyhedrix/e_klintro.htm