A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. Uniform polyhedra are semi-regular but the faces and vertices need not be convex. A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

Exclusive of the infinite sets of prisms and antiprisms, there are 75 uniform polyhedra. In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

The convex uniform polyhedra include the five Platonic solids, the thirteen Archimedean solids, and the convex prisms and antiprisms. In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting only on each of its vertices. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...

The non-convex uniform polyhedra include the four Kepler-Poinsot solids, 53 additional non-convex uniform polyhedra, and the non-convex prisms, antiprisms and crossed antiprisms. A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...

Naming Schemes

There are 3 common naming schemes:

1. Schläfli symbol {p,q} - compact form of vertex figure for regular polyhedra. p=face sides, q=faces/vertex.
2. Vertex Configuration - List of faces on each vertex, usually separated by periods.
3. Wythoff Symbols

In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...

Convex Forms [18+2n]

Convex Regular Solids [5] {p,q}

There are 5 Platonic Solids: In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting only on each of its vertices. ...

1. tetrahedron (3|2 3) - {3,3}
2. octahedron (4|2 3) - {3,4}
3. cube (3|2 4) - {4,3}
4. icosahedron (5|2 3) - {3,5}
5. dodecahedron (3|2 5) - {5,3}

Convex Semiregular [13+2n]

The semiregular polyhedra consist of 13 Archimedean solids and two infinite sets - the prisms and antiprisms. In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

Convex prisms [n]

There are an infinite number of prisms; an n-sided regular prism is bound by two regular n-sided polygons and n squares in between, n=3,4,5.... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...

4.4.3

4.4.4

4.4.5

4.4.6

Image File history File links Triangular_prism. ... Image File history File links Simple_cube. ... Image File history File links Pentagonal_prism. ... Image File history File links Hexagonal_prism. ...

Convex antiprism [n]

There are an infinite number of antiprisms; an n-sided regular antiprism is bound by two regular n-sided polygons and 2n equilateral triangles in between, n=3,4,5.... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

• n-agonal antiprisms: n=3,4,5...

3.3.3.3

3.3.3.4

3.3.3.5

3.3.3.6

3.3.3.17

Image File history File links Regular_octahedron. ... Image File history File links Square_antiprism. ... Image File history File links Pentagonal_antiprism. ... Image File history File links Hexagonal_antiprism. ... Download high resolution version (877x527, 53 KB)Heptadecagonal antiprism, made by me using POV-Ray, see image:poly. ...

Archimedean solids [13]

There are 13 Archimedean solids: In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...

1. truncated tetrahedron (2 3|3) - 3.6.6
2. truncated cube (2 3|4) - 3.8.8
3. truncated dodecahedron (2 3|5) - 3.10.10
4. truncated octahedron (2 4|3) - 4.6.6
5. truncated cuboctahedron (2 3 4|) - 4.6.8
6. truncated icosidodecahedron (2 3 5|) - 4.6.10
7. truncated icosahedron (2 5|3) - 5.6.6
8. cuboctahedron (2|3 4) - 3.4.3.4
9. rhombicuboctahedron (3 4|2) - 3.4.4.4
10. rhombicosidodecahedron (3 5|2) - 3.4.5.4
11. icosidodecahedron (2|3 5) - 3.5.3.5
12. snub cube (|2 3 4) - 3.3.3.3.4
13. snub dodecahedron (|2 3 5) - 3.3.3.3.5

3.6.6	3.8.8	3.10.10	4.6.6	4.6.8	4.6.10	5.6.6
3.4.3.4	3.4.4.4	3.4.5.4	3.5.3.5
3.3.3.3.4	3.3.3.3.5

The truncated tetrahedron is an Archimedean solid. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... The truncated dodecahedron is an Archimedean solid. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... The truncated icosahedron is an Archimedean solid. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... The snub cube, or snub cuboctahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. ... Download high resolution version (867x773, 50 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (819x855, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (876x802, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (855x868, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (868x869, 105 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (878x874, 80 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (818x804, 71 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (858x871, 113 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (841x861, 76 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (837x817, 96 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (853x866, 121 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ...

Stellated (concave) Forms [57+3n]

Concave Regular (Kepler-Poinsot solids) [4]

There are 4 Kepler-Poinsot solids: A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...

1. small stellated dodecahedron (5|2 5/2) - {5/2,5}
2. great stellated dodecahedron (3|2 5/2) - {5/2,3}
3. great dodecahedron (5/2|2 5) - {3,5/2}
4. great icosahedron (5/2|2 3) - {5,5/2}

Concave Semiregular [53]

There are 53 Uniform concave Semiregular polyhedra:

These appear much more complicated than the convex forms, but really reuse the same vertices of the convex forms. They still represent a fixed sequence of polygon faces going around each vertex. The difference is that the polygons can be stellation forms, like the pentagram. A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ...

• For example the stellated truncated hexahedron has a vertex figure of 8/3.8/3.3. This simply means a vertex is defined by two octagrams {8/3}, one triangle {3}. The octagram {8/3} is defined by the points of an octagon, but skipping two points, and going around the circle three times to complete.
• A somewhat funny polygon, {3/2} is a "backwards" triangle. (Connecting the points of a triangle by skipping every other one.) This means if you're building a vertex in a counter-clockwise order, that you should go clockwise momentarily to add this polygon.
• Some vertex figures themselves are concave. For example, (3.5.3.5.3.5)/2 means 6 the vertex contains alternating triangles and square going around twice (720 degrees).

1. octahemioctahedron (3/2 3|3) - 6.3/2.6.3
2. tetrahemihexahedron (3/2 3|2) -  4.3/2.4.3
3. small cubicuboctahedron (3/2 4|4) - 8.3/2.8.4
4. great cubicuboctahedron (3 4|4/3) - 8/3.3.8/3.4
5. cubohemioctahedron (4/3 4|3) -  6.4/3.6.4
6. cubitruncated cuboctahedron (4/3 3 4|) - 8/3.6.8
7. great rhombicuboctahedron (3/2 4|2) - 4.3/2.4.4
8. small rhombihexahedron (3/2 2 4|) - 8.4.8/7.4/3
9. stellated truncated hexahedron (2 3|4/3) - 8/3.8/3.3
10. great truncated cuboctahedron (4/3 2 3|) - 8/3.4.6
11. great rhombihexahedron (4/3 3/2 2|) - 4.8/3.4/3.8/5
12. small ditrigonal icosidodecahedron (3|5/2 3) - 5/2.3.5/2.3.5/2.3
13. small icosicosidodecahedron (5/2 3|3) - 6.5/2.6.3
14. small snub icosicosidodecahedron (|5/2 3 3) - 3.5/2.3.3.3.3
15. small dodecicosidodecahedron (3/2 5|5) - 10.3/2.10.5
16. dodecadodecahedron (2|5/2 5) - 5/2.5.5/2.5
17. truncated great dodecahedron (2 5/2|5) - 10.10.5/2
18. rhombidodecadodecahedron (5/2 5|2) - 4.5/2.4.5
19. small rhombidodecahedron (2 5/2 5|) - 10.4.10/9.4/3
20. snub dodecadodecahedron (|2 5/2 5) - 3.3.5/2.3.5
21. ditrigonal dodecadodecahedron (3|5/3 5) - 5/3.5.5/3.5.5/3.5
22. great ditrigonal dodecicosidodecahedron (3 5|5/3) - 10/3.3.10/3.3
23. small ditrigonal dodecicosidodecahedron (5/3 3|5) - 10.5/3.10.3
24. icosidodecadodecahedron (5/3 5|3) - 6.5/3.6.5
25. icositruncated dodecadodecahedron (5/3 3 5|) - 10/3.6.10
26. snub icosidodecadodecahedron (|5/3 3 5) - 3.5/3.3.3.3.5
27. great ditrigonal icosidodecahedron (3/2|3 5) - (3.5.3.5.3.5)/2
28. great icosicosidodecahedron (3/2 5|3)
29. small icosihemidodecahedron (3/2 3|5)
30. small dodecicosahedron (3/2 3 5|)
31. small dodecahemidodecahedron (5/4 5|5)
32. great icosidodecahedron (2|5/2 3)
33. great truncated icosahedron (2 5/2|3)
34. rhombicosahedron (2 5/2 3|)
35. great snub icosidodecahedron (|2 5/2 3)
36. small stellated truncated dodecahedron (2 5|5/3)
37. truncated dodecadodecahedron (5/3 2 5|)
38. inverted snub dodecadodecahedron (|5/3 2 5)
39. great dodecicosidodecahedron (5/2 3|5/3)
40. small dodecahemicosahedron (5/3 5/2|3)
41. great dodecicosahedron (5/3 5/2 3|)
42. great snub dodecicosidodecahedron (|5/3 5/2 3)
43. great dodecahemicosahedron (5/4 5|3)
44. great stellated truncated dodecahedron (2 3|5/3)
45. great rhombicosidodecahedron (5/3 3|2)
46. great truncated icosidodecahedron (5/3 2 3|)
47. great inverted snub icosidodecahedron (|5/3 2 3)
48. great dodecahemidodecahedron (5/3 5/2|5/3)
49. great icosihemidodecahedron (3/2 3|5/3)
50. small retrosnub icosicosidodecahedron (|3/2 3/2 5/2)
51. great rhombidodecahedron (3/2 5/3 2|)
52. great retrosnub icosidodecahedron (|3/2 5/3 2)
53. great dirhombicosidodecahedron (|3/2 5/3 3 5/2)

In geometry, the Tetrahemihexahedron is a concave uniform polyhedron, indexed as U4. ...

Concave Prisms [n]

1. n-agrammic prisms

4.4.5/2

Image File history File links Pentagram_prism. ...

Concave Antiprism [n]

1. n-agrammic antiprism

3.3.3.5/2

Image File history File links Pentagram_antiprism. ...

Concave Crossed Antiprism [n]

1. n-agrammic crossed antiprism

3.3.3.5/3

Image File history File links Pentagram_crossantiprism. ...

See Also

• Polyhedron
• Archimedean solid
• Kepler-Poinsot solid

A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...

External Links

• Uniform Solution for Uniform Polyhedra
• The Uniform Polyhedra
• Virtual Polyhedra Uniform Polyhedra
• Eric W. Weisstein. "Uniform Polyhedron." From MathWorld--A Wolfram Web Resource.