In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a uniform polyhedron is a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
which has
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s as
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
and is
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
(
transitive on its
vertices, isogonal, i.e. there is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
mapping any vertex onto any other). It follows that all vertices are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
, and the polyhedron has a high degree of
reflectional and
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
.
Uniform polyhedra can be divided between
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
forms with convex
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
faces and star forms. Star forms have either regular
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
or
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s or both.
This list includes these:
* all 75 nonprismatic
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
;
* a few representatives of the infinite sets of
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
s and
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
s;
* one
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
polyhedron, Skilling's figure with overlapping edges.
It was proven in that there are only 75
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
other than the infinite families of
prisms and
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
s. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.
Not included are:
* The
uniform polyhedron compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...
s.
* 40 potential
uniform polyhedra with degenerate vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s which have overlapping edges (not counted by
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
);
* The uniform tilings (infinite polyhedra)
** 11 Euclidean
convex uniform tilings;
** 28 Euclidean
nonconvex or apeirogonal uniform tilings;
** Infinite number of
uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
.
* Any
polygons
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
or
4-polytopes
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
Indexing
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
*
''CCoxeter et al., 1954, showed the
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
*
''WWenninger, 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.
*
''KKaleido, 1993: The 80 figures were grouped by symmetry: 1–5 as representatives of the infinite families of prismatic forms with
dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
, 6–9 with
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
, 10–26 with
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
, 27–80 with
icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...
.
*
''UMathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.
Names of polyhedra by number of sides
There are generic
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
names for the most common
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
. The 5
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s are called a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
,
hexahedron
A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.
There ...
,
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
,
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
and
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a
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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
.
Table of polyhedra
The convex forms are listed in order of degree of
s from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.
There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.
Convex uniform polyhedra
Uniform star polyhedra
The forms containing only convex faces are listed first, followed by the forms with star faces. Again infinitely many prisms and antiprisms exist; they are listed here up to the 8-sided ones.
The uniform polyhedra | 3 3, | , | 3, | 3 , and | () (3) have some faces occurring as coplanar pairs. (Coxeter et al. 1954, pp. 423, 425, 426; Skilling 1975, p. 123)
Special case
The great disnub dirhombidodecahedron has 240 of its 360 edges coinciding in space in 120 pairs. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.
Column key
* Uniform indexing: U01–U80 (Tetrahedron first, Prisms at 76+)
* Kaleido software indexing: K01–K80 (K
''n'' = U
''n''–5 for ''n'' = 6 to 80) (prisms 1–5, Tetrahedron etc. 6+)
*
Magnus Wenninger
Father Magnus J. Wenninger Order of Saint Benedict, OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction.
Early life ...
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 characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
, . Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
* Density: the
Density (polytope)
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions,
representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It ...
represents the number of windings of a polyhedron around its center. This is left blank for non-
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
polyhedra and
hemipolyhedra
In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other p ...
(polyhedra with faces passing through their centers), for which the density is not well-defined.
* 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.
See also
*
List of uniform polyhedra by vertex figure
There are many relations among the uniform polyhedra..
Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.
Others share the same vertices and edges as other polyhedron.
The grouping below exhibit some of these r ...
*
List of uniform polyhedra by Wythoff symbol
There are many relations among the uniform polyhedra.
Here they are grouped by the Wythoff symbol.
Key
Regular
All the faces are identical, each edge is identical and each vertex is identical.
They all have a Wythoff symbol of the form p ...
*
List of uniform polyhedra by Schwarz triangle
There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non- dihedral a ...
References
*
*
*
*
*
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://www.mathconsult.ch/showroom/unipoly
*
** https://web.archive.org/web/20171110075259/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)
** https://www.math.technion.ac.il/~rl/kaleido
*** https://web.archive.org/web/20110927223146/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
* Also
** http://www.polyedergarten.de/polyhedrix/e_klintro.htm
{{Nonconvex polyhedron navigator
Uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
ja:一様多面体の一覧