List Of Uniform Polyhedra
   HOME

TheInfoList



OR:

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a uniform polyhedron is a
polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
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 ...
s as faces and is vertex-transitive ( 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'' me ...
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 modu ...
, 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 (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
. 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 polytop ...
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 ...
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, Decagram (geometry)#Related figures, certain notable ones can ...
faces or
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s or both. This list includes these: * all 75 nonprismatic
uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
; * a few representatives of the infinite sets of prisms and
antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s; * one degenerate 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—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...
other than the infinite families of prisms and
antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
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 t ...
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 general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s which have overlapping edges (not counted by Coxeter); * 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. * Any
polygons In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon' ...
or 4-polytopes


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 polytop ...
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, 6–9 with
tetrahedral symmetry image:tetrahedron.svg, 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 co ...
, 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 polyhedr ...
. * ''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 a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
names for the most common
polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
. The 5
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s are called a
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
,
hexahedron A hexahedron (: hexahedra or hexahedrons) or sexahedron (: 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 are seven ...
,
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
,
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
.


Table of polyhedra

The convex forms are listed in order of degree of
vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
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 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 and education Born to ...
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's ...
, . Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero. * Density: the Density (polytope) represents the number of windings of a polyhedron around its center. This is left blank for non- orientable polyhedra and hemipolyhedra (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 * List of uniform polyhedra by Wythoff symbol * List of uniform polyhedra by Schwarz triangle


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—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...
ja:一様多面体の一覧