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 has

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

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 ...

(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 mod ...

. Uniform

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 ...

may be regular (if also face- and

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...

), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be

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 ...

, so many of the uniform polyhedra are also star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra: *Infinite classes: ** prisms, **

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. * Convex exceptional: ** 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 e ...

s: regular convex polyhedra, ** 13

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s: 2 quasiregular and 11 semiregular convex polyhedra. * Star (nonconvex) exceptional: ** 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra, ** 53

uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

: 14 quasiregular and 39 semiregular. Hence 5 + 13 + 4 + 53 = 75. There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the

great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antipr ...

(Skilling's figure).

Dual polyhedra In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

to uniform polyhedra are

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

(isohedral) and have regular

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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

s, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan s ...

. The concept of uniform polyhedron is a special case of the concept of

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...

, which also applies to shapes in higher-dimensional (or lower-dimensional) space.

Definition

define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra. gave a rather complicated definition of a polyhedron, while gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional

abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: *Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra. *Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. *Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra. *Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction. *Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra.

History

Regular convex polyhedra

* The

s date back to the classical Greeks and were studied by the

Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

(c. 424 – 348 BC), Theaetetus (c. 417 BC – 369 BC),

Timaeus of Locri Timaeus of Locri (; grc, Τίμαιος ὁ Λοκρός, Tímaios ho Lokrós; la, Timaeus Locrus) is a character in two of Plato's dialogues, ''Timaeus (dialogue), Timaeus'' and ''Critias (dialogue), Critias''. In both, he appears as a Ancient G ...

(ca. 420–380 BC) and

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

(fl. 300 BC). The

Etruscans The Etruscan civilization () was developed by a people of Etruria in ancient Italy with a common language and culture who formed a federation of city-states. After conquering adjacent lands, its territory covered, at its greatest extent, rou ...

discovered the regular dodecahedron before 500 BC.

Nonregular uniform convex polyhedra

* The

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

was known by

. * Archimedes (287 BC – 212 BC) discovered all of the 13

s. His original book on the subject was lost, but

Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

(c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra. *

Piero della Francesca Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...

(1415 – 1492) rediscovered the five truncations of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron, and included illustrations and calculations of their metric properties in his book ''

De quinque corporibus regularibus ''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is ...

''. He also discussed the cuboctahedron in a different book. * Luca Pacioli plagiarized Francesca's work in ''

De divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da ...

'' in 1509, adding the

rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...

, calling it a ''icosihexahedron'' for its 26 faces, which was drawn by

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...

. * Johannes Kepler (1571–1630) was the first to publish the complete list of

s, in 1619. He also identified the infinite families of uniform prisms and antiprisms.

Regular star polyhedra

Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...

(1619) discovered two of the regular Kepler–Poinsot polyhedra, the

small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...

and

great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

. *

Louis Poinsot Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a c ...

(1809) discovered the other two, the

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

and

great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...

. *The set of four were proven complete by Augustin-Louis Cauchy in 1813, and named by Arthur Cayley in 1859.

Other 53 nonregular star polyhedra

* Of the remaining 53,

Edmund Hess Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. See also * Schläfli–Hess polychoron * Hess polytope References * ''Regular Polytopes In mathematics, a re ...

(1878) discovered two, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra. * The geometer H.S.M. Coxeter discovered the remaining twelve in collaboration with J. C. P. Miller (1930–1932) but did not publish. M.S. Longuet-Higgins and

H.C. Longuet-Higgins Hugh Christopher Longuet-Higgins (April 11, 1923 – March 27, 2004) was a British scholar and teacher. He was the Professor of Theoretical Chemistry at the University of Cambridge for 13 years until 1967 when he moved to the University of Edin ...

independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. * published the list of uniform polyhedra. * proved their conjecture that the list was complete. * In 1974,

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 Ge ...

published his book ''Polyhedron models'', which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson. * independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility (the

). * In 1987,

Edmond Bonan Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy. Biography After completing his undergraduate studies ...

drew all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca: almost of them were shown during the International Stereoscopic Union Congress held at the Congress Theatre, Eastbourne, United Kingdom.. * In 1993, Zvi Har'El (1949–2008) produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper ''Uniform Solution for Uniform Polyhedra'', counting figures 1-80. * Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system. * In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its

Wythoff symbol In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform pol ...

Uniform star polyhedra

The 57 nonprismatic nonconvex forms, with exception of the

great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as . It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices. This is ...

, are compiled by Wythoff constructions within

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...

Convex forms by Wythoff construction

The convex uniform polyhedra can be named by

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

operations on the regular form. In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group. Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The

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 ...

is a regular polyhedron, and a triangular antiprism. The

is also a ''rectified tetrahedron''. Many polyhedra are repeated from different construction sources, and are colored differently. The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune hav ...

s and

dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

s which are degenerate polyhedra. These symmetry groups are formed from the reflectional

point groups in three dimensions In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometrie ...

, each represented by a fundamental triangle (''p'' ''q'' ''r''), where ''p'' > 1, ''q'' > 1, ''r'' > 1 and . *

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 ...

(3 3 2) – order 24 *

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 ...

(4 3 2) – order 48 *

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 ...

(5 3 2) – order 120 *

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, g ...

(''n'' 2 2), for ''n'' = 3,4,5,... – order 4''n'' The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides. Along with the prisms and their

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, g ...

, the spherical Wythoff construction process adds two ''regular'' classes which become degenerate as polyhedra : the ''

dihedra A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

'' and the ''

hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...

'', the first having only two faces, and the second only two vertices. The truncation of the regular ''hosohedra'' creates the prisms. Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form. For the infinite set of prismatic forms, they are indexed in four families: #

Hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...

''H''_2... (only as spherical tilings) #

Dihedra A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

''D''_2... (only as spherical tilings) # Prisms ''P''_3... (truncated hosohedra) #

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 ''A''_3... (snub prisms)

Summary tables

And a sampling of dihedral symmetries: (The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)

(3 3 2) T_d tetrahedral symmetry

The

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 ...

of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation. The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

A₂ or ,3 as well as a

Coxeter diagram 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 t ...

: . There are 24 triangles, visible in the faces of the tetrakis hexahedron, and in the alternately colored triangles on a sphere: : Tetrakishexahedron

(4 3 2) O_h octahedral symmetry

The

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 ...

of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above. The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the

B₂ or ,3 as well as a

: . There are 48 triangles, visible in the faces of the

disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is fa ...

, and in the alternately colored triangles on a sphere: : Disdyakisdodecahedron

(5 3 2) I_h icosahedral symmetry

The

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 ...

of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above. The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the

G₂ or ,3 as well as a

: . There are 120 triangles, visible in the faces of the disdyakis triacontahedron, and in the alternately colored triangles on a sphere: Disdyakistriacontahedron

(p 2 2) Prismatic ,2 I₂(p) family (D_''p''h dihedral symmetry)

The

of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere. The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the

I₂(p) or ,2 as well as a prismatic

: . Below are the first five dihedral symmetries: D₂ ... D₆. The dihedral symmetry D_p has order ''4n'', represented the faces of a bipyramid, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.

(2 2 2) Dihedral symmetry

There are 8 fundamental triangles, visible in the faces of the

square bipyramid 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 ...

(Octahedron) and alternately colored triangles on a sphere: :

(3 2 2) D_3h dihedral symmetry

There are 12 fundamental triangles, visible in the faces of the

hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitiv ...

and alternately colored triangles on a sphere: : Hexagonale bipiramide

(4 2 2) D_4h dihedral symmetry

There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a sphere: :

(5 2 2) D_5h dihedral symmetry

There are 20 fundamental triangles, visible in the faces of the

decagonal bipyramid In geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If a decagonal bipyramid is to be face-transitive, all faces must be isosceles triangles. It is an icosahedron, but not the regular one ...

and alternately colored triangles on a sphere: :

(6 2 2) D_6h dihedral symmetry

There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere.

Wythoff construction operators

Notes

References

* Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900

* * * * * *

External links

*
Uniform Solution for Uniform PolyhedraThe Uniform Polyhedra
Uniform Polyhedra
Uniform polyhedron gallery
''Has a visual chart of all 75'' {{Polytopes Uniform polyhedra, *