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 ...

, Conway polyhedron notation, invented by

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

and promoted by

George W. Hart George William Hart (born 1955) is an American sculptor and geometer. Before retiring, he was an associate professor of Electrical Engineering at Columbia University in New York City and then an interdepartmental research professor at Stony B ...

, is used to describe

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 ...

based on a seed polyhedron modified by various prefix

operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...

s. Conway and Hart extended the idea of using operators, like

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

as defined by

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 ...

, to build related polyhedra of the same symmetry. For example, represents a

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

, and , parsed as , is ( topologically) a

truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...

. The simplest operator dual swaps

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

and

face 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 aff ...

elements; e.g., a dual

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 ...

is an

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 ...

: . Applied in a series, these operators allow many

higher order polyhedra Higher may refer to: Music * The Higher, a 2002–2012 American pop rock band Albums * ''Higher'' (Ala Boratyn album) or the title song, 2007 * ''Higher'' (Ezio album) or the title song, 2000 * ''Higher'' (Harem Scarem album) or the title song ...

to be generated. Conway defined the operators (ambo), (

bevel A bevelled edge (UK) or beveled edge (US) is an edge of a structure that is not perpendicular to the faces of the piece. The words bevel and chamfer overlap in usage; in general usage they are often interchanged, while in technical usage they ...

), ( dual), (expand), (gyro), (join), (kis), (meta), (ortho), (

snub A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" ...

), and (

truncate In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

), while Hart added ( reflect) and (propellor). Later implementations named further operators, sometimes referred to as "extended" operators. Conway's basic operations are sufficient to generate the Archimedean and

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 sol ...

s from the

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. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation (), while a truncation after ambo produces bevel (). Polyhedra can be studied topologically, in terms of how their vertices, edges, and faces connect together, or geometrically, in terms of the placement of those elements in space. Different implementations of these operators may create polyhedra that are geometrically different but topologically equivalent. These topologically equivalent polyhedra can be thought of as one of many embeddings of a

polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-con ...

on the sphere. Unless otherwise specified, in this article (and in the literature on Conway operators in general) topology is the primary concern. Polyhedra with

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

0 (i.e. topologically equivalent to a sphere) are often put into

canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...

to avoid ambiguity.

Operators

In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a

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 ...

is an ''ambo cube'', (See fourth row in table, "a = ambo".) i.e. , and a

is . Repeated application of an operator can be denoted with an exponent: ''j²'' = ''o''. In general, Conway operators are not

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

. Individual operators can be visualized in terms of

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

s (or chambers), as below. Each right triangle is a

. Each white chamber is a rotated version of the others, and so is each colored chamber. For achiral operators, the colored chambers are a reflection of the white chambers, and all are transitive. In group terms, achiral operators correspond to

dihedral group 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 ...

s where ''n'' is the number of sides of a face, while chiral operators correspond to

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

s lacking the reflective symmetry of the dihedral groups. Achiral and

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively. LSPs should be understood as local operations that preserve symmetry, not operations that preserve local symmetry. Again, these are symmetries in a topological sense, not a geometric sense: the exact angles and edge lengths may differ. Hart introduced the reflection operator ''r'', that gives the mirror image of the polyhedron. This is not strictly a LOPSP, since it does not preserve orientation: it reverses it, by exchanging white and red chambers. ''r'' has no effect on achiral polyhedra aside from orientation, and ''rr = S'' returns the original polyhedron. An overline can be used to indicate the other chiral form of an operator: = ''rsr''. An operation is irreducible if it cannot be expressed as a composition of operators aside from ''d'' and ''r''. The majority of Conway's original operators are irreducible: the exceptions are ''e'', ''b'', ''o'', and ''m''.

Matrix representation

The relationship between the number of vertices, edges, and faces of the seed and the polyhedron created by the operations listed in this article can be expressed as a matrix

\mathbf_x

. When ''x'' is the operator,

v,e,f

are the vertices, edges, and faces of the seed (respectively), and

v',e',f'

are the vertices, edges, and faces of the result, then :

\mathbf_x \begin v \\ e \\ f \end = \begin v' \\ e' \\ f' \end

. The matrix for the composition of two operators is just the product of the matrixes for the two operators. Distinct operators may have the same matrix, for example, ''p'' and ''l''. The edge count of the result is an integer multiple ''d'' of that of the seed: this is called the inflation rate, or the edge factor. The simplest operators, the

identity operator Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), a ...

''S'' and the dual operator ''d'', have simple matrix forms: :

\mathbf_S = \begin
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  0 & 0 & 1
  \end = \mathbf_3

\mathbf_d = \begin
  0 & 0 & 1 \\
  0 & 1 & 0 \\
  1 & 0 & 0
  \end

Two dual operators cancel out; ''dd'' = ''S'', and the square of

\mathbf_d

is the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

. When applied to other operators, the dual operator corresponds to horizontal and vertical reflections of the matrix. Operators can be grouped into groups of four (or fewer if some forms are the same) by identifying the operators ''x'', ''xd'' (operator of dual), ''dx'' (dual of operator), and ''dxd'' (conjugate of operator). In this article, only the matrix for ''x'' is given, since the others are simple reflections.

Number of operators

The number of LSPs for each inflation rate is

2, 2, 4, 6, 6, 20, 28, 58, 82, \cdots

starting with inflation rate 1. However, not all LSPs necessarily produce a polyhedron whose edges and vertices form a 3-connected graph, and as a consequence of

Steinitz's theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar grap ...

do not necessarily produce a convex polyhedron from a convex seed. The number of 3-connected LSPs for each inflation rate is

2, 2, 4, 6, 4, 20, 20, 54, 64, \cdots

Original operations

Strictly, seed (''S''), needle (''n''), and zip (''z'') were not included by Conway, but they are related to original Conway operations by duality so are included here. From here on, operations are visualized on cube seeds, drawn on the surface of that cube. Blue faces cross edges of the seed, and pink faces lie over vertices of the seed. There is some flexibility in the exact placement of vertices, especially with chiral operators.

Seeds

Any polyhedron can serve as a seed, as long as the operations can be executed on it. Common seeds have been assigned a letter. The

s are represented by the first letter of their name ( Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron); the prisms (P_''n'') for ''n''-gonal forms; antiprisms (A_''n''); cupolae (U_''n''); anticupolae (V_''n''); and pyramids (Y_''n''). Any Johnson solid can be referenced as J_''n'', for ''n''=1..92. All of the five Platonic solids can be generated from prismatic generators with zero to two operators: *

Triangular pyramid 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 ...

: ''Y''₃ (A tetrahedron is a special pyramid) ** '' T'' = ''Y''₃ ** '' O'' = ''aT'' (ambo tetrahedron) ** '' C'' = ''jT'' (join tetrahedron) ** '' I'' = ''sT'' (snub tetrahedron) ** '' D'' = ''gT'' (gyro tetrahedron) * Triangular antiprism: ''A''₃ (An octahedron is a special antiprism) ** ''O'' = ''A''₃ ** ''C'' = ''dA''₃ *

Square prism In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...

: ''P''₄ (A cube is a special prism) ** ''C'' = ''P''₄ *

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''₅ ** ''I'' = ''k''₅''A''₅ (A special

gyroelongated dipyramid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson ...

) ** ''D'' = ''t''₅''dA''₅ (A special

truncated trapezohedron In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism. T ...

) The regular Euclidean tilings can also be used as seeds: * ''Q'' = Quadrille =

Square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the s ...

* ''H'' = Hextille =

Hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathemat ...

= ''dΔ'' * ''Δ'' = Deltille =

Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

= ''dH''

Extended operations

These are operations created after Conway's original set. Note that many more operations exist than have been named; just because an operation is not here does not mean it does not exist (or is not an LSP or LOPSP). To simplify, only irreducible operators are included in this list: others can be created by composing operators together.

Indexed extended operations

A number of operators can be grouped together by some criteria, or have their behavior modified by an index. These are written as an operator with a subscript: ''x_n''.

Augmentation

Augmentation operations retain original edges. They may be applied to any independent subset of faces, or may be converted into a ''join''-form by removing the original edges. Conway notation supports an optional index to these operators: 0 for the join-form, or 3 or higher for how many sides affected faces have. For example, ''k''₄''Y''₄=O: taking a square-based pyramid and gluing another pyramid to the square base gives an octahedron. The truncate operator ''t'' also has an index form ''t_n'', indicating that only vertices of a certain degree are truncated. It is equivalent to ''dk_nd''. Some of the extended operators can be created in special cases with ''k_n'' and ''t_n'' operators. For example, a

chamfered cube In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operatio ...

, ''cC'', can be constructed as ''t''₄''daC'', as a

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

, ''daC'' or ''jC'', with its degree-4 vertices truncated. A lofted cube, ''lC'' is the same as ''t''₄''kC''. A quinto-dodecahedron, ''qD'' can be constructed as ''t''₅''daaD'' or ''t''₅''deD'' or ''t''₅''oD'', a deltoidal hexecontahedron, ''deD'' or ''oD'', with its degree-5 vertices truncated.

Meta/Bevel

Meta adds vertices at the center and along the edges, while bevel adds faces at the center, seed vertices, and along the edges. The index is how many vertices or faces are added along the edges. Meta (in its non-indexed form) is also called

cantitruncation In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...

omnitruncation In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...

. Note that 0 here does not mean the same as for augmentation operations: it means zero vertices (or faces) are added along the edges.

Medial

Medial is like meta, except it does not add edges from the center to each seed vertex. The index 1 form is identical to Conway's ortho and expand operators: expand is also called

cantellation In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...

and

expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

. Note that ''o'' and ''e'' have their own indexed forms, described below. Also note that some implementations start indexing at 0 instead of 1.

Goldberg-Coxeter

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction. The GC construction can be thought of as taking a triangular section of a triangular lattice, or a square section of a square lattice, and laying that over each face of the polyhedron. This construction can be extended to any face by identifying the chambers of the triangle or square (the "master polygon"). Operators in the triangular family can be used to produce the

Goldberg polyhedra In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ...

and

geodesic polyhedra A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhed ...

: see

List of geodesic polyhedra and Goldberg polyhedra This is a list of selected geodesic polyhedron, geodesic polyhedra and Goldberg polyhedron, Goldberg polyhedra, two infinite classes of polyhedron, polyhedra. Geodesic polyhedra and Goldberg polyhedra are polyhedral dual, duals of each other. The g ...

for formulas. The two families are the triangular GC family, ''c_a,b'' and ''u_a,b'', and the quadrilateral GC family, ''e_a,b'' and ''o_a,b''. Both the GC families are indexed by two integers

a \ge 1

and

b \ge 0

. They possess many nice qualities: * The indexes of the families have a relationship with certain

Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...

s over the complex numbers: the

Eisenstein integers In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form :z = a + b\omega , where and are integers and :\omega = \fr ...

for the triangular GC family, and the

Gaussian integers In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...

for the quadrilateral GC family. * Operators in the ''x'' and ''dxd'' columns within the same family commute with each other. The operators are divided into three classes (examples are written in terms of ''c'' but apply to all 4 operators): * Class I: . Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. ''c''_''a'',0 = ''c_a''. * Class II: . Also achiral. Can be decomposed as ''c_a,a'' = ''c_ac''_1,1 * Class III: All other operators. These are chiral, and ''c_a,b'' and ''c_b,a'' are the chiral pairs of each other. Of the original Conway operations, the only ones that do not fall into the GC family are ''g'' and ''s'' (gyro and snub). Meta and bevel (''m'' and ''b'') can be expressed in terms of one operator from the triangular family and one from the quadrilateral family.

Triangular

By basic number theory, for any values of ''a'' and ''b'',

T \not\equiv 2\ (\mathrm\ 3)

Quadrilateral

Examples

Archimedean and Catalan solids

Conway's original set of operators can create all of the

Archimedean solids 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 compose ...

and

Catalan solids Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #1 ...

, using the

Platonic solids 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 ...

as seeds. (Note that the ''r'' operator is not necessary to create both chiral forms.) Image:truncated tetrahedron.png,

Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...

''tT'' Image:cuboctahedron.png,

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 ...

''aC'' = ''aO'' = ''eT'' Image:truncated hexahedron.png,

Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

''tC'' Image:truncated octahedron.png,

Truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

''tO'' = ''bT'' Image:small rhombicuboctahedron.png,

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 eac ...

''eC'' = ''eO'' Image:Great rhombicuboctahedron.png,

''bC'' = ''bO'' Image:snub hexahedron.png,

snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...

''sC'' = ''sO'' Image:icosidodecahedron.png,

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...

''aD'' = ''aI'' Image:truncated dodecahedron.png,

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 This polyhedron can be formed from a regular dodecahedron by truncat ...

''tD'' Image:truncated icosahedron.png,

truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. ...

''tI'' Image:small rhombicosidodecahedron.png, rhombicosidodecahedron
''eD'' = ''eI'' Image:Great rhombicosidodecahedron.png,

truncated icosidodecahedron In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Wooda ...

''bD'' = ''bI'' Image:snub dodecahedron ccw.png,

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. The snub dodecahedron has 92 faces (the most ...

''sD'' = ''sI'' Image:triakistetrahedron.jpg,

Triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be see ...

''kT'' Image:rhombicdodecahedron.jpg,

Rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

''jC'' = ''jO'' = ''oT'' Image:triakisoctahedron.jpg,

Triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangular ...

''kO'' Image:tetrakishexahedron.jpg, Tetrakis hexahedron
''kC'' = ''mT'' Image:deltoidalicositetrahedron.jpg,

Deltoidal icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...

''oC'' = ''oO'' Image:disdyakisdodecahedron.jpg,

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 ...

''mC'' = ''mO'' Image:pentagonalicositetrahedronccw.jpg, Pentagonal icositetrahedron
''gC'' = ''gO'' Image:rhombictriacontahedron.jpg,

Rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

''jD'' = ''jI'' Image:triakisicosahedron.jpg,

Triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 po ...

''kI'' Image:Pentakisdodecahedron.jpg,

Pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that i ...

''kD'' Image:Deltoidalhexecontahedron.jpg, Deltoidal hexecontahedron
''oD'' = ''oI'' Image:Disdyakistriacontahedron.jpg, Disdyakis triacontahedron
''mD'' = ''mI'' Image:Pentagonalhexecontahedronccw.jpg,

Pentagonal hexecontahedron In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catala ...

''gD'' = ''gI''

Composite operators

The

, ''tI'', can be used as a seed to create some more visually-pleasing polyhedra, although these are neither

nor

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 congruent ...

. File:Uniform polyhedron-53-t12.svg, ''tI'' File:Rectified truncated icosahedron.png, ''atI'' File:truncated truncated icosahedron.png, ''ttI'' File:Conway polyhedron Dk6k5tI.png, ''ztI'' = ''ttD'' File:Expanded truncated icosahedron.png, ''etI'' File:Truncated rectified truncated icosahedron.png, ''btI'' File:Snub rectified truncated icosahedron.png, ''stI'' File:Pentakisdodecahedron.jpg, ''dtI'' = ''nI'' = ''kD'' File:Joined truncated icosahedron.png, ''jtI'' File:kissed kissed dodecahedron.png, ''ntI'' = ''kkD'' File:Conway polyhedron K6k5tI.png, ''ktI'' File:ortho truncated icosahedron.png, ''otI'' File:Meta_truncated_icosahedron.png, ''mtI'' File:Gyro_truncated_icosahedron.png, ''gtI''

On the plane

Each of the convex uniform tilings and their duals can be created by applying Conway operators to the

regular tilings This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ' ...

''Q'', ''H'', and ''Δ''. File:1-uniform_n5.svg,

''Q'' = ''dQ'' = ''aQ'' = ''eQ''
= ''jQ'' = ''oQ'' File:1-uniform_n2.svg,

Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling, semiregular tiling by regular polygons of the Euclidean plane with one square (geometry), square and two octagons on each vertex (geometry), vertex. This is the only edge-to-edge ti ...

''tQ'' = ''bQ'' File:1-uniform_2_dual.svg,

Tetrakis square tiling In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed b ...

''kQ'' = ''mQ'' File:1-uniform_n9.svg,

Snub square tiling In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is ''s''. Conway calls it a snub quadrille, constructed by a snub operation applie ...

''sQ'' File:1-uniform_9_dual.svg,

Cairo pentagonal tiling In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called Ma ...

''gQ'' File:1-uniform_n1.svg,

''H'' = ''dΔ'' = ''tΔ'' File:1-uniform_n7.svg,

Trihexagonal tiling In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2. ...

''aH'' = ''aΔ'' File:1-uniform_n4.svg,

Truncated hexagonal tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a he ...

''tH'' File:1-uniform_n6.svg,

Rhombitrihexagonal tiling In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 2 ...

''eH'' = ''eΔ'' File:1-uniform_n3.svg,

Truncated trihexagonal tiling In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''. Names Uniform colorings The ...

''bH'' = ''bΔ'' File:1-uniform_n10.svg,

Snub trihexagonal tiling In geometry, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a ...

''sH'' = ''sΔ'' File:1-uniform_1_dual.svg,

Triangle tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular Tessellation, tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of ...

''Δ'' = ''dH'' = ''kH'' File:1-uniform_7_dual.svg,

Rhombille tiling In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape a ...

''jΔ'' = ''jH'' File:1-uniform_4_dual.svg,

Triakis triangular tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a he ...

''kΔ'' File:1-uniform_6_dual.svg,

Deltoidal trihexagonal tiling In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 200 ...

''oΔ'' = ''oH'' File:1-uniform_3_dual.svg,

Kisrhombille tiling In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''. Names Uniform colorings The ...

''mΔ'' = ''mH'' File:1-uniform_10_dual.svg, Floret pentagonal tiling
''gΔ'' = ''gH''

On a torus

Conway operators can also be applied to

toroidal polyhedra In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedr ...

and polyhedra with multiple holes. File:Toroidal monohedron.png, A 1x1 regular square torus, _1,0 File:Torus map 4x4.png, A regular 4x4 square torus, _4,0 File:First truncated square tiling on torus24x12.png, tQ24×12 projected to torus File:Truncated square tiling on torus24x12.png, taQ24×12 projected to torus File:Conway_torus_ActQ24x8.png, actQ24×8 projected to torus File:Truncated hexagonal tiling torus24x12.png, tH24×12 projected to torus File:Truncated trihexagonal tiling on torus24x8.png, taH24×8 projected to torus Conway torus kH24-12.png, kH24×12 projected to torus

References

External links

polyHédronisme
generates polyhedra in HTML5 canvas, taking Conway notation as input {{Polyhedron navigator Elementary geometry Polyhedra Mathematical notation John Horton Conway