Conway polyhedron notation

In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a cuboctahedron is an ambo cube, i.e. , and a truncated cuboctahedron is . Repeated application of an operator can be denoted with an exponent: j2 = o. In general, Conway operators are not commutative. Individual operators can be visualized in terms of fundamental domains (or chambers), as below. Each right triangle is a fundamental domain. 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 groups where n is the number of sides of a face, while chiral operators correspond to cyclic groups lacking the reflective symmetry of the dihedral groups. Achiral and chiral 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. == 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 Platonic solids are represented by the first letter of their name (Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron); the prisms (Pn) for n-gonal forms; antiprisms (An); cupolae (Un); anticupolae (Vn); and pyramids (Yn). Any Johnson solid can be referenced as Jn, for n=1..92. All of the five Platonic solids can be generated from prismatic generators with zero to two operators: • Triangular pyramid: Y3 (A tetrahedron is a special pyramid) • T = Y3 • O = aT (ambo tetrahedron) • C = jT (join tetrahedron) • I = sT (snub tetrahedron) • D = gT (gyro tetrahedron) • Triangular antiprism: A3 (An octahedron is a special antiprism) • O = A3 • C = dA3 • Square prism: P4 (A cube is a special prism) • C = P4 • Pentagonal antiprism: A5 • I = k5A5 (A special gyroelongated dipyramid) • D = t5dA5 (A special truncated trapezohedron) The regular Euclidean tilings can also be used as seeds: • Q = Quadrille = Square tiling • H = Hextille = Hexagonal tiling = dΔ • Δ = Deltille = Triangular tiling = 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). In addition, only irreducible operators are included in this list; many 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. 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 and geodesic polyhedra: see List of geodesic polyhedra and Goldberg polyhedra for formulas. The two families are the triangular GC family, ca,b and ua,b, and the quadrilateral GC family, ea,b and oa,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 domains over the complex numbers: the Eisenstein integers for the triangular GC family, and the Gaussian integers 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. ca,0 = ca. • Class II: . Also achiral. Can be decomposed as ca,a = cac1,1 • Class III: All other operators. These are chiral, and ca,b and cb,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{mod}\ 3). Quadrilateral ==Examples==

Archimedean and Catalan solids Conway's original set of operators can create all of the Archimedean solids and Catalan solids, using the Platonic solids as seeds. (Note that the r operator is not necessary to create both chiral forms.) Image:truncated tetrahedron.png| Truncated tetrahedrontT Image:cuboctahedron.png| CuboctahedronaC = aO = eT Image:truncated hexahedron.png| Truncated cubetC Image:truncated octahedron.png| Truncated octahedrontO = bT Image:small rhombicuboctahedron.png| RhombicuboctahedroneC = eO Image:Great rhombicuboctahedron.png| truncated cuboctahedronbC = bO Image:snub hexahedron.png| snub cubesC = sO Image:icosidodecahedron.png| icosidodecahedronaD = aI Image:truncated dodecahedron.png| truncated dodecahedrontD Image:truncated icosahedron.png| truncated icosahedrontI Image:small rhombicosidodecahedron.png| rhombicosidodeca­hedroneD = eI Image:Great rhombicosidodecahedron.png| truncated icosidodecahedronbD = bI Image:snub dodecahedron ccw.png| snub dodecahedronsD = sI Image:triakistetrahedron.svg| Triakis tetrahedronkT Image:rhombicdodecahedron.jpg| Rhombic dodecahedronjC = jO = oT Image:triakisoctahedron.jpg| Triakis octahedronkO Image:tetrakishexahedron.jpg| Tetrakis hexahedronkC = mT Image:deltoidalicositetrahedron.jpg| Deltoidal icositetrahedronoC = oO Image:disdyakisdodecahedron.jpg| Disdyakis dodecahedronmC = mO Image:pentagonalicositetrahedronccw.jpg| Pentagonal icositetrahedrongC = gO Image:rhombictriacontahedron.svg| Rhombic triacontahedronjD = jI Image:triakisicosahedron.jpg| Triakis icosahedronkI Image:Pentakisdodecahedron.jpg| Pentakis dodecahedronkD Image:Deltoidalhexecontahedron.jpg| Deltoidal hexecontahedronoD = oI Image:Disdyakistriacontahedron.jpg| Disdyakis triacontahedronmD = mI Image:Pentagonalhexecontahedronccw.jpg| Pentagonal hexecontahedrongD = gI Composite operators The truncated icosahedron, tI, can be used as a seed to create some more visually-pleasing polyhedra, although these are neither vertex nor face-transitive. File:Uniform polyhedron-53-t12.svg|tI File:Rectified truncated icosahedron1.svg|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 Q, H, and Δ. File:1-uniform_n5.svg|Square tilingQ = dQ = aQ = eQ= jQ = oQ File:1-uniform_n2.svg|Truncated square tilingtQ = bQ File:1-uniform_2_dual.svg|Tetrakis square tilingkQ = mQ File:1-uniform_n9.svg|Snub square tilingsQ File:1-uniform_9_dual.svg|Cairo pentagonal tilinggQ File:1-uniform_n1.svg|Hexagonal tilingH = dΔ = tΔ File:1-uniform_n7.svg|Trihexagonal tilingaH = aΔ File:1-uniform_n4.svg|Truncated hexagonal tilingtH File:1-uniform_n6.svg|Rhombitrihexagonal tilingeH = eΔ File:1-uniform_n3.svg|Truncated trihexagonal tilingbH = bΔ File:1-uniform_n10.svg|Snub trihexagonal tilingsH = sΔ File:1-uniform_1_dual.svg|Triangle tilingΔ = dH = kH File:1-uniform_7_dual.svg|Rhombille tilingjΔ = jH File:1-uniform_4_dual1.svg|Triakis triangular tilingkΔ File:1-uniform_6_dual.svg|Deltoidal trihexagonal tilingoΔ = oH File:1-uniform_3_dual.svg|Kisrhombille tilingmΔ = mH File:1-uniform_10_dual.svg|Floret pentagonal tilinggΔ = gH On a torus Conway operators can also be applied to toroidal polyhedra and polyhedra with multiple holes. File:Toroidal monohedron.png|A 1x1 regular square torus, {4,4}1,0 File:Torus map 4x4.png|A regular 4x4 square torus, {4,4}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 == See also ==