Conway polyhedron notation

Method of describing higher order polyhedra

In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, tC represents a truncated cube, and taC, parsed as t(aC), is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e. g., a dual cube is an octahedron: 1=dC = O. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators a (ambo), b (bevel), d (dual), e (expand), g (gyro), j (join), k (kis), m (meta), o (ortho), s (snub), and t (truncate), while Hart added r (reflect) and p (propellor). Later implementations named further operators, sometimes referred to as "extended" operators. Conway's basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation (1=aa = e), while a truncation after ambo produces bevel (1=ta = b). 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 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 0 (i. e. topologically equivalent to a sphere) are often put into canonical form to avoid ambiguity.

+Triangular Goldberg Coxeter operators a b Class Edge factor T = a2 + ab + b2 Matrix Master triangle x xd dx dxd 1 0 I 1 u1 = Sdc1 = S 2 0 I 4 u2 = u dc du c2 = c 3 0 I 9 u3 = nn nk zt c3 = zz 4 0 I 16 u4 = uu uud = dcc duu = ccdc4 = cc 5 0 I 25 u5u5d = dc5du5 = c5d c5 6 0 I 36 u6 = unn unk cztu6 = czz 7 0 I 49 u7 = u2,1u1,2 = vrv vrvd = dwrw dvrv = wrwd c7 = c2,1c1,2 = wrw 8 0 I 64 u8 = u3 u3d = dc3 du3 = c3d c8 = c3 9 0 I 81 u9 = n4 n3k = kz3 tn3 = z3t c9 = z4 1 1 II 3 u1,1 = n k t c1,1 = z 2 1 III 7 v = u2,1 vd = dw dv = wd w = c2,1 3 1 III 13 u3,1u3,1d = dc3,1du3,1 = c3,1d c3,1 3 2 III 19 u3,2 u3,2d = dc3,2 du3,2 = c3,2d c3,2 4 3 III 37 u4,3 u4,3d = dc4,3 du4,3 = c4,3dc4,3 5 4 III 61 u5,4u5,4d = dc5,4du5,4 = c5,4dc5,4 6 5 III 91 u6,5 = u1,2u1,3u6,5d = dc6,5du6,5 = c6,5dc6,5=c1,2c1,3 7 6 III 127 u7,6u7,6d = dc7,6du7,6 = c7,6dc7,6 8 7 III 169 u8,7 = u3,12u8,7d = dc8,7du8,7 = c8,7dc8,7 = c3,12 9 8 III 217 u9,8 = u2,1u5,1u9,8d = dc9,8du9,8 = c9,8dc9,8 = c2,1c5,1 I, II, or III ua,bua,bd = dca,bdua,b = ca,bdca,b I or III ua,bua,bd = dca,bdua,b = ca,bdca,b
By basic number theory, for any values of a and b, .

+Quadrilateral Goldberg Coxeter operators a b Class Edge factor T = a2 + b2 Matrix Master square x xd dx dxd 1 0 I 1 o1 = Se1 = do1 = dd = S 2 0 I 4 o2 = o = j2 e2 = e = a2 3 0 I 9 o3 e3 o3 4 0 I 16 o4 = oo = j4 e4 = ee = a4 5 0 I 25 o5 = o2,1o1,2 = prpe5 = e2,1e1,2o5= dprpd 6 0 I 36 o6 = o2o3e6 = e2e3 7 0 I 49 o7e7o7 8 0 I 64 o8 = o3 = j6 e8 = e3 = a6 9 0 I 81 o9 = o32 e9 = e32o9 10 0 I 100 o10 = oo2,1o1,2e10 = ee2,1e1,2 1 1 II 2 o1,1 = j e1,1 = a 2 2 II 8 o2,2 = j3 e2,2 = a3 1 2 III 5 o1,2 = p e1,2 = dp = pd p I, II, or III T even oa,bea,b I or III T odd oa,bea,boa,b

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.)

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.

Each of the convex uniform tilings and their duals can be created by applying Conway operators to the regular tilings Q, H, and Δ.

Conway operators can also be applied to toroidal polyhedra and polyhedra with multiple holes.