Conway polyhedron notation

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a> and <a href="/facts/Topology/2EqW47cU">topology</a>, Conway polyhedron notation, invented by <a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a> and promoted by <a href="/facts/George_W._Hart/uvnDa47L">George W. Hart</a>, is used to describe <a href="/facts/Polyhedron/UC0p9FTF">polyhedra</a> based on a seed polyhedron modified by various prefix <a href="/facts/Operation_(mathematics)/Yplv602Q">operations</a>.
Conway and Hart extended the idea of using operators, like <a href="/facts/Truncation_(geometry)/DYb589Je">truncation</a> as defined by <a href="/facts/Kepler/wk6GTN3C">Kepler</a>, to build related polyhedra of the same symmetry. For example, tC represents a <a href="/facts/Truncated_cube/XVAUG4fD">truncated cube</a>, and taC, parsed as t(aC), is (<a href="/facts/Topology/2EqW47cU">topologically</a>) a <a href="/facts/Truncated_cuboctahedron/nd2ltxcm">truncated cuboctahedron</a>. The simplest operator <a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a> swaps <a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a> and <a href="/facts/Face_(geometry)/T2E6BTU4">face</a> elements; e.g., a dual <a href="/facts/Cube/65lUaAqN">cube</a> is an <a href="/facts/Octahedron/G38q92xW">octahedron</a>: dC = O. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators a (ambo), b (<a href="/facts/Bevel/ErgYGVs9">bevel</a>), d (<a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a>), e (expand), g (gyro), j (join), k (kis), m (meta), o (ortho), s (<a href="/facts/Snub_(geometry)/T8opxlLu">snub</a>), and t (<a href="/facts/Truncation_(geometry)/DYb589Je">truncate</a>), while Hart added r (<a href="/facts/Reflection_(mathematics)/5JHWnrql">reflect</a>) and p (propellor). Later implementations named further operators, sometimes referred to as "extended" operators. Conway's basic operations are sufficient to generate the <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean</a> and <a href="/facts/Catalan_solid/R8D2cP7D">Catalan solids</a> from the <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a>. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation (aa = e), while a truncation after ambo produces bevel (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 <a href="/facts/Graph_embedding/IyBA3ttM">embeddings</a> of a <a href="/facts/Polyhedral_graph/IKPCDH6r">polyhedral graph</a> 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 <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> 0 (i.e. topologically equivalent to a sphere) are often put into <a href="/facts/Canonical_polyhedron/xHuG5dCI">canonical form</a> to avoid ambiguity.

Conway polyhedron notation open-in-new

Conway polyhedron notation