Conway polyhedron notation
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.
[1][2] Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry.
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).
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).
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.
In Conway's notation, operations on polyhedra are applied like functions, from right to left.
Individual operators can be visualized in terms of fundamental domains (or chambers), as below.
Hart introduced the reflection operator r, that gives the mirror image of the polyhedron.
[6] 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.
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 do not necessarily produce a convex polyhedron from a convex seed.
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).
A number of operators can be grouped together by some criteria, or have their behavior modified by an index.
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, a chamfered cube, cC, can be constructed as t4daC, as a rhombic dodecahedron, daC or jC, with its degree-4 vertices truncated.
A quinto-dodecahedron, qD can be constructed as t5daaD or t5deD or t5oD, a deltoidal hexecontahedron, deD or oD, with its degree-5 vertices truncated.
[4] Medial is like meta, except it does not add edges from the center to each seed vertex.
[16][17] 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").
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.
