User:Tomruen/Semioperators

Coxeter operators
Coxeter/Johnson operators are sometimes useful to mix with Conway operators. For clarity in Conway notation these operations are given uppercase symbolic letter. Coxeter's t-notation defines active rings as indices a Coxeter-Dynkin diagram. So here a capital T with indices 0,1,2 define the uniform operators from a regular seed. The zero index cab ne see to represent vertices, 1 represents edges, and 2 represents faces. With T = T0,1 is an ordinary truncation, and R = T1 is a full truncation, or rectify, the same as Conway's ambo operator. For example, r{4,3} or t1{4,3} is Coxeter's name for a cuboctahedron, a rectified cube is RC, the same as Conway's ambo cube, aC.

Semioperators
.

Coxeter's semi or demi operator, H for Half, reduces faces into half as many sides, and quadrilateral faces into digons, with two coinciding edges, which may or may not be replaced by a single edge. For example, a half cube, h{4,3}, also called a demicube, is HC, representing one of two tetrahedra. Ho reduces an ortho to ambo/Rectify.

Other semi-operators can be defined using the H operator. Conway calls Coxeter's Snub operation S, a semi-snub, defined as Ht. Conway's snub operator s is defined as SR. For example, SRC is a snub cube, sr{4,3}. Coxeter's snub octahedron, s{3,4} can be defined as SO, a pyritohedral symmetry construction of the regular icosahedron. It also is consistent with the Johnson solid snub square antiprism as SA4.

A semi-gyro operator, G, is defined as here dHt. This allows Conway's gyro g to be defined as GR. For example, GRC is a gyro-cube, gC or a pentagonal icositetrahedron. And GO defines a pyritohedron with pyritohedral symmetry, while gT, a gyro tetrahedron defines the same topological polyhedron with tetrahedral symmetry.

Both of these operators, S and G, require an even-valence seed polyhedra. In all of these semi-operations, there are two choices of alternated vertices within the half operator. These two construction are not topologically identical in the general case. For example, HjC ambiguously defines either a cube or octahedron, depending on which set of vertices are taken.

Other operators only apply to polyhedra with all even-sided faces. The simplest is the semi-join operator, as the conjugate operator of half, dHd.

A semi-ortho operator, F, is a conjugate operator to semi-snub. It adds a vertex in the center of the faces, and bisects all edges, but only connects new edges from each center to half of the edges, creating new hexagonal faces. Original square faces do not require the central vertex and need only a single edge across the face, creating pairs of pentagons. For example, a dodecahedron, tetartoid, can be constructed as FC.

A semi-expand operator, E, is defined as Htd or Hz. This creates triangular faces. For example, EC created a pyritohedral symmetry construction of a regular pseudoicosahedron.