# Conway polyhedron notation

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Conway polyhedron notation is a textual notation for a certain class of convex polyhedra. Introduced by Conway, the applications of the notation to date have been mainly artistic, with the sculptor George Hart helping to popularize it.

The notation begins with "seed" polyhedron and and modifies it with a series of "operators" to obtain the desired polyhedron. These operators include standard ones such duals, truncation, and rectification, but also some more unusual ones. The notation itself is a string of mostly letters, representing the seed polyhedron as an uppercase letter on the right-hand side of the string, and operators as lowercase letters that are applied from right to left. For example, gtC is a cube (C) subjected to truncation (t) and then a "gyro" operation (g).

The notation can be straightforwardly generalized for nonconvex polyhedra by e.g. starting with a nonconvex seed. It has not been applied to dimensions other than 3.

## Description

 Polyhedron Tetrahedron Octahedron Cube Icosahedron Dodecahedron n -gonal prism n -gonal antiprism n -gonal pyramid Seed letter T O C I D Pn An Yn
Commonly-used operators in Conway notation (Pink and blue regions are cut away; pink is related to original vertices and blue is related to original edges)
Operator name Image Conway notation Notes
(Seed)

(Example: cube)

C = dd
dual dC Faces replaced with vertices.

Vertices replaced with faces.

truncate tC Cuts off each vertex, with the cuts independent of one another.

A number after the t can mean "only apply this operator to vertices surrounded by this many edges."

ambo aC Rectification.

Cuts off each vertex, to the extent that the vertices of the cuts just touch each other.

expand; explode eC Cantellation, which is the 3D form of expansion.

= aa, = do

bevel bC The 3D form of omnitruncation.

= ta, = dm

reflect rC Produces the mirror image of a chiral polyhedron.

Does not change achiral polyhedra.

kis kC Puts a vertex in the middle of each face, and connects it with edges to the existing vertices.

This adds a pyramid to each face.

This operator creates the tetrakis hexahedron from the cube.

A number after the k can mean "only apply this operator to faces with this many edges."

The operator that creates the rhombic dodecahedron from the cube (or octahedron).

= da

ortho oC Basically turns each n -gon face into n  quadrilaterals.

The operator that creates the deltoidal icositetrahedron from the cube (or octahedron).

= daa, = de

meta mC The operator that creates the

disdyakis dodecahedron from the cube (or octahedron).

= kj, = db

snub sC Snubbing. Produces a chiral polyhedron.

= dg

gyro gC Basically turns each n -gon face into n  pentagons. Chiral.

The operator that creates the pentagonal icositetrahedron from the cube (or octahedron).

= ds

The Platonic solids can be represented as Conway operators applied to prisms, antiprisms, and pyramids.

 T = Y3 O = aY3 A3 dP4 C = jY3 dA3 P4 I = sY3 k5A5 D = gY3 t5dA5

The Archimedean and Catalan solids can also be represented as Conway operators applied to Platonic solids. (Applying these to the tetrahedron might result in a Platonic solid.)

truncate rectify truncate dual cantellate omnitrunate snub Catalan solid corresponding to one of these
t a td e; aa b; ta s d- followed by this operator(s)
kis of dual "rhombic" solid kis "deltoidal" solid disdyakis "pentagonal" solid Archimedean solid corresponding to one of these
kd; dt j; da k; dtd o; daa m; dta g; ds d- followed by this operator(s)

The wide variety of possibilities by which new faces, edges, and vertices can be created and by which old ones can be removed may give rise to many different operators from different sources. Some are not default or standardized, and might have conflicting notation or might not be found in all implementations of the Conway notation.

Examples of less-official operators in Conway notation
Operator name Image Notes
(Seed)
zip Bitruncation.

The operator that creates the truncated octahedron from the cube.

= dk, = td

needle The operator that creates the triakis octahedron from the cube.

= kd, = dt

chamfer The operator that creates the chamfered cube from the cube.
propellor
whirl
loft Equivalent to augmenting the faces with prisms.
lace Equivalent to augmenting the faces with antiprisms.
quinto