# Conway polyhedron notation

This example chart shows how 11 new forms can be derived from the cube using 3 operations. The new polyhedra are shown as maps on the surface of the cube so the topological changes are more apparent. Vertices are marked in all forms with circles.
This chart adds 3 more operations: George Hart's p=propellor operator that adds quadrilaterals, g=gyro operation that creates pentagons, and a c=Chamfer operation that replaces edges with hexagons

Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.

The seed polyhedra are the Platonic solids, represented by the first letter of their name (T,O,C,I,D); the prisms (Pn), antiprisms (An) and pyramids (Yn). Any convex polyhedron can serve as a seed, as long as the operations can be executed on it.

John Conway extended the idea of using operators, like truncation defined by Kepler, to build related polyhedra of the same symmetry. His descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. Applied in a series, these operators allow many higher order polyhedra to be generated.

## Operations on polyhedra

Elements are given from the seed (v,e,f) to the new forms, assuming seed is a convex polyhedron: (a topological sphere, Euler characteristic = 2) An example image is given for each operation, based on a cubic seed. The basic operations are sufficient to generate the reflective uniform polyhedra and theirs duals. Some basic operations can be made as composites of others.

Special forms

The kis operator has a variation, kn, which only adds pyramids to n-sided faces.
The truncate operator has a variation, tn, which only truncates order-n vertices.

The operators are applied like functions from right to left. For example a cuboctahedron is an ambo cube, i.e. t(C) = aC, and a truncated cuboctahedron is t(a(C)) = t(aC) = taC.

Basic operations
OperatorExampleNameAlternate
construction
verticesedgesfacesDescription
Seed vef Seed form
ddual fevdual of the seed polyhedron - each vertex creates a new face
aambo e2e2 + eNew vertices are added mid-edges, while old vertices are removed. (rectify)
jjoin dae + 22eeThe seed is augmented with pyramids at a height high enough so that 2 coplanar triangles from 2 different pyramids share an edge.
ttruncate dkd2e3ee + 2truncate all vertices.
conjugate kis
kkis dtde + 23e2eraises a pyramid on each face.
i -- dk2e3ee + 2Dual of kis. (bitruncation)
n-- kd e + 23e2eKis of dual
eexpand aa = aj 2e4e2e + 2Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate)
oortho de = ja = jj2e + 24e2eEach n-gon faces are divided into n quadrilaterals.
bbevel ta4e6e2e + 2New faces are added in place of edges and vertices. (cantitruncation)
mmeta db = kj2e + 26e4e n-gon faces are divided into 2n triangles

### Extended operators

These extended operators can't be created in general from the basic operations above. Some can be created in special cases with k and t operators only applied to specific sided faces and vertices. For example a chamfered cube, cC, can be constructed as t4daC, as a rhombic dodecahedron, daC or jC, with its valence-4 vertices truncated. And a quinto-dodecahedron, qD can be constructed as t5daaD or t5deD or t5oD, a deltoidal hexecontahedron, deD or oD, with its valence-5 vertices truncated.

Extended operations
OperatorExampleNameAlternate
construction
verticesedgesfacesDescription
Seed vefSeed form
cchamfer v + 2e 4ef + eAn edge-truncation. New hexagonal faces are added in place of edges.
- - dc f + e4ev + 2e Dual of chamfer
q quinto v+3e 6e f+2e Ortho followed by truncation of vertices centered on original faces. This create 2 new pentagons for every original edge.
- - dq f+2e 6e v+3e Dual of quinto

### Chiral extended operators

These extended operators can't be created in general from the basic operations above. The gyro/snub operations are needed to generate the uniform polyhedra and duals with rotational symmetry.

Geometric artist George W. Hart created an operation he called a propellor, and another reflect to create mirror images of the rotated forms.

• p – "propellor" (A rotation operator that creates quadrilaterals at the vertices). This operation is self-dual: dpX=pdX.
• r – "reflect" – makes the mirror image of the seed; it has no effect unless the seed was made with s or g.

The half operator, h, from Coxeter, reduces square faces into digons, with two coinciding edges, which may or may not be replaced by a single edge. If digons remain, subsequently truncation operations can expand digons into square faces.

Chiral extended operations
OperatorExampleNameAlternate
construction
verticesedgesfacesDescription
Seed vefSeed form
rreflect
(Hart)
vef Mirror image for chiral forms
hhalf * v/2ef+v/2Alternation, remove half vertices,
limited to seed polyhedra with even-sided faces
ppropellor
(Hart)
v + 2e5ef + 2eA face rotation that creates quadrilaterals at vertices (self-dual)
- - dp = pdf + 2e5ev + 2e
ssnub dg = hta2e5e3e + 2"expand and twist" – each vertex creates a new face and each edge creates two new triangles
ggyro ds3e + 25e2eEach n-gon face is divided into n pentagons.
wwhirl v+4e7ef+2e Gyro followed by truncation of vertices centered on original faces.
This create 2 new hexagons for every original edge
- - dwf+2e7ev+4e Dual of whirl

## Generating regular seeds

All of the five regular polyhedra can be generated from prismatic generators with zero to two operators:

The regular Euclidean tilings can also be used as seeds:

## Examples

The cube can generate all the convex uniform polyhedra with octahedral symmetry. The first row generates the Archimedean solids and the second row the Catalan solids, the second row forms being duals of the first. Comparing each new polyhedron with the cube, each operation can be visually understood. (Two polyhedron forms don't have single operator names given by Conway.)

Cube
"seed"
ambo truncate bitruncate expand bevel

C

aC = djC

tC = dkdC

tdC = dkC

eC = aaC = doC

bC = taC = dmC = dkjC
dual join dual truncate kis ortho meta

dC

jC

kdC = dtC

kC

oC

mC
Extended operations
snub propellor chamfer whirl

sC

pC

cC

wC
gyro dual propeller dual chamfer dual whirl

gC = dsC

dpC

dcC

dwC
 T tT aT tdT eT bT sT dT dtT jT kT oT mT gT

The truncated icosahedron as a nonregular seed creates more polyhedra which are not vertex or face uniform.

Truncated icosahedron seed
"seed" ambo truncate bitruncate expand bevel

tI

atI

ttI

tdtI

etI

btI
dual join kis ortho meta

dtI

jtI

kdtI

ktI

otI

mtI
Extended operations
snub propellor chamfer quinto whirl

stI

ptI

ctI

qtI

wtI
gyro dual propeller dual chamfer dual quinto dual whirl

gtI

dptI

dctI

dqtI

dwtI

## Geometric coordinates of derived forms

In general the seed polyhedron can be considered a tiling of a surface since the operators represent topological operations so the exact geometric positions of the vertices of the derived forms are not defined in general. A convex regular polyhedron seed can be considered a tiling on a sphere, and so the derived polyhedron can equally be assumed to be positioned on the surface of a sphere. Similar a regular tiling on a plane, such as a hexagonal tiling can be a seed tiling for derived tilings. Nonconvex polyhedra can become seeds if a related topological surface is defined to constrain the positions of the vertices. For example toroidal polyhedra can derive other polyhedra with point on the same torus surface.

### Euclidean examples

 H tH aH tdH = H eH taH = bH sH dH dtH daH = jH dtdH = kH deH = oH dtaH = mH dsH = gH

### Hyperbolic examples

Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"
truncate ambo bitruncate expand bevel snub
dual dual kis join kis ortho meta gyro

## Other polyhedra

Iterating operators on simple forms can produce progressively larger polyhedra, maintaining the fundamental symmetry of the seed element. The vertices are assumed to be on the same spherical radius. Some generated forms can exist as spherical tilings, but fail to produce polyhedra with planar faces.

### Icosahedral symmetry

Rhombic:

Triangular:

Dual triangular:

Triangular chiral:

Dual triangular chiral:

## References

• George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings