Conway polyhedron notation

{{Short description|Method of describing higher-order polyhedra}}

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.{{cite book

| last1 = Conway | first1 = John | author1-link = John Horton Conway

| last2 = Burgiel | first2 = Heidi

| last3 = Goodman-Strauss | first3 = Chaim | author3-link = Chaim Goodman-Strauss

| contribution = Chapter 21: Naming Archimedean and Catalan polyhedra and tilings

| isbn = 978-1-56881-220-5

| page = 288

| publisher = AK Peters

| title = The Symmetries of Things | title-link = The Symmetries of Things

| year = 2008}}{{mathworld|id=ConwayPolyhedronNotation|title=Conway Polyhedron Notation}}

Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, {{mvar|tC}} represents a truncated cube, and {{mvar|taC}}, parsed as {{math|t(aC)}}, is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.g., a dual cube is an octahedron: {{math|1=dC = O}}. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators {{mvar|a}} (ambo), {{mvar|b}} (bevel), {{mvar|d}} (dual), {{mvar|e}} (expand), {{mvar|g}} (gyro), {{mvar|j}} (join), {{mvar|k}} (kis), {{mvar|m}} (meta), {{mvar|o}} (ortho), {{mvar|s}} (snub), and {{mvar|t}} (truncate), while Hart added {{mvar|r}} (reflect) and {{mvar|p}} (propellor).{{cite web|url=http://www.georgehart.com/virtual-polyhedra/conway_notation.html|author=George W. Hart|title=Conway Notation for Polyhedra|website=Virtual Polyhedra|date=1998}} Later implementations named further operators, sometimes referred to as "extended" operators.{{cite web|url=http://www.antiprism.com/programs/conway.html|author=Adrian Rossiter|title=conway - Conway Notation transformations|website=Antiprism Polyhedron Modelling Software}}{{cite web|url=https://levskaya.github.io/polyhedronisme/|author=Anselm Levskaya|title=polyHédronisme}} Conway's basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation ({{math|1=aa = e}}), while a truncation after ambo produces bevel ({{math|1=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 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.

Operators

In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a cuboctahedron is an ambo cube,{{cite web|url=http://www.georgehart.com/virtual-polyhedra/conway_notation.html |title=Conway Notation for Polyhedra| website=Virtual Polyhedra| date=1998| first=George| last=Hart}} (See fourth row in table, "a = ambo".) i.e. {{tmath|1=a(C) = aC}}, and a truncated cuboctahedron is {{tmath|1=t(a(C)) = t(aC) = taC}}. Repeated application of an operator can be denoted with an exponent: j² = o. In general, Conway operators are not commutative.

Individual operators can be visualized in terms of fundamental domains (or chambers), as below. Each right triangle is a fundamental domain. Each white chamber is a rotated version of the others, and so is each colored chamber. For achiral operators, the colored chambers are a reflection of the white chambers, and all are transitive. In group terms, achiral operators correspond to dihedral groups {{math|D{{sub|n}}}} where n is the number of sides of a face, while chiral operators correspond to cyclic groups {{math|C{{sub|n}}}} lacking the reflective symmetry of the dihedral groups. Achiral and chiral operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively.{{Cite journal|year=2017|title=Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=473|issue=2206|pages=20170267|arxiv=1705.02848|last1=Brinkmann|first1=G.|last2=Goetschalckx|first2=P.|last3=Schein|first3=S.|doi=10.1098/rspa.2017.0267|bibcode=2017RSPSA.47370267B|s2cid=119171258}}{{cite arXiv|last1=Goetschalckx|first1=Pieter|last2=Coolsaet|first2=Kris|last3=Van Cleemput|first3=Nico|date=2020-04-12|title=Generation of Local Symmetry-Preserving Operations|class=math.CO|eprint=1908.11622}}{{cite arXiv|last1=Goetschalckx|first1=Pieter|last2=Coolsaet|first2=Kris|last3=Van Cleemput|first3=Nico|date=2020-04-11|title=Local Orientation-Preserving Symmetry Preserving Operations on Polyhedra|class=math.CO|eprint=2004.05501}}

LSPs should be understood as local operations that preserve symmetry, not operations that preserve local symmetry. Again, these are symmetries in a topological sense, not a geometric sense: the exact angles and edge lengths may differ.

class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"

|+Fundamental domains of faces with $n$ sides

!3 (Triangle)

!4 (Square)

!5 (Pentagon)

!6 (Hexagon)

align = "center"

align = "center"

| colspan="4" |The fundamental domains for polyhedron groups. The groups are $D_3,D_4,D_5,D_6$ for achiral polyhedra, and $C_3,C_4,C_5,C_6$ for chiral polyhedra.

Hart introduced the reflection operator r, that gives the mirror image of the polyhedron. 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. An overline can be used to indicate the other chiral form of an operator: {{overline|s}} = rsr.

An operation is irreducible if it cannot be expressed as a composition of operators aside from d and r. The majority of Conway's original operators are irreducible: the exceptions are e, b, o, and m.

= Matrix representation =

class="wikitable floatright"
x \| $\begin{bmatrix}$ a & b & c \\ 0 & d & 0 \\ a' & b' & c' \end{bmatrix}=\mathbf{M}_x
xd \| $\begin{bmatrix}$ c & b & a \\ 0 & d & 0 \\ c' & b' & a' \end{bmatrix}=\mathbf{M}_x\mathbf{M}_d
dx \| $\begin{bmatrix}$ a' & b' & c' \\ 0 & d & 0 \\ a & b & c \end{bmatrix}=\mathbf{M}_d\mathbf{M}_x
dxd \| $\begin{bmatrix}$ c' & b' & a' \\ 0 & d & 0 \\ c & b & a \end{bmatrix} = \mathbf{M}_d\mathbf{M}_x\mathbf{M}_d

class="wikitable floatright"

| $\begin{bmatrix}$

a & b & c \\

0 & d & 0 \\

a' & b' & c'

\end{bmatrix}=\mathbf{M}_x

| $\begin{bmatrix}$

c & b & a \\

0 & d & 0 \\

c' & b' & a'

\end{bmatrix}=\mathbf{M}_x\mathbf{M}_d

| $\begin{bmatrix}$

a' & b' & c' \\

0 & d & 0 \\

a & b & c

\end{bmatrix}=\mathbf{M}_d\mathbf{M}_x

dxd

| $\begin{bmatrix}$

c' & b' & a' \\

0 & d & 0 \\

c & b & a

\end{bmatrix} = \mathbf{M}_d\mathbf{M}_x\mathbf{M}_d

The relationship between the number of vertices, edges, and faces of the seed and the polyhedron created by the operations listed in this article can be expressed as a matrix $\mathbf{M}_x$ . When x is the operator, $v,e,f$ are the vertices, edges, and faces of the seed (respectively), and $v',e',f'$ are the vertices, edges, and faces of the result, then

: $\mathbf{M}_x \begin{bmatrix} v \\ e \\ f \end{bmatrix} = \begin{bmatrix} v' \\ e' \\ f' \end{bmatrix}$ .

The matrix for the composition of two operators is just the product of the matrixes for the two operators. Distinct operators may have the same matrix, for example, p and l. The edge count of the result is an integer multiple d of that of the seed: this is called the inflation rate, or the edge factor.

The simplest operators, the identity operator S and the dual operator d, have simple matrix forms:

: $\mathbf{M}_S = \begin{bmatrix}$

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{bmatrix} = \mathbf{I}_3, $\mathbf{M}_d = \begin{bmatrix}$

0 & 0 & 1 \\

0 & 1 & 0 \\

1 & 0 & 0

\end{bmatrix}

Two dual operators cancel out; dd = S, and the square of $\mathbf{M}_d$ is the identity matrix. When applied to other operators, the dual operator corresponds to horizontal and vertical reflections of the matrix. Operators can be grouped into groups of four (or fewer if some forms are the same) by identifying the operators x, xd (operator of dual), dx (dual of operator), and dxd (conjugate of operator). In this article, only the matrix for x is given, since the others are simple reflections.

= Number of operators =

The number of LSPs for each inflation rate is $2, 2, 4, 6, 6, 20, 28, 58, 82, \cdots$ starting with inflation rate 1. 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 number of 3-connected LSPs for each inflation rate is $2, 2, 4, 6, 4, 20, 20, 54, 64, \cdots$ .

Original operations

Strictly, seed (S), needle (n), and zip (z) were not included by Conway, but they are related to original Conway operations by duality so are included here.

From here on, operations are visualized on cube seeds, drawn on the surface of that cube. Blue faces cross edges of the seed, and pink faces lie over vertices of the seed. There is some flexibility in the exact placement of vertices, especially with chiral operators.

Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd	Notes
class="wikitable sortable" style="text-align: center" \|+Original Conway operators
1 \| $\begin{bmatrix}$ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \| 100px Seed: S \|colspan="2"\| 100px Dual: d \| 100px Seed: dd = S \| Dual replaces each face with a vertex, and each vertex with a face.
2 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 0 \end{bmatrix} \|colspan="2"\| 100px Join: j \|colspan="2"\| 100px Ambo: a \| Join creates quadrilateral faces. Ambo creates degree-4 vertices, and is also called rectification, or the medial graph in graph theory.{{mathworld\|id=Rectification\|title=Rectification}}
3 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px Kis: k \| 100px Needle: n \| 100px Zip: z \| 100px Truncate: t \| Kis raises a pyramid on each face, and is also called akisation, Kleetope, cumulation,{{mathworld\|id=Cumulation\|title=Cumulation}} accretion, or pyramid-augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges.{{mathworld\|id=Truncation\|title=Truncation}} Zip is also called bitruncation.
4 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 4 & 0 \\ 0 & 2 & 0 \end{bmatrix} \|colspan=2\| 100px Ortho: o = jj \|colspan=2\| 100px Expand: e = aa \|
5 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 5 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px Gyro: g \| gd = rgr \| sd = rsr \| 100px Snub: s \| Chiral operators. See Snub (geometry). Contrary to Hart, gd is not the same as g: it is its chiral pair.{{cite web\|title=Antiprism - Chirality issue in conway\|url=https://groups.google.com/forum/#!topic/antiprism/6NcYnKP4mTc}}
6 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 6 & 0 \\ 0 & 4 & 0 \end{bmatrix} \|colspan="2"\| 100px Meta: m = kj \|colspan="2"\| 100px Bevel: b = ta \|

Seeds

Any polyhedron can serve as a seed, as long as the operations can be executed on it. Common seeds have been assigned a letter.

The Platonic solids are represented by the first letter of their name (Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron); the prisms (P_n) for n-gonal forms; antiprisms (A_n); cupolae (U_n); anticupolae (V_n); and pyramids (Y_n). Any Johnson solid can be referenced as J_n, for n=1..92.

All of the five Platonic solids can be generated from prismatic generators with zero to two operators:{{cite journal|url=http://www.mi.sanu.ac.rs/vismath/zefiro2008/__generation_of_icosahedron_by_5tetrahedra.htm|title=Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra|author=Livio Zefiro|journal=Vismath|year=2008}}

Triangular pyramid: Y₃ (A tetrahedron is a special pyramid)
T = Y₃
O = aT (ambo tetrahedron)
C = jT (join tetrahedron)
I = sT (snub tetrahedron)
D = gT (gyro tetrahedron)
Triangular antiprism: A₃ (An octahedron is a special antiprism)
O = A₃
C = dA₃
Square prism: P₄ (A cube is a special prism)
C = P₄
Pentagonal antiprism: A₅
I = k₅A₅ (A special gyroelongated dipyramid)
D = t₅dA₅ (A special truncated trapezohedron)

The regular Euclidean tilings can also be used as seeds:

Q = Quadrille = Square tiling
H = Hextille = Hexagonal tiling = dΔ
Δ = Deltille = Triangular tiling = dH

Extended operations

These are operations created after Conway's original set. Note that many more operations exist than have been named; just because an operation is not here does not mean it does not exist (or is not an LSP or LOPSP). In addition, only irreducible operators are

included in this list; many others can be created by composing operators together.

class="wikitable sortable" style="text-align: center" \|+Irreducible extended operators
Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd	Notes
4 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 4 & 0 \\ 0 & 1 & 1 \end{bmatrix} \| 100px Chamfer: c \| 100px cd = du \| 100px dc = ud \| 100px Subdivide: u \| Chamfer is the join-form of l. See Chamfer (geometry).
5 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 5 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px Propeller: p \|colspan="2"\| 100px dp = pd \| 100px dpd = p \| Chiral operators. The propeller operator was developed by George Hart.{{cite conference\|url=http://www.georgehart.com/propello/propello.html\|author=George W. Hart\|title=Sculpture based on Propellorized Polyhedra\|conference=Proceedings of MOSAIC 2000\|location=Seattle, WA\|date=August 2000\| pages=61–70 }}
5 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 5 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px Loft: l \| 100px ld \| 100px dl \| 100px dld \|
6 \| $\begin{bmatrix}$ 1 & 3 & 0 \\ 0 & 6 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px Quinto: q \| 100px qd \| 100px dq \| 100px dqd \|
6 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 6 & 0 \\ 0 & 3 & 1 \end{bmatrix} \|100px Join-lace: L₀ \|120x120px L₀d \|100px dL₀ \|100px dL₀d \| See below for explanation of join notation.
7 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 7 & 0 \\ 0 & 4 & 1 \end{bmatrix} \| 100px Lace: L \| 100px Ld \| 100px dL \| 100px dLd \|
7 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 7 & 0 \\ 0 & 4 & 0 \end{bmatrix} \| 100px Stake: K \| 100px Kd \| 100px dK \| 100px dKd \|
7 \| $\begin{bmatrix}$ 1 & 4 & 0 \\ 0 & 7 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px Whirl: w \| wd = dv \| 100px vd = dw \| Volute: v \| Chiral operators.
8 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 8 & 0 \\ 0 & 5 & 0 \end{bmatrix} \|100px Join-kis-kis: {{tmath\|1=(kk)_0}} \|100px {{tmath\|1=(kk)_0d}} \|100px {{tmath\|1=d(kk)_0}} \|100px {{tmath\|1=d(kk)_0d}} \|Sometimes named J. See below for explanation of join notation. The non-join-form, kk, is not irreducible.
10 \| $\begin{bmatrix}$ 1 & 3 & 1 \\ 0 & 10 & 0 \\ 0 & 6 & 0 \end{bmatrix} \| 100px Cross: X \| 100px Xd \| 100px dX \| 100px dXd \|

class="wikitable sortable" style="text-align: center"

|+Irreducible extended operators

Edge factor

Matrix

\mathbf{M}_x

dxd

Notes

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 4 & 0 \\

0 & 1 & 1

\end{bmatrix}

| 100px
Chamfer: c

| 100px
cd = du

| 100px
dc = ud

| 100px
Subdivide: u

| Chamfer is the join-form of l. See Chamfer (geometry).

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 5 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px
Propeller: p

|colspan="2"| 100px
dp = pd

| 100px
dpd = p

| Chiral operators. The propeller operator was developed by George Hart.{{cite conference|url=http://www.georgehart.com/propello/propello.html|author=George W. Hart|title=Sculpture based on Propellorized Polyhedra|conference=Proceedings of MOSAIC 2000|location=Seattle, WA|date=August 2000| pages=61–70 }}

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 5 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px
Loft: l

| 100px
ld

| 100px
dl

| 100px
dld

| $\begin{bmatrix}$

1 & 3 & 0 \\

0 & 6 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px
Quinto: q

| 100px
qd

| 100px
dq

| 100px
dqd

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 6 & 0 \\

0 & 3 & 1

\end{bmatrix}

|100px
Join-lace: L₀

|120x120px
L₀d

|100px
dL₀

|100px
dL₀d

| See below for explanation of join notation.

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 7 & 0 \\

0 & 4 & 1

\end{bmatrix}

| 100px
Lace: L

| 100px
Ld

| 100px
dL

| 100px
dLd

| $\begin{bmatrix}$

1 & 2 & 1 \\

0 & 7 & 0 \\

0 & 4 & 0

\end{bmatrix}

| 100px
Stake: K

| 100px
Kd

| 100px
dK

| 100px
dKd

| $\begin{bmatrix}$

1 & 4 & 0 \\

0 & 7 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px
Whirl: w

| wd = dv

| 100px
vd = dw

| Volute: v

| Chiral operators.

| $\begin{bmatrix}$

1 & 2 & 1 \\

0 & 8 & 0 \\

0 & 5 & 0

\end{bmatrix}

|100px
Join-kis-kis: {{tmath|1=(kk)_0}}

|100px
{{tmath|1=(kk)_0d}}

|100px
{{tmath|1=d(kk)_0}}

|100px
{{tmath|1=d(kk)_0d}}

|Sometimes named J. See below for explanation of join notation. The non-join-form, kk, is not irreducible.

| $\begin{bmatrix}$

1 & 3 & 1 \\

0 & 10 & 0 \\

0 & 6 & 0

\end{bmatrix}

| 100px
Cross: X

| 100px
Xd

| 100px
dX

| 100px
dXd

Indexed extended operations

A number of operators can be grouped together by some criteria, or have their behavior modified by an index. These are written as an operator with a subscript: x_n.

=Augmentation=

Augmentation operations retain original edges. 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, k₄Y₄=O: taking a square-based pyramid and gluing another pyramid to the square base gives an octahedron.

class=wikitable style="text-align: center"
Augmentation operator \|x \|80px k \|80px l \|80px L \|80px K \|80px (kk)
Corresponding join-form operator \|x₀ \|80px k₀ = j \|80px l₀ = c \|80px L₀ \|80px K₀ = jk \|80px (kk)₀
align="center" !Augmentation \| \|Pyramid	Prism	Antiprism

class=wikitable style="text-align: center"

Augmentation operator

|80px

(kk)

Corresponding

join-form operator

|x₀

|80px
k₀ = j

|80px
l₀ = c

|80px
L₀

|80px
K₀ = jk

|80px
(kk)₀

align="center"

!Augmentation

|Pyramid

Prism

Antiprism

The truncate operator t also has an index form t_n, indicating that only vertices of a certain degree are truncated. It is equivalent to dk_nd.

Some of the extended operators can be created in special cases with k_n and t_n operators. 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 lofted cube, lC is the same as t₄kC. 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.

=Meta/Bevel=

Meta adds vertices at the center and along the edges, while bevel adds faces at the center, seed vertices, and along the edges. The index is how many vertices or faces are added along the edges. Meta (in its non-indexed form) is also called cantitruncation or omnitruncation. Note that 0 here does not mean the same as for augmentation operations: it means zero vertices (or faces) are added along the edges.

n	Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd
class="wikitable sortable" style="text-align: center" \|+Meta/Bevel operators
0 \| 3 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 2 & 0 \end{bmatrix} \|100px k = m₀ \| 100px n \|100px z = b₀ \| 100px t
1 \| 6 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 6 & 0 \\ 0 & 4 & 0 \end{bmatrix} \|colspan="2"\| 100px m = m₁ = kj \|colspan="2"\| 100px b = b₁ = ta
2 \|9 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 9 & 0 \\ 0 & 6 & 0 \end{bmatrix} \|100px m₂ \|100px m₂d \|100px b₂ \|100px b₂d
3 \|12 \| $\begin{bmatrix}$ 1 & 3 & 1 \\ 0 & 12 & 0 \\ 0 & 8 & 0 \end{bmatrix} \|100px m₃ \|m₃d \|b₃ \|b₃d
n \| 3n+3 \| $\begin{bmatrix}$ 1 & n & 1 \\ 0 & 3n+3 & 0 \\ 0 & 2n+2 & 0 \end{bmatrix} \|m_n \|m_nd \|b_n \|b_nd

=Medial=

Medial is like meta, except it does not add edges from the center to each seed vertex. The index 1 form is identical to Conway's ortho and expand operators: expand is also called cantellation and expansion. Note that o and e have their own indexed forms, described below. Also note that some implementations start indexing at 0 instead of 1.

n	Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd
class="wikitable sortable" style="text-align: center" \|+Medial operators
1 \| 4 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 4 & 0 \\ 0 & 2 & 0 \end{bmatrix} \|colspan=2\| 100px M₁ = o = jj \|colspan=2\| 100px e = aa
2 \|7 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 7 & 0 \\ 0 & 4 & 0 \end{bmatrix} \|100px Medial: M = M₂ \|100px Md \|100px dM \|100px dMd
n \|3n+1 \| $\begin{bmatrix}$ 1 & n & 1 \\ 0 & 3n+1 & 0 \\ 0 & 2n & 0 \end{bmatrix} \|M_n \|M_nd \|dM_n \|dM_nd

=Goldberg-Coxeter=

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction.{{cite journal |author1=Deza, M. |author-link=Michel Deza |author2=Dutour, M|title=Goldberg–Coxeter constructions for 3-and 4-valent plane graphs|journal=The Electronic Journal of Combinatorics |volume=11|year=2004|pages=#R20|doi=10.37236/1773|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r20|doi-access=free}}{{cite book|author1=Deza, M.-M.|author2=Sikirić, M. D.|author3=Shtogrin, M. I. | year=2015 |title=Geometric Structure of Chemistry-Relevant Graphs: Zigzags and Central Circuits|publisher=Springer|pages=131–148 |chapter=Goldberg–Coxeter Construction and Parameterization|chapter-url=https://books.google.com/books?id=HLi4CQAAQBAJ&q=goldberg-coxeter&pg=PA130 |isbn=9788132224495}} 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"). Operators in the triangular family can be used to produce the Goldberg polyhedra and geodesic polyhedra: see List of geodesic polyhedra and Goldberg polyhedra for formulas.

The two families are the triangular GC family, c_a,b and u_a,b, and the quadrilateral GC family, e_a,b and o_a,b. Both the GC families are indexed by two integers $a \ge 1$ and $b \ge 0$ . They possess many nice qualities:

The indexes of the families have a relationship with certain Euclidean domains over the complex numbers: the Eisenstein integers for the triangular GC family, and the Gaussian integers for the quadrilateral GC family.
Operators in the x and dxd columns within the same family commute with each other.

The operators are divided into three classes (examples are written in terms of c but apply to all 4 operators):

Class I: {{tmath|1=b=0}}. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. c_a,0 = c_a.
Class II: {{tmath|1=a=b}}. Also achiral. Can be decomposed as c_a,a = c_ac_1,1
Class III: All other operators. These are chiral, and c_a,b and c_b,a are the chiral pairs of each other.

Of the original Conway operations, the only ones that do not fall into the GC family are g and s (gyro and snub). Meta and bevel (m and b) can be expressed in terms of one operator from the triangular family and one from the quadrilateral family.

== Triangular ==

class="wikitable sortable" style="text-align: center" \|+Triangular Goldberg-Coxeter operators
a	b	Class	Edge factor T = a² + ab + b²	Matrix $\mathbf{M}_x$	Master triangle	x	xd	dx	dxd
1 \| 0 \| I \| 1 \| $\begin{bmatrix}$ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \|100px \|100px u₁ = S \|colspan=2\|100px d \|100px c₁ = S
2 \| 0 \| I \| 4 \| $\begin{bmatrix}$ 1 & 1 & 0 \\ 0 & 4 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px \| 100px u₂ = u \| 100px dc \| 100px du \| 100px c₂ = c
3 \| 0 \| I \| 9 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 9 & 0 \\ 0 & 6 & 0 \end{bmatrix} \| 100px \| 100px u₃ = nn \| 100px nk \| 100px zt \| 100px c₃ = zz
4 \| 0 \| I \| 16 \| $\begin{bmatrix}$ 1 & 5 & 0 \\ 0 & 16 & 0 \\ 0 & 10 & 1 \end{bmatrix} \| 100px \|100px u₄ = uu \| uud = dcc \| duu = ccd \|c₄ = cc
5 \| 0 \| I \| 25 \| $\begin{bmatrix}$ 1 & 8 & 0 \\ 0 & 25 & 0 \\ 0 & 16 & 1 \end{bmatrix} \| 100px \|100px u₅ \|u₅d = dc₅ \|du₅ = c₅d \| c₅
6 \| 0 \| I \| 36 \| $\begin{bmatrix}$ 1 & 11 & 1 \\ 0 & 36 & 0 \\ 0 & 24 & 0 \end{bmatrix} \| 100px \|100px u₆ = unn \| unk \| czt \|u₆ = czz
7 \| 0 \| I \| 49 \| $\begin{bmatrix}$ 1 & 16 & 0 \\ 0 & 49 & 0 \\ 0 & 32 & 1 \end{bmatrix} \| 100px \|100px u₇ = u_2,1u_1,2 = vrv \| vrvd = dwrw \| dvrv = wrwd \| c₇ = c_2,1c_1,2 = wrw
8 \| 0 \| I \| 64 \| $\begin{bmatrix}$ 1 & 21 & 0 \\ 0 & 64 & 0 \\ 0 & 42 & 1 \end{bmatrix} \| 100px \|100px u₈ = u³ \| u³d = dc³ \| du³ = c³d \| c₈ = c³
9 \| 0 \| I \| 81 \| $\begin{bmatrix}$ 1 & 26 & 1 \\ 0 & 81 & 0 \\ 0 & 54 & 0 \end{bmatrix} \| 100px \|100px u₉ = n⁴ \| n³k = kz³ \| tn³ = z³t \| c₉ = z⁴
1 \| 1 \| II \| 3 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px \|100px u_1,1 = n \| 100px k \| 100px t \| 100px c_1,1 = z
2 \| 1 \| III \| 7 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 7 & 0 \\ 0 & 4 & 1 \end{bmatrix} \| 100px \| v = u_2,1 \| 100px vd = dw \| dv = wd \| 100px w = c_2,1
3 \| 1 \| III \| 13 \| $\begin{bmatrix}$ 1 & 4 & 0 \\ 0 & 13 & 0 \\ 0 & 8 & 1 \end{bmatrix} \| 100px \|u_3,1 \|u_3,1d = dc_3,1 \|du_3,1 = c_3,1d \| 100px c_3,1
3 \| 2 \| III \| 19 \| $\begin{bmatrix}$ 1 & 6 & 0 \\ 0 & 19 & 0 \\ 0 & 12 & 1 \end{bmatrix} \| 100px \| u_3,2 \| u_3,2d = dc_3,2 \| du_3,2 = c_3,2d \| 100px c_3,2
4 \| 3 \| III \| 37 \| $\begin{bmatrix}$ 1 & 12 & 0 \\ 0 & 37 & 0 \\ 0 & 24 & 1 \end{bmatrix} \| 100px \| u_4,3 \| u_4,3d = dc_4,3 \| du_4,3 = c_4,3d \|100px c_4,3
5 \| 4 \| III \| 61 \| $\begin{bmatrix}$ 1 & 20 & 0 \\ 0 & 61 & 0 \\ 0 & 40 & 1 \end{bmatrix} \| 100px \|u_5,4 \|u_5,4d = dc_5,4 \|du_5,4 = c_5,4d \|100px c_5,4
6 \| 5 \| III \| 91 \| $\begin{bmatrix}$ 1 & 30 & 0 \\ 0 & 91 & 0 \\ 0 & 60 & 1 \end{bmatrix} \| 100px \|u_6,5 = u_1,2u_1,3 \|u_6,5d = dc_6,5 \|du_6,5 = c_6,5d \|100px c_6,5=c_1,2c_1,3
7 \| 6 \| III \| 127 \| $\begin{bmatrix}$ 1 & 42 & 0 \\ 0 & 127 & 0 \\ 0 & 84 & 1 \end{bmatrix} \| 100px \|u_7,6 \|u_7,6d = dc_7,6 \|du_7,6 = c_7,6d \|100px c_7,6
8 \| 7 \| III \| 169 \| $\begin{bmatrix}$ 1 & 56 & 0 \\ 0 & 169 & 0 \\ 0 & 112 & 1 \end{bmatrix} \| 100px \|u_8,7 = u_3,1² \|u_8,7d = dc_8,7 \|du_8,7 = c_8,7d \|100px c_8,7 = c_3,1²
9 \| 8 \| III \| 217 \| $\begin{bmatrix}$ 1 & 72 & 0 \\ 0 & 217 & 0 \\ 0 & 144 & 1 \end{bmatrix} \| 100px \|u_9,8 = u_2,1u_5,1 \|u_9,8d = dc_9,8 \|du_9,8 = c_9,8d \|100px c_9,8 = c_2,1c_5,1
colspan=2\| $a \equiv b$ $\ (\mathrm{mod}\ 3)$ \| I, II, or III \| $T \equiv 0\$ $(\mathrm{mod}\ 3)$ \| $\begin{bmatrix}$ 1 & \frac{T}{3}-1 & 1 \\ 0 & T & 0 \\ 0 & \frac{2}{3}T & 0 \end{bmatrix} \|... \|u_a,b \|u_a,bd = dc_a,b \|du_a,b = c_a,bd \|c_a,b
colspan=2\| $a \not \equiv b$ $\ (\mathrm{mod}\ 3)$ \| I or III \| $T \equiv 1$ $\ (\mathrm{mod}\ 3)$ \| $\begin{bmatrix}$ 1 & \frac{T-1}{3} & 0 \\ 0 & T & 0 \\ 0 & 2\frac{T-1}{3} & 1 \end{bmatrix} \|... \|u_a,b \|u_a,bd = dc_a,b \|du_a,b = c_a,bd \|c_a,b

class="wikitable sortable" style="text-align: center"

|+Triangular Goldberg-Coxeter operators

Class

Edge factor
T = a² + ab + b²

Matrix

\mathbf{M}_x

Master triangle

dxd

| 0

| I

| 1

| $\begin{bmatrix}$

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{bmatrix}

|100px

|100px
u₁ = S

|colspan=2|100px
d

|100px
c₁ = S

| 0

| I

| 4

| $\begin{bmatrix}$

1 & 1 & 0 \\

0 & 4 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px

| 100px
u₂ = u

| 100px
dc

| 100px
du

| 100px
c₂ = c

| 0

| I

| 9

| $\begin{bmatrix}$

1 & 2 & 1 \\

0 & 9 & 0 \\

0 & 6 & 0

\end{bmatrix}

| 100px

| 100px
u₃ = nn

| 100px
nk

| 100px
zt

| 100px
c₃ = zz

| 0

| I

| 16

| $\begin{bmatrix}$

1 & 5 & 0 \\

0 & 16 & 0 \\

0 & 10 & 1

\end{bmatrix}

| 100px

|100px
u₄ = uu

| uud = dcc

| duu = ccd

|c₄ = cc

| 0

| I

| 25

| $\begin{bmatrix}$

1 & 8 & 0 \\

0 & 25 & 0 \\

0 & 16 & 1

\end{bmatrix}

| 100px

|100px
u₅

|u₅d = dc₅

|du₅ = c₅d

| c₅

| 0

| I

| 36

| $\begin{bmatrix}$

1 & 11 & 1 \\

0 & 36 & 0 \\

0 & 24 & 0

\end{bmatrix}

| 100px

|100px
u₆ = unn

| unk

| czt

|u₆ = czz

| 0

| I

| 49

| $\begin{bmatrix}$

1 & 16 & 0 \\

0 & 49 & 0 \\

0 & 32 & 1

\end{bmatrix}

| 100px

|100px
u₇ = u_2,1u_1,2 = vrv

| vrvd = dwrw

| dvrv = wrwd

| c₇ = c_2,1c_1,2 = wrw

| 0

| I

| 64

| $\begin{bmatrix}$

1 & 21 & 0 \\

0 & 64 & 0 \\

0 & 42 & 1

\end{bmatrix}

| 100px

|100px
u₈ = u³

| u³d = dc³

| du³ = c³d

| c₈ = c³

| 0

| I

| 81

| $\begin{bmatrix}$

1 & 26 & 1 \\

0 & 81 & 0 \\

0 & 54 & 0

\end{bmatrix}

| 100px

|100px
u₉ = n⁴

| n³k = kz³

| tn³ = z³t

| c₉ = z⁴

| 1

| II

| 3

| $\begin{bmatrix}$

1 & 0 & 1 \\

0 & 3 & 0 \\

0 & 2 & 0

\end{bmatrix}

| 100px

|100px
u_1,1 = n

| 100px
k

| 100px
t

| 100px
c_1,1 = z

| 1

| III

| 7

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 7 & 0 \\

0 & 4 & 1

\end{bmatrix}

| 100px

| v = u_2,1

| 100px
vd = dw

| dv = wd

| 100px
w = c_2,1

| 1

| III

| 13

| $\begin{bmatrix}$

1 & 4 & 0 \\

0 & 13 & 0 \\

0 & 8 & 1

\end{bmatrix}

| 100px

|u_3,1

|u_3,1d = dc_3,1

|du_3,1 = c_3,1d

| 100px
c_3,1

| 2

| III

| 19

| $\begin{bmatrix}$

1 & 6 & 0 \\

0 & 19 & 0 \\

0 & 12 & 1

\end{bmatrix}

| 100px

| u_3,2

| u_3,2d = dc_3,2

| du_3,2 = c_3,2d

| 100px
c_3,2

| 3

| III

| 37

| $\begin{bmatrix}$

1 & 12 & 0 \\

0 & 37 & 0 \\

0 & 24 & 1

\end{bmatrix}

| 100px

| u_4,3

| u_4,3d = dc_4,3

| du_4,3 = c_4,3d

|100px
c_4,3

| 4

| III

| 61

| $\begin{bmatrix}$

1 & 20 & 0 \\

0 & 61 & 0 \\

0 & 40 & 1

\end{bmatrix}

| 100px

|u_5,4

|u_5,4d = dc_5,4

|du_5,4 = c_5,4d

|100px
c_5,4

| 5

| III

| 91

| $\begin{bmatrix}$

1 & 30 & 0 \\

0 & 91 & 0 \\

0 & 60 & 1

\end{bmatrix}

| 100px

|u_6,5 = u_1,2u_1,3

|u_6,5d = dc_6,5

|du_6,5 = c_6,5d

|100px
c_6,5=c_1,2c_1,3

| 6

| III

| 127

| $\begin{bmatrix}$

1 & 42 & 0 \\

0 & 127 & 0 \\

0 & 84 & 1

\end{bmatrix}

| 100px

|u_7,6

|u_7,6d = dc_7,6

|du_7,6 = c_7,6d

|100px
c_7,6

| 7

| III

| 169

| $\begin{bmatrix}$

1 & 56 & 0 \\

0 & 169 & 0 \\

0 & 112 & 1

\end{bmatrix}

| 100px

|u_8,7 = u_3,1²

|u_8,7d = dc_8,7

|du_8,7 = c_8,7d

|100px
c_8,7 = c_3,1²

| 8

| III

| 217

| $\begin{bmatrix}$

1 & 72 & 0 \\

0 & 217 & 0 \\

0 & 144 & 1

\end{bmatrix}

| 100px

|u_9,8 = u_2,1u_5,1

|u_9,8d = dc_9,8

|du_9,8 = c_9,8d

|100px
c_9,8 = c_2,1c_5,1

colspan=2|

a \equiv b

\ (\mathrm{mod}\ 3)

| I, II, or III

| $T \equiv 0\$ $(\mathrm{mod}\ 3)$

| $\begin{bmatrix}$

1 & \frac{T}{3}-1 & 1 \\

0 & T & 0 \\

0 & \frac{2}{3}T & 0

\end{bmatrix}

|...

|u_a,b

|u_a,bd = dc_a,b

|du_a,b = c_a,bd

|c_a,b

colspan=2|

a \not \equiv b

\ (\mathrm{mod}\ 3)

| I or III

| $T \equiv 1$ $\ (\mathrm{mod}\ 3)$

| $\begin{bmatrix}$

1 & \frac{T-1}{3} & 0 \\

0 & T & 0 \\

0 & 2\frac{T-1}{3} & 1

\end{bmatrix}

|...

|u_a,b

|u_a,bd = dc_a,b

|du_a,b = c_a,bd

|c_a,b

By basic number theory, for any values of a and b, $T \not\equiv 2\ (\mathrm{mod}\ 3)$ .

== Quadrilateral ==

class="wikitable sortable" style="text-align: center" \|+Quadrilateral Goldberg-Coxeter operators
a	b	Class	Edge factor T = a² + b²	Matrix $\mathbf{M}_x$	Master square	x	xd	dx	dxd
1 \| 0 \| I \| 1 \| $\begin{bmatrix}$ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \| 100px \|100px o₁ = S \|colspan=2\|100px e₁ = d \|100px o₁ = dd = S
2 \| 0 \| I \| 4 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 4 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px \|colspan=2\| 100px o₂ = o = j² \|colspan=2\| 100px e₂ = e = a²
3 \| 0 \| I \| 9 \| $\begin{bmatrix}$ 1 & 4 & 0 \\ 0 & 9 & 0 \\ 0 & 4 & 1 \end{bmatrix} \| 100px \| 100px o₃ \|colspan=2\| 100px e₃ \| 100px o₃
4 \| 0 \| I \| 16 \| $\begin{bmatrix}$ 1 & 7 & 1 \\ 0 & 16 & 0 \\ 0 & 8 & 0 \end{bmatrix} \| 100px \|colspan=2\|100px o₄ = oo = j⁴ \|colspan=2\| 100px e₄ = ee = a⁴
5 \| 0 \| I \| 25 \| $\begin{bmatrix}$ 1 & 12 & 0 \\ 0 & 25 & 0 \\ 0 & 12 & 1 \end{bmatrix} \| 100px \|100px o₅ = o_2,1o_1,2 = prp \|colspan=2\|e₅ = e_2,1e_1,2 \|100px o₅= dprpd
6 \| 0 \| I \| 36 \| $\begin{bmatrix}$ 1 & 17 & 1 \\ 0 & 36 & 0 \\ 0 & 18 & 0 \end{bmatrix} \| 100px \|colspan=2\|100px o₆ = o₂o₃ \|colspan=2\|e₆ = e₂e₃
7 \| 0 \| I \| 49 \| $\begin{bmatrix}$ 1 & 24 & 0 \\ 0 & 49 & 0 \\ 0 & 24 & 1 \end{bmatrix} \| 100px \|100px o₇ \|colspan=2\|e₇ \|100px o₇
8 \| 0 \| I \| 64 \| $\begin{bmatrix}$ 1 & 31 & 1 \\ 0 & 64 & 0 \\ 0 & 32 & 0 \end{bmatrix} \| 100px \|colspan=2\|100px o₈ = o³ = j⁶ \|colspan=2\| e₈ = e³ = a⁶
9 \| 0 \| I \| 81 \| $\begin{bmatrix}$ 1 & 40 & 0 \\ 0 & 81 & 0 \\ 0 & 40 & 1 \end{bmatrix} \| 100px \|100px o₉ = o₃² \|colspan=2\| e₉ = e₃² \|100px o₉
10 \| 0 \| I \| 100 \| $\begin{bmatrix}$ 1 & 49 & 1 \\ 0 & 100 & 0 \\ 0 & 50 & 0 \end{bmatrix} \| 100px \|colspan=2\|100px o₁₀ = oo_2,1o_1,2 \|colspan=2\|e₁₀ = ee_2,1e_1,2
1 \| 1 \| II \| 2 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 0 \end{bmatrix} \| 100px \|colspan=2\| 100px o_1,1 = j \|colspan=2\| 100px e_1,1 = a
2 \| 2 \| II \| 8 \| $\begin{bmatrix}$ 1 & 3 & 1 \\ 0 & 8 & 0 \\ 0 & 4 & 0 \end{bmatrix} \| 100px \|colspan=2\| 100px o_2,2 = j³ \|colspan=2\| 100px e_2,2 = a³
1 \| 2 \| III \| 5 \| $\begin{bmatrix}$ 1 & 2 & 0 \\ 0 & 5 & 0 \\ 0 & 2 & 1 \end{bmatrix} \| 100px \| 100px o_1,2 = p \|colspan=2\| 100px e_1,2 = dp = pd \| 100px p
colspan=2\| $a \equiv b$ $\ (\mathrm{mod}\ 2)$ \| I, II, or III \| T even \| $\begin{bmatrix}$ 1 & \frac{T}{2}-1 & 1 \\ 0 & T & 0 \\ 0 & \frac{T}{2} & 0 \end{bmatrix} \|... \|colspan=2\|o_a,b \|colspan=2\|e_a,b
colspan=2\| $a \not\equiv b$ $\ (\mathrm{mod}\ 2)$ \| I or III \| T odd \| $\begin{bmatrix}$ 1 & \frac{T-1}{2} & 0 \\ 0 & T & 0 \\ 0 & \frac{T-1}{2} & 1 \end{bmatrix} \|... \|o_a,b \|colspan=2\|e_a,b \|o_a,b

class="wikitable sortable" style="text-align: center"

|+Quadrilateral Goldberg-Coxeter operators

Class

Edge factor
T = a² + b²

Matrix

\mathbf{M}_x

Master square

dxd

| 0

| I

| 1

| $\begin{bmatrix}$

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{bmatrix}

| 100px

|100px
o₁ = S

|colspan=2|100px
e₁ = d

|100px
o₁ = dd = S

| 0

| I

| 4

| $\begin{bmatrix}$

1 & 1 & 1 \\

0 & 4 & 0 \\

0 & 2 & 0

\end{bmatrix}

| 100px

|colspan=2| 100px
o₂ = o = j²

|colspan=2| 100px
e₂ = e = a²

| 0

| I

| 9

| $\begin{bmatrix}$

1 & 4 & 0 \\

0 & 9 & 0 \\

0 & 4 & 1

\end{bmatrix}

| 100px

| 100px
o₃

|colspan=2| 100px
e₃

| 100px
o₃

| 0

| I

| 16

| $\begin{bmatrix}$

1 & 7 & 1 \\

0 & 16 & 0 \\

0 & 8 & 0

\end{bmatrix}

| 100px

|colspan=2|100px
o₄ = oo = j⁴

|colspan=2| 100px
e₄ = ee = a⁴

| 0

| I

| 25

| $\begin{bmatrix}$

1 & 12 & 0 \\

0 & 25 & 0 \\

0 & 12 & 1

\end{bmatrix}

| 100px

|100px
o₅ = o_2,1o_1,2 = prp

|colspan=2|e₅ = e_2,1e_1,2

|100px
o₅= dprpd

| 0

| I

| 36

| $\begin{bmatrix}$

1 & 17 & 1 \\

0 & 36 & 0 \\

0 & 18 & 0

\end{bmatrix}

| 100px

|colspan=2|100px
o₆ = o₂o₃

|colspan=2|e₆ = e₂e₃

| 0

| I

| 49

| $\begin{bmatrix}$

1 & 24 & 0 \\

0 & 49 & 0 \\

0 & 24 & 1

\end{bmatrix}

| 100px

|100px
o₇

|colspan=2|e₇

|100px
o₇

| 0

| I

| 64

| $\begin{bmatrix}$

1 & 31 & 1 \\

0 & 64 & 0 \\

0 & 32 & 0

\end{bmatrix}

| 100px

|colspan=2|100px
o₈ = o³ = j⁶

|colspan=2| e₈ = e³ = a⁶

| 0

| I

| 81

| $\begin{bmatrix}$

1 & 40 & 0 \\

0 & 81 & 0 \\

0 & 40 & 1

\end{bmatrix}

| 100px

|100px
o₉ = o₃²

|colspan=2|
e₉ = e₃²

|100px
o₉

| 0

| I

| 100

| $\begin{bmatrix}$

1 & 49 & 1 \\

0 & 100 & 0 \\

0 & 50 & 0

\end{bmatrix}

| 100px

|colspan=2|100px
o₁₀ = oo_2,1o_1,2

|colspan=2|e₁₀ = ee_2,1e_1,2

| 1

| II

| 2

| $\begin{bmatrix}$

1 & 0 & 1 \\

0 & 2 & 0 \\

0 & 1 & 0

\end{bmatrix}

| 100px

|colspan=2| 100px
o_1,1 = j

|colspan=2| 100px
e_1,1 = a

| 2

| II

| 8

| $\begin{bmatrix}$

1 & 3 & 1 \\

0 & 8 & 0 \\

0 & 4 & 0

\end{bmatrix}

| 100px

|colspan=2| 100px
o_2,2 = j³

|colspan=2| 100px
e_2,2 = a³

| 2

| III

| 5

| $\begin{bmatrix}$

1 & 2 & 0 \\

0 & 5 & 0 \\

0 & 2 & 1

\end{bmatrix}

| 100px

| 100px
o_1,2 = p

|colspan=2| 100px
e_1,2 = dp = pd

| 100px
p

colspan=2|

a \equiv b

\ (\mathrm{mod}\ 2)

| I, II, or III

| T even

| $\begin{bmatrix}$

1 & \frac{T}{2}-1 & 1 \\

0 & T & 0 \\

0 & \frac{T}{2} & 0

\end{bmatrix}

|...

|colspan=2|o_a,b

|colspan=2|e_a,b

colspan=2|

a \not\equiv b

\ (\mathrm{mod}\ 2)

| I or III

| T odd

| $\begin{bmatrix}$

1 & \frac{T-1}{2} & 0 \\

0 & T & 0 \\

0 & \frac{T-1}{2} & 1

\end{bmatrix}

|...

|o_a,b

|colspan=2|e_a,b

|o_a,b

Examples

=Archimedean and Catalan solids=

Conway's original set of operators can create all of the Archimedean solids and Catalan solids, using the Platonic solids as seeds. (Note that the r operator is not necessary to create both chiral forms.)

Image:truncated tetrahedron.png| Truncated tetrahedron
tT

Image:cuboctahedron.png| Cuboctahedron
aC = aO = eT

Image:truncated hexahedron.png| Truncated cube
tC

Image:truncated octahedron.png| Truncated octahedron
tO = bT

Image:small rhombicuboctahedron.png| Rhombicuboctahedron
eC = eO

Image:Great rhombicuboctahedron.png| truncated cuboctahedron
bC = bO

Image:snub hexahedron.png| snub cube
sC = sO

Image:icosidodecahedron.png| icosidodecahedron
aD = aI

Image:truncated dodecahedron.png| truncated dodecahedron
tD

Image:truncated icosahedron.png| truncated icosahedron
tI

Image:small rhombicosidodecahedron.png| rhombicosidodecahedron
eD = eI

Image:Great rhombicosidodecahedron.png| truncated icosidodecahedron
bD = bI

Image:snub dodecahedron ccw.png| snub dodecahedron
sD = sI

Image:triakistetrahedron.jpg| Triakis tetrahedron
kT

Image:rhombicdodecahedron.jpg| Rhombic dodecahedron
jC = jO = oT

Image:triakisoctahedron.jpg| Triakis octahedron
kO

Image:tetrakishexahedron.jpg| Tetrakis hexahedron
kC = mT

Image:deltoidalicositetrahedron.jpg| Deltoidal icositetrahedron
oC = oO

Image:disdyakisdodecahedron.jpg| Disdyakis dodecahedron
mC = mO

Image:pentagonalicositetrahedronccw.jpg| Pentagonal icositetrahedron
gC = gO

Image:rhombictriacontahedron.svg| Rhombic triacontahedron
jD = jI

Image:triakisicosahedron.jpg| Triakis icosahedron
kI

Image:Pentakisdodecahedron.jpg| Pentakis dodecahedron
kD

Image:Deltoidalhexecontahedron.jpg| Deltoidal hexecontahedron
oD = oI

Image:Disdyakistriacontahedron.jpg| Disdyakis triacontahedron
mD = mI

Image:Pentagonalhexecontahedronccw.jpg| Pentagonal hexecontahedron
gD = gI

=Composite operators=

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.

File:Uniform polyhedron-53-t12.svg|tI

File:Rectified truncated icosahedron.png|atI

File:truncated truncated icosahedron.png|ttI

File:Conway polyhedron Dk6k5tI.png|ztI = ttD

File:Expanded truncated icosahedron.png|etI

File:Truncated rectified truncated icosahedron.png|btI

File:Snub rectified truncated icosahedron.png|stI

File:Pentakisdodecahedron.jpg|dtI = nI = kD

File:Joined truncated icosahedron.png|jtI

File:kissed kissed dodecahedron.png|ntI = kkD

File:Conway polyhedron K6k5tI.png|ktI

File:ortho truncated icosahedron.png|otI

File:Meta_truncated_icosahedron.png|mtI

File:Gyro_truncated_icosahedron.png|gtI

=On the plane=

Each of the convex uniform tilings and their duals can be created by applying Conway operators to the regular tilings Q, H, and Δ.

File:1-uniform_n5.svg|Square tiling
Q = dQ = aQ = eQ
= jQ = oQ

File:1-uniform_n2.svg|Truncated square tiling
tQ = bQ

File:1-uniform_2_dual.svg|Tetrakis square tiling
kQ = mQ

File:1-uniform_n9.svg|Snub square tiling
sQ

File:1-uniform_9_dual.svg|Cairo pentagonal tiling
gQ

File:1-uniform_n1.svg|Hexagonal tiling
H = dΔ = tΔ

File:1-uniform_n7.svg|Trihexagonal tiling
aH = aΔ

File:1-uniform_n4.svg|Truncated hexagonal tiling
tH

File:1-uniform_n6.svg|Rhombitrihexagonal tiling
eH = eΔ

File:1-uniform_n3.svg|Truncated trihexagonal tiling
bH = bΔ

File:1-uniform_n10.svg|Snub trihexagonal tiling
sH = sΔ

File:1-uniform_1_dual.svg|Triangle tiling
Δ = dH = kH

File:1-uniform_7_dual.svg|Rhombille tiling
jΔ = jH

File:1-uniform_4_dual.svg|Triakis triangular tiling
kΔ

File:1-uniform_6_dual.svg|Deltoidal trihexagonal tiling
oΔ = oH

File:1-uniform_3_dual.svg|Kisrhombille tiling
mΔ = mH

File:1-uniform_10_dual.svg|Floret pentagonal tiling
gΔ = gH

=On a torus=

Conway operators can also be applied to toroidal polyhedra and polyhedra with multiple holes.

File:Toroidal monohedron.png|A 1x1 regular square torus, {4,4}_1,0

File:Torus map 4x4.png|A regular 4x4 square torus, {4,4}_4,0

File:First truncated square tiling on torus24x12.png|tQ24×12 projected to torus

File:Truncated square tiling on torus24x12.png|taQ24×12 projected to torus

File:Conway_torus_ActQ24x8.png|actQ24×8 projected to torus

File:Truncated hexagonal tiling torus24x12.png|tH24×12 projected to torus

File:Truncated trihexagonal tiling on torus24x8.png|taH24×8 projected to torus

Conway torus kH24-12.png|kH24×12 projected to torus

References

External links

[https://levskaya.github.io/polyhedronisme/ polyHédronisme]: generates polyhedra in HTML5 canvas, taking Conway notation as input

Category:Elementary geometry

Category:Polyhedra

Category:Mathematical notation

Category:John Horton Conway

Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd	Notes
class="wikitable sortable" style="text-align: center" \|+Original Conway operators
1 \| $\begin{bmatrix}$ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \| 100px Seed: S \|colspan="2"\| 100px Dual: d \| 100px Seed: dd = S \| Dual replaces each face with a vertex, and each vertex with a face.
2 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 0 \end{bmatrix} \|colspan="2"\| 100px Join: j \|colspan="2"\| 100px Ambo: a \| Join creates quadrilateral faces. Ambo creates degree-4 vertices, and is also called rectification, or the medial graph in graph theory.{{mathworld\|id=Rectification\|title=Rectification}}
3 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px Kis: k \| 100px Needle: n \| 100px Zip: z \| 100px Truncate: t \| Kis raises a pyramid on each face, and is also called akisation, Kleetope, cumulation,{{mathworld\|id=Cumulation\|title=Cumulation}} accretion, or pyramid-augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges.{{mathworld\|id=Truncation\|title=Truncation}} Zip is also called bitruncation.
4 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 4 & 0 \\ 0 & 2 & 0 \end{bmatrix} \|colspan=2\| 100px Ortho: o = jj \|colspan=2\| 100px Expand: e = aa \|
5 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 5 & 0 \\ 0 & 2 & 0 \end{bmatrix} \| 100px Gyro: g \| gd = rgr \| sd = rsr \| 100px Snub: s \| Chiral operators. See Snub (geometry). Contrary to Hart, gd is not the same as g: it is its chiral pair.{{cite web\|title=Antiprism - Chirality issue in conway\|url=https://groups.google.com/forum/#!topic/antiprism/6NcYnKP4mTc}}
6 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 6 & 0 \\ 0 & 4 & 0 \end{bmatrix} \|colspan="2"\| 100px Meta: m = kj \|colspan="2"\| 100px Bevel: b = ta \|

n	Edge factor	Matrix $\mathbf{M}_x$	x	xd	dx	dxd
class="wikitable sortable" style="text-align: center" \|+Meta/Bevel operators
0 \| 3 \| $\begin{bmatrix}$ 1 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 2 & 0 \end{bmatrix} \|100px k = m₀ \| 100px n \|100px z = b₀ \| 100px t
1 \| 6 \| $\begin{bmatrix}$ 1 & 1 & 1 \\ 0 & 6 & 0 \\ 0 & 4 & 0 \end{bmatrix} \|colspan="2"\| 100px m = m₁ = kj \|colspan="2"\| 100px b = b₁ = ta
2 \|9 \| $\begin{bmatrix}$ 1 & 2 & 1 \\ 0 & 9 & 0 \\ 0 & 6 & 0 \end{bmatrix} \|100px m₂ \|100px m₂d \|100px b₂ \|100px b₂d
3 \|12 \| $\begin{bmatrix}$ 1 & 3 & 1 \\ 0 & 12 & 0 \\ 0 & 8 & 0 \end{bmatrix} \|100px m₃ \|m₃d \|b₃ \|b₃d
n \| 3n+3 \| $\begin{bmatrix}$ 1 & n & 1 \\ 0 & 3n+3 & 0 \\ 0 & 2n+2 & 0 \end{bmatrix} \|m_n \|m_nd \|b_n \|b_nd

Conway polyhedron notation

Operators

= Matrix representation =

= Number of operators =

Original operations

Seeds

Extended operations

Indexed extended operations

=Augmentation=

=Meta/Bevel=

=Medial=

=Goldberg-Coxeter=

== Triangular ==

== Quadrilateral ==

Examples

=Archimedean and Catalan solids=

=Composite operators=

=On the plane=

=On a torus=

See also

References

External links