Conway polyhedron notation

File:Conway relational chart.png

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.

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.

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, Template:Mvar represents a truncated cube, and Template:Mvar, parsed as Template:Math, is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.g., a dual cube is an octahedron: Template:Math. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators Template:Mvar (ambo), Template:Mvar (bevel), Template:Mvar (dual), Template:Mvar (expand), Template:Mvar (gyro), Template:Mvar (join), Template:Mvar (kis), Template:Mvar (meta), Template:Mvar (ortho), Template:Mvar (snub), and Template:Mvar (truncate), while Hart added Template:Mvar (reflect) and Template:Mvar (propellor). Later implementations named further operators, sometimes referred to as "extended" operators. 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 (Template:Math), while a truncation after ambo produces bevel (Template:Math).

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, i.e. Template:Tmath, and a truncated cuboctahedron is Template:Tmath. 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 Template:Math where n is the number of sides of a face, while chiral operators correspond to cyclic groups Template:Math 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. 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.

Fundamental domains of faces with $n$ sides

3 (Triangle)	4 (Square)	5 (Pentagon)	6 (Hexagon)
File:Triangle chambers.svg	File:Quadrilateral chambers.svg	File:Pentagon chambers.svg	File:Hexagon chambers.svg
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: 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

x	$\left[\begin{array}{ccc}a& b& c\\ 0& d& 0\\ a^{\prime }& b^{\prime }& c^{\prime }\end{array}\right]={\mathbf{𝐌}}_x$
xd	$\left[\begin{array}{ccc}c& b& a\\ 0& d& 0\\ c^{\prime }& b^{\prime }& a^{\prime }\end{array}\right]={\mathbf{𝐌}}_x{\mathbf{𝐌}}_d$
dx	$\left[\begin{array}{ccc}{a}^{\prime }& b^{\prime }& c^{\prime }\\ 0& d& 0\\ a& b& c\end{array}\right]={\mathbf{𝐌}}_d{\mathbf{𝐌}}_x$
dxd	$\left[\begin{array}{ccc}{c}^{\prime }& b^{\prime }& a^{\prime }\\ 0& d& 0\\ c& b& a\end{array}\right]={\mathbf{𝐌}}_d{\mathbf{𝐌}}_x{\mathbf{𝐌}}_d$

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 $v^{\prime },e^{\prime },f^{\prime }$ are the vertices, edges, and faces of the result, then

${\mathbf{𝐌}}_x\left[\begin{array}{c}v\\ e\\ f\end{array}\right]=\left[\begin{array}{c}{v}^{\prime }\\ e^{\prime }\\ f^{\prime }\end{array}\right]$.

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:

${\mathbf{𝐌}}_S=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]={\mathbf{𝐈}}_3$, ${\mathbf{𝐌}}_d=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

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.

Original Conway operators

Edge factor	Matrix ${\mathbf{𝐌}}_x$	x	xd	dx	dxd	Notes
1	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	File:Conway C.png Seed: S	File:Conway dC.png Dual: d		File:Conway C.png Seed: dd = S	Dual replaces each face with a vertex, and each vertex with a face.
2	$\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 0& 1& 0\end{array}\right]$	File:Conway jC.png Join: j		File:Conway aC.png Ambo: a		Join creates quadrilateral faces. Ambo creates degree-4 vertices, and is also called rectification, or the medial graph in graph theory.
3	$\left[\begin{array}{ccc}1& 0& 1\\ 0& 3& 0\\ 0& 2& 0\end{array}\right]$	File:Conway kC.png Kis: k	File:Conway kdC.png Needle: n	File:Conway dkC.png Zip: z	File:Conway tC.png Truncate: t	Kis raises a pyramid on each face, and is also called akisation, Kleetope, cumulation, accretion, or pyramid-augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges. Zip is also called bitruncation.
4	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 4& 0\\ 0& 2& 0\end{array}\right]$	File:Conway oC.png Ortho: o = jj		File:Conway eC.png Expand: e = aa
5	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 5& 0\\ 0& 2& 0\end{array}\right]$	File:Conway gC.png Gyro: g	gd = rgr	sd = rsr	File:Conway sC.png Snub: s	Chiral operators. See Snub (geometry). Contrary to Hart, gd is not the same as g: it is its chiral pair.
6	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 6& 0\\ 0& 4& 0\end{array}\right]$	File:Conway mC.png Meta: m = kj		File:Conway bC.png 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: Template:Div col

• 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.

Irreducible extended operators

Edge factor	Matrix ${\mathbf{𝐌}}_x$	x	xd	dx	dxd	Notes
4	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 4& 0\\ 0& 1& 1\end{array}\right]$	File:Conway cC.png Chamfer: c	File:Conway duC.png cd = du	File:Conway dcC.png dc = ud	File:Conway uC.png Subdivide: u	Chamfer is the join-form of l. See Chamfer (geometry).
5	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 5& 0\\ 0& 2& 1\end{array}\right]$	File:Conway pC.png Propeller: p	File:Conway dpC.png dp = pd		File:Conway pC.png dpd = p	Chiral operators. The propeller operator was developed by George Hart.
5	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 5& 0\\ 0& 2& 1\end{array}\right]$	File:Conway lC.png Loft: l	File:Conway ldC.png ld	File:Conway dlC.png dl	File:Conway dldC.png dld
6	$\left[\begin{array}{ccc}1& 3& 0\\ 0& 6& 0\\ 0& 2& 1\end{array}\right]$	File:Conway qC.png Quinto: q	File:Conway qdC.png qd	File:Conway dqC.png dq	File:Conway dqdC.png dqd
6	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 6& 0\\ 0& 3& 1\end{array}\right]$	File:Conway L0C.png Join-lace: L0	File:Conway Diagram L0d.png L0d	File:Conway dL0C.png dL0	File:Conway dL0d.png dL0d	See below for explanation of join notation.
7	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 7& 0\\ 0& 4& 1\end{array}\right]$	File:Conway LC.png Lace: L	File:Conway L0dC.png Ld	File:Conway dLC.png dL	File:Conway dLdC.png dLd
7	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 7& 0\\ 0& 4& 0\end{array}\right]$	File:Conway KC.png Stake: K	File:Conway KdC.png Kd	File:Conway dKC.png dK	File:Conway dKdC.png dKd
7	$\left[\begin{array}{ccc}1& 4& 0\\ 0& 7& 0\\ 0& 2& 1\end{array}\right]$	File:Conway wC.png Whirl: w	wd = dv	File:Conway dwC.png vd = dw	Volute: v	Chiral operators.
8	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 8& 0\\ 0& 5& 0\end{array}\right]$	File:Conway (kk)0C.png Join-kis-kis: Template:Tmath	File:Conway (kk)0dC.png Template:Tmath	File:Conway d(kk)0C.png Template:Tmath	File:Conway d(kk)0dC.png Template:Tmath	Sometimes named J. See below for explanation of join notation. The non-join-form, kk, is not irreducible.
10	$\left[\begin{array}{ccc}1& 3& 1\\ 0& 10& 0\\ 0& 6& 0\end{array}\right]$	File:Conway XC.png Cross: X	File:Conway XdC.png Xd	File:Conway dXC.png dX	File:Conway dXdC.png 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.

Augmentation operator	x	File:Conway kC.png k	File:Conway lC.png l	File:Conway LC.png L	File:Conway KC.png K	File:Conway kkC.png (kk)
Corresponding join-form operator	x0	File:Conway jC.png k0 = j	File:Conway cC.png l0 = c	File:Conway L0C.png L0	File:Conway K0C.png K0 = jk	File:Conway (kk)0C.png (kk)0
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.

Meta/Bevel operators

n	Edge factor	Matrix ${\mathbf{𝐌}}_x$	x	xd	dx	dxd
0	3	$\left[\begin{array}{ccc}1& 0& 1\\ 0& 3& 0\\ 0& 2& 0\end{array}\right]$	File:Conway kC.png k = m0	File:Conway kdC.png n	File:Conway dkC.png z = b0	File:Conway tC.png t
1	6	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 6& 0\\ 0& 4& 0\end{array}\right]$	File:Conway mC.png m = m1 = kj		File:Conway bC.png b = b1 = ta
2	9	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 9& 0\\ 0& 6& 0\end{array}\right]$	File:Conway m3C.png m2	File:Conway m3dC.png m2d	File:Conway b3C.png b2	File:Conway dm3dC.png b2d
3	12	$\left[\begin{array}{ccc}1& 3& 1\\ 0& 12& 0\\ 0& 8& 0\end{array}\right]$	File:Conway m4C.png m3	m3d	b3	b3d
n	3n+3	$\left[\begin{array}{ccc}1& n& 1\\ 0& 3n+3& 0\\ 0& 2n+2& 0\end{array}\right]$	mn	mnd	bn	bnd

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.

Medial operators

n	Edge factor	Matrix ${\mathbf{𝐌}}_x$	x	xd	dx	dxd
1	4	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 4& 0\\ 0& 2& 0\end{array}\right]$	File:Conway oC.png M1 = o = jj		File:Conway eC.png e = aa
2	7	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 7& 0\\ 0& 4& 0\end{array}\right]$	File:Conway MC.png Medial: M = M2	File:Conway MdC.png Md	File:Conway dMC.png dM	File:Conway dMdC.png dMd
n	3n+1	$\left[\begin{array}{ccc}1& n& 1\\ 0& 3n+1& 0\\ 0& 2n& 0\end{array}\right]$	Mn	Mnd	dMn	dMnd

Goldberg-Coxeter

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction. 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: Template:Tmath. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. c_a,0 = c_a.
• Class II: Template:Tmath. 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

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	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	File:Subdivided triangle 01 00.svg	File:Conway C.png u1 = S	File:Conway dC.png d		File:Conway C.png c1 = S
2	0	I	4	$\left[\begin{array}{ccc}1& 1& 0\\ 0& 4& 0\\ 0& 2& 1\end{array}\right]$	File:Subdivided triangle 02 00.svg	File:Conway uC.png u2 = u	File:Conway dcC.png dc	File:Conway duC.png du	File:Conway cC.png c2 = c
3	0	I	9	$\left[\begin{array}{ccc}1& 2& 1\\ 0& 9& 0\\ 0& 6& 0\end{array}\right]$	File:Subdivided triangle 03 00.svg	File:Conway ktC.png u3 = nn	File:Conway dtkC.png nk	File:Conway dktC.png zt	File:Conway tkC.png c3 = zz
4	0	I	16	$\left[\begin{array}{ccc}1& 5& 0\\ 0& 16& 0\\ 0& 10& 1\end{array}\right]$	File:Subdivided triangle 04 00.svg	File:Conway u4C.png u4 = uu	uud = dcc	duu = ccd	c4 = cc
5	0	I	25	$\left[\begin{array}{ccc}1& 8& 0\\ 0& 25& 0\\ 0& 16& 1\end{array}\right]$	File:Subdivided triangle 05 00.svg	File:Conway u5C.png u5	u5d = dc5	du5 = c5d	c5
6	0	I	36	$\left[\begin{array}{ccc}1& 11& 1\\ 0& 36& 0\\ 0& 24& 0\end{array}\right]$	File:Subdivided triangle 06 00.svg	File:Conway u6C.png u6 = unn	unk	czt	u6 = czz
7	0	I	49	$\left[\begin{array}{ccc}1& 16& 0\\ 0& 49& 0\\ 0& 32& 1\end{array}\right]$	File:Subdivided triangle 07 00.svg	File:Conway u7.png u7 = u2,1u1,2 = vrv	vrvd = dwrw	dvrv = wrwd	c7 = c2,1c1,2 = wrw
8	0	I	64	$\left[\begin{array}{ccc}1& 21& 0\\ 0& 64& 0\\ 0& 42& 1\end{array}\right]$	File:Subdivided triangle 08 00.svg	File:Conway u8C.png u8 = u3	u3d = dc3	du3 = c3d	c8 = c3
9	0	I	81	$\left[\begin{array}{ccc}1& 26& 1\\ 0& 81& 0\\ 0& 54& 0\end{array}\right]$	File:Subdivided triangle 09 00.svg	File:Conway u9C.png u9 = n4	n3k = kz3	tn3 = z3t	c9 = z4
1	1	II	3	$\left[\begin{array}{ccc}1& 0& 1\\ 0& 3& 0\\ 0& 2& 0\end{array}\right]$	File:Subdivided triangle 01 01.svg	File:Conway kdC.png u1,1 = n	File:Conway kC.png k	File:Conway tC.png t	File:Conway dkC.png c1,1 = z
2	1	III	7	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 7& 0\\ 0& 4& 1\end{array}\right]$	File:Subdivided triangle 02 01.svg	v = u2,1	File:Conway dwC.png vd = dw	dv = wd	File:Conway wC.png w = c2,1
3	1	III	13	$\left[\begin{array}{ccc}1& 4& 0\\ 0& 13& 0\\ 0& 8& 1\end{array}\right]$	File:Subdivided triangle 03 01.svg	u3,1	u3,1d = dc3,1	du3,1 = c3,1d	File:Conway w3C.png c3,1
3	2	III	19	$\left[\begin{array}{ccc}1& 6& 0\\ 0& 19& 0\\ 0& 12& 1\end{array}\right]$	File:Subdivided triangle 03 02.svg	u3,2	u3,2d = dc3,2	du3,2 = c3,2d	File:Conway w3-2.png c3,2
4	3	III	37	$\left[\begin{array}{ccc}1& 12& 0\\ 0& 37& 0\\ 0& 24& 1\end{array}\right]$	File:Subdivided triangle 04 03.svg	u4,3	u4,3d = dc4,3	du4,3 = c4,3d	File:Conway w4-3C.png c4,3
5	4	III	61	$\left[\begin{array}{ccc}1& 20& 0\\ 0& 61& 0\\ 0& 40& 1\end{array}\right]$	File:Subdivided triangle 05 04.svg	u5,4	u5,4d = dc5,4	du5,4 = c5,4d	File:Conway w5-4C.png c5,4
6	5	III	91	$\left[\begin{array}{ccc}1& 30& 0\\ 0& 91& 0\\ 0& 60& 1\end{array}\right]$	File:Subdivided triangle 06 05.svg	u6,5 = u1,2u1,3	u6,5d = dc6,5	du6,5 = c6,5d	File:Conway w6-5C.png c6,5=c1,2c1,3
7	6	III	127	$\left[\begin{array}{ccc}1& 42& 0\\ 0& 127& 0\\ 0& 84& 1\end{array}\right]$	File:Subdivided triangle 07 06.svg	u7,6	u7,6d = dc7,6	du7,6 = c7,6d	File:Conway w7C.png c7,6
8	7	III	169	$\left[\begin{array}{ccc}1& 56& 0\\ 0& 169& 0\\ 0& 112& 1\end{array}\right]$	File:Subdivided triangle 08 07.svg	u8,7 = u3,12	u8,7d = dc8,7	du8,7 = c8,7d	File:Conway w8C.png c8,7 = c3,12
9	8	III	217	$\left[\begin{array}{ccc}1& 72& 0\\ 0& 217& 0\\ 0& 144& 1\end{array}\right]$	File:Subdivided triangle 09 08.svg	u9,8 = u2,1u5,1	u9,8d = dc9,8	du9,8 = c9,8d	File:Conway w9C.png c9,8 = c2,1c5,1
$a\equiv b$$(\mathrm{m}\mathrm{o}\mathrm{d}3)$		I, II, or III	$T\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d}3)$	$\left[\begin{array}{ccc}1& \frac{T}{3}-1& 1\\ 0& T& 0\\ 0& \frac{2}{3}T& 0\end{array}\right]$	...	ua,b	ua,bd = dca,b	dua,b = ca,bd	ca,b
$a\equiv ̸b$$(\mathrm{m}\mathrm{o}\mathrm{d}3)$		I or III	$T\equiv 1$$(\mathrm{m}\mathrm{o}\mathrm{d}3)$	$\left[\begin{array}{ccc}1& \frac{T-1}{3}& 0\\ 0& T& 0\\ 0& 2\frac{T-1}{3}& 1\end{array}\right]$	...	ua,b	ua,bd = dca,b	dua,b = ca,bd	ca,b

By basic number theory, for any values of a and b, $T\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}3)$.

Quadrilateral

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	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	File:Subdivided square 01 00.svg	File:Conway C.png o1 = S	File:Conway dC.png e1 = d		File:Conway C.png o1 = dd = S
2	0	I	4	$\left[\begin{array}{ccc}1& 1& 1\\ 0& 4& 0\\ 0& 2& 0\end{array}\right]$	File:Subdivided square 02 00.svg	File:Conway oC.png o2 = o = j2		File:Conway eC.png e2 = e = a2
3	0	I	9	$\left[\begin{array}{ccc}1& 4& 0\\ 0& 9& 0\\ 0& 4& 1\end{array}\right]$	File:Subdivided square 03 00.svg	File:Conway o3C.png o3	File:Conway e3C.png e3		File:Conway o3C.png o3
4	0	I	16	$\left[\begin{array}{ccc}1& 7& 1\\ 0& 16& 0\\ 0& 8& 0\end{array}\right]$	File:Subdivided square 04 00.svg	File:Conway deeC.png o4 = oo = j4		File:Conway eeC.png e4 = ee = a4
5	0	I	25	$\left[\begin{array}{ccc}1& 12& 0\\ 0& 25& 0\\ 0& 12& 1\end{array}\right]$	File:Subdivided square 05 00.svg	File:Conway o5C.png o5 = o2,1o1,2 = prp	e5 = e2,1e1,2		File:Conway o5C.png o5= dprpd
6	0	I	36	$\left[\begin{array}{ccc}1& 17& 1\\ 0& 36& 0\\ 0& 18& 0\end{array}\right]$	File:Subdivided square 06 00.svg	File:Conway o6C.png o6 = o2o3		e6 = e2e3
7	0	I	49	$\left[\begin{array}{ccc}1& 24& 0\\ 0& 49& 0\\ 0& 24& 1\end{array}\right]$	File:Subdivided square 07 00.svg	File:Conway o7C.png o7	e7		File:Conway o7C.png o7
8	0	I	64	$\left[\begin{array}{ccc}1& 31& 1\\ 0& 64& 0\\ 0& 32& 0\end{array}\right]$	File:Subdivided square 08 00.svg	File:Conway o8C.png o8 = o3 = j6		e8 = e3 = a6
9	0	I	81	$\left[\begin{array}{ccc}1& 40& 0\\ 0& 81& 0\\ 0& 40& 1\end{array}\right]$	File:Subdivided square 09 00.svg	File:Conway o9C.png o9 = o32	e9 = e32		File:Conway o9C.png o9
10	0	I	100	$\left[\begin{array}{ccc}1& 49& 1\\ 0& 100& 0\\ 0& 50& 0\end{array}\right]$	File:Subdivided square 10 00.svg	File:Conway o10C.png o10 = oo2,1o1,2		e10 = ee2,1e1,2
1	1	II	2	$\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 0& 1& 0\end{array}\right]$	File:Subdivided square 01 01.svg	File:Conway jC.png o1,1 = j		File:Conway aC.png e1,1 = a
2	2	II	8	$\left[\begin{array}{ccc}1& 3& 1\\ 0& 8& 0\\ 0& 4& 0\end{array}\right]$	File:Subdivided square 02 02.svg	File:Conway daaaC.png o2,2 = j3		File:Conway aaaC.png e2,2 = a3
1	2	III	5	$\left[\begin{array}{ccc}1& 2& 0\\ 0& 5& 0\\ 0& 2& 1\end{array}\right]$	File:Subdivided square 01 02.svg	File:Conway pC.png o1,2 = p	File:Conway dpC.png e1,2 = dp = pd		File:Conway pC.png p
$a\equiv b$$(\mathrm{m}\mathrm{o}\mathrm{d}2)$		I, II, or III	T even	$\left[\begin{array}{ccc}1& \frac{T}{2}-1& 1\\ 0& T& 0\\ 0& \frac{T}{2}& 0\end{array}\right]$	...	oa,b		ea,b
$a\equiv ̸b$$(\mathrm{m}\mathrm{o}\mathrm{d}2)$		I or III	T odd	$\left[\begin{array}{ccc}1& \frac{T-1}{2}& 0\\ 0& T& 0\\ 0& \frac{T-1}{2}& 1\end{array}\right]$	...	oa,b	ea,b		oa,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.)

• Archimedean
• Truncated tetrahedron tT
  Truncated tetrahedron
  tT
• Cuboctahedron aC = aO = eT
  Cuboctahedron
  aC = aO = eT
• Truncated cube tC
  Truncated cube
  tC
• Truncated octahedron tO = bT
  Truncated octahedron
  tO = bT
• Rhombicuboctahedron eC = eO
  Rhombicuboctahedron
  eC = eO
• truncated cuboctahedron bC = bO
  truncated cuboctahedron
  bC = bO
• snub cube sC = sO
  snub cube
  sC = sO
• icosidodecahedron aD = aI
  icosidodecahedron
  aD = aI
• truncated dodecahedron tD
  truncated dodecahedron
  tD
• truncated icosahedron tI
  truncated icosahedron
  tI
• rhombicosidodeca­hedron eD = eI
  rhombicosidodeca­hedron
  eD = eI
• truncated icosidodecahedron bD = bI
  truncated icosidodecahedron
  bD = bI
• snub dodecahedron sD = sI
  snub dodecahedron
  sD = sI

• Catalan
• Triakis tetrahedron kT
  Triakis tetrahedron
  kT
• Rhombic dodecahedron jC = jO = oT
  Rhombic dodecahedron
  jC = jO = oT
• Triakis octahedron kO
  Triakis octahedron
  kO
• Tetrakis hexahedron kC = mT
  Tetrakis hexahedron
  kC = mT
• Deltoidal icositetrahedron oC = oO
  Deltoidal icositetrahedron
  oC = oO
• Disdyakis dodecahedron mC = mO
  Disdyakis dodecahedron
  mC = mO
• Pentagonal icositetrahedron gC = gO
  Pentagonal icositetrahedron
  gC = gO
• Rhombic triacontahedron jD = jI
  Rhombic triacontahedron
  jD = jI
• Triakis icosahedron kI
  Triakis icosahedron
  kI
• Pentakis dodecahedron kD
  Pentakis dodecahedron
  kD
• Deltoidal hexecontahedron oD = oI
  Deltoidal hexecontahedron
  oD = oI
• Disdyakis triacontahedron mD = mI
  Disdyakis triacontahedron
  mD = mI
• Pentagonal hexecontahedron gD = gI
  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.

• tI
  tI
• atI
  atI
• ttI
  ttI
• ztI = ttD
  ztI = ttD
• etI
  etI
• btI
  btI
• stI
  stI

• Duals
• dtI = nI = kD
  dtI = nI = kD
• jtI
  jtI
• ntI = kkD
  ntI = kkD
• ktI
  ktI
• otI
  otI
• mtI
  mtI
• gtI
  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.

• Square tiling Q = dQ = aQ = eQ = jQ = oQ
  Square tiling
  Q = dQ = aQ = eQ
  = jQ = oQ
• Truncated square tiling tQ = bQ
  Truncated square tiling
  tQ = bQ
• Tetrakis square tiling kQ = mQ
  Tetrakis square tiling
  kQ = mQ
• Snub square tiling sQ
  Snub square tiling
  sQ
• Cairo pentagonal tiling gQ
  Cairo pentagonal tiling
  gQ

• Hexagonal tiling H = dΔ = tΔ
  Hexagonal tiling
  H = dΔ = tΔ
• Trihexagonal tiling aH = aΔ
  Trihexagonal tiling
  aH = aΔ
• Truncated hexagonal tiling tH
  Truncated hexagonal tiling
  tH
• Rhombitrihexagonal tiling eH = eΔ
  Rhombitrihexagonal tiling
  eH = eΔ
• Truncated trihexagonal tiling bH = bΔ
  Truncated trihexagonal tiling
  bH = bΔ
• Snub trihexagonal tiling sH = sΔ
  Snub trihexagonal tiling
  sH = sΔ

• Triangle tiling Δ = dH = kH
  Triangle tiling
  Δ = dH = kH
• Rhombille tiling jΔ = jH
  Rhombille tiling
  jΔ = jH
• Triakis triangular tiling kΔ
  Triakis triangular tiling
  kΔ
• Deltoidal trihexagonal tiling oΔ = oH
  Deltoidal trihexagonal tiling
  oΔ = oH
• Kisrhombille tiling mΔ = mH
  Kisrhombille tiling
  mΔ = mH
• Floret pentagonal tiling gΔ = gH
  Floret pentagonal tiling
  gΔ = gH

On a torus

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

• A 1x1 regular square torus, {4,4}1,0
  A 1x1 regular square torus, {4,4}_1,0
• A regular 4x4 square torus, {4,4}4,0
  A regular 4x4 square torus, {4,4}_4,0
• tQ24×12 projected to torus
  tQ24×12 projected to torus
• taQ24×12 projected to torus
  taQ24×12 projected to torus
• actQ24×8 projected to torus
  actQ24×8 projected to torus
• tH24×12 projected to torus
  tH24×12 projected to torus
• taH24×8 projected to torus
  taH24×8 projected to torus
• kH24×12 projected to torus
  kH24×12 projected to torus

See also

• Symmetrohedron
• Zonohedron
• Schläfli symbol

External links

• polyHédronisme: generates polyhedra in HTML5 canvas, taking Conway notation as input