Matrix representation

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x	${\displaystyle {\begin{bmatrix}a&b&c\\0&d&0\\a'&b'&c'\end{bmatrix}}=\mathbf {M} _{x}}$
xd	${\displaystyle {\begin{bmatrix}c&b&a\\0&d&0\\c'&b'&a'\end{bmatrix}}=\mathbf {M} _{x}\mathbf {M} _{d}}$
dx	${\displaystyle {\begin{bmatrix}a'&b'&c'\\0&d&0\\a&b&c\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}}$
dxd	${\displaystyle {\begin{bmatrix}c'&b'&a'\\0&d&0\\c&b&a\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}\mathbf {M} _{d}}$

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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 ${\displaystyle \mathbf {M} _{x}}$. When x is the operator, ${\displaystyle v,e,f}$ are the vertices, edges, and faces of the seed (respectively), and ${\displaystyle v',e',f'}$ are the vertices, edges, and faces of the result, then

${\displaystyle \mathbf {M} _{x}{\begin{bmatrix}v\\e\\f\end{bmatrix}}={\begin{bmatrix}v'\\e'\\f'\end{bmatrix}}}$.

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.^[7]

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

${\displaystyle \mathbf {M} _{S}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=\mathbf {I} _{3}}$, ${\displaystyle \mathbf {M} _{d}={\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}}$

Two dual operators cancel out; dd = S, and the square of ${\displaystyle \mathbf {M} _{d}}$ 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 ${\displaystyle 2,2,4,6,6,20,28,58,82,\cdots }$ 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 ${\displaystyle 2,2,4,6,4,20,20,54,64,\cdots }$.^[8]

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.

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Operator	k	l	L	K	(kk)
x
x0	k0 = j	l0 = c	L0	K0 = jk	${\displaystyle (kk)_{0}}$
Augmentation	Pyramid	Prism	Antiprism

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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.^[4]

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Meta/Bevel operators

n	Edge factor	Matrix ${\displaystyle \mathbf {M} _{x}}$	x	xd	dx	dxd
0	3	${\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}$	k = m0	n	z = b0	t
1	6	${\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}}$	m = m1 = kj		b = b1 = ta
2	9	${\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}$	m2	m2d	b2	b2d
3	12	${\displaystyle {\begin{bmatrix}1&3&1\\0&12&0\\0&8&0\end{bmatrix}}}$	m3	m3d	b3	b3d
n	3n+3	${\displaystyle {\begin{bmatrix}1&n&1\\0&3n+3&0\\0&2n+2&0\end{bmatrix}}}$	mn	mnd	bn	bnd

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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.^[4]

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Medial operators

n	Edge factor	Matrix ${\displaystyle \mathbf {M} _{x}}$	x	xd	dx	dxd
1	4	${\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}$	M1 = o = jj		e = aa
2	7	${\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}}$	Medial: M = M2	Md	dM	dMd
n	3n+1	${\displaystyle {\begin{bmatrix}1&n&1\\0&3n+1&0\\0&2n&0\end{bmatrix}}}$	Mn	Mnd	dMn	dMnd

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Goldberg-Coxeter

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction.^[16][17] 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").^[7] 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 ${\displaystyle a\geq 1}$ and ${\displaystyle b\geq 0}$. 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: ${\displaystyle b=0}$. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. c_a,0 = c_a.
• Class II: ${\displaystyle a=b}$. 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

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Triangular Goldberg-Coxeter operators

a	b	Class	Edge factor T = a2 + ab + b2	Matrix ${\displaystyle \mathbf {M} _{x}}$	Master triangle	x	xd	dx	dxd
1	0	I	1	${\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}$		u1 = S	d		c1 = S
2	0	I	4	${\displaystyle {\begin{bmatrix}1&1&0\\0&4&0\\0&2&1\end{bmatrix}}}$		u2 = u	dc	du	c2 = c
3	0	I	9	${\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}$		u3 = nn	nk	zt	c3 = zz
4	0	I	16	${\displaystyle {\begin{bmatrix}1&5&0\\0&16&0\\0&10&1\end{bmatrix}}}$		u4 = uu	uud = dcc	duu = ccd	c4 = cc
5	0	I	25	${\displaystyle {\begin{bmatrix}1&8&0\\0&25&0\\0&16&1\end{bmatrix}}}$		u5	u5d = dc5	du5 = c5d	c5
6	0	I	36	${\displaystyle {\begin{bmatrix}1&11&1\\0&36&0\\0&24&0\end{bmatrix}}}$		u6 = unn	unk	czt	u6 = czz
7	0	I	49	${\displaystyle {\begin{bmatrix}1&16&0\\0&49&0\\0&32&1\end{bmatrix}}}$		u7 = u2,1u1,2 = vrv	vrvd = dwrw	dvrv = wrwd	c7 = c2,1c1,2 = wrw
8	0	I	64	${\displaystyle {\begin{bmatrix}1&21&0\\0&64&0\\0&42&1\end{bmatrix}}}$		u8 = u3	u3d = dc3	du3 = c3d	c8 = c3
9	0	I	81	${\displaystyle {\begin{bmatrix}1&26&1\\0&81&0\\0&54&0\end{bmatrix}}}$		u9 = n4	n3k = kz3	tn3 = z3t	c9 = z4
1	1	II	3	${\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}$		u1,1 = n	k	t	c1,1 = z
2	1	III	7	${\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}}$		v = u2,1	vd = dw	dv = wd	w = c2,1
3	1	III	13	${\displaystyle {\begin{bmatrix}1&4&0\\0&13&0\\0&8&1\end{bmatrix}}}$		u3,1	u3,1d = dc3,1	du3,1 = c3,1d	c3,1
3	2	III	19	${\displaystyle {\begin{bmatrix}1&6&0\\0&19&0\\0&12&1\end{bmatrix}}}$		u3,2	u3,2d = dc3,2	du3,2 = c3,2d	c3,2
4	3	III	37	${\displaystyle {\begin{bmatrix}1&12&0\\0&37&0\\0&24&1\end{bmatrix}}}$		u4,3	u4,3d = dc4,3	du4,3 = c4,3d	c4,3
5	4	III	61	${\displaystyle {\begin{bmatrix}1&20&0\\0&61&0\\0&40&1\end{bmatrix}}}$		u5,4	u5,4d = dc5,4	du5,4 = c5,4d	c5,4
6	5	III	91	${\displaystyle {\begin{bmatrix}1&30&0\\0&91&0\\0&60&1\end{bmatrix}}}$		u6,5 = u1,2u1,3	u6,5d = dc6,5	du6,5 = c6,5d	c6,5=c1,2c1,3
7	6	III	127	${\displaystyle {\begin{bmatrix}1&42&0\\0&127&0\\0&84&1\end{bmatrix}}}$		u7,6	u7,6d = dc7,6	du7,6 = c7,6d	c7,6
8	7	III	169	${\displaystyle {\begin{bmatrix}1&56&0\\0&169&0\\0&112&1\end{bmatrix}}}$		u8,7 = u3,12	u8,7d = dc8,7	du8,7 = c8,7d	c8,7 = c3,12
9	8	III	217	${\displaystyle {\begin{bmatrix}1&72&0\\0&217&0\\0&144&1\end{bmatrix}}}$		u9,8 = u2,1u5,1	u9,8d = dc9,8	du9,8 = c9,8d	c9,8 = c2,1c5,1
${\displaystyle a\equiv b}$${\displaystyle \ (\mathrm {mod} \ 3)}$		I, II, or III	${\displaystyle T\equiv 0\ }$${\displaystyle (\mathrm {mod} \ 3)}$	${\displaystyle {\begin{bmatrix}1&{\frac {T}{3}}-1&1\\0&T&0\\0&{\frac {2}{3}}T&0\end{bmatrix}}}$	...	ua,b	ua,bd = dca,b	dua,b = ca,bd	ca,b
${\displaystyle a\not \equiv b}$${\displaystyle \ (\mathrm {mod} \ 3)}$		I or III	${\displaystyle T\equiv 1}$${\displaystyle \ (\mathrm {mod} \ 3)}$	${\displaystyle {\begin{bmatrix}1&{\frac {T-1}{3}}&0\\0&T&0\\0&2{\frac {T-1}{3}}&1\end{bmatrix}}}$	...	ua,b	ua,bd = dca,b	dua,b = ca,bd	ca,b

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

Quadrilateral

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Quadrilateral Goldberg-Coxeter operators

a	b	Class	Edge factor T = a2 + b2	Matrix ${\displaystyle \mathbf {M} _{x}}$	Master square	x	xd	dx	dxd
1	0	I	1	${\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}$		o1 = S	e1 = d		o1 = dd = S
2	0	I	4	${\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}$		o2 = o = j2		e2 = e = a2
3	0	I	9	${\displaystyle {\begin{bmatrix}1&4&0\\0&9&0\\0&4&1\end{bmatrix}}}$		o3	e3		o3
4	0	I	16	${\displaystyle {\begin{bmatrix}1&7&1\\0&16&0\\0&8&0\end{bmatrix}}}$		o4 = oo = j4		e4 = ee = a4
5	0	I	25	${\displaystyle {\begin{bmatrix}1&12&0\\0&25&0\\0&12&1\end{bmatrix}}}$		o5 = o2,1o1,2 = prp	e5 = e2,1e1,2		o5= dprpd
6	0	I	36	${\displaystyle {\begin{bmatrix}1&17&1\\0&36&0\\0&18&0\end{bmatrix}}}$		o6 = o2o3		e6 = e2e3
7	0	I	49	${\displaystyle {\begin{bmatrix}1&24&0\\0&49&0\\0&24&1\end{bmatrix}}}$		o7	e7		o7
8	0	I	64	${\displaystyle {\begin{bmatrix}1&31&1\\0&64&0\\0&32&0\end{bmatrix}}}$		o8 = o3 = j6		e8 = e3 = a6
9	0	I	81	${\displaystyle {\begin{bmatrix}1&40&0\\0&81&0\\0&40&1\end{bmatrix}}}$		o9 = o32	e9 = e32		o9
10	0	I	100	${\displaystyle {\begin{bmatrix}1&49&1\\0&100&0\\0&50&0\end{bmatrix}}}$		o10 = oo2,1o1,2		e10 = ee2,1e1,2
1	1	II	2	${\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}}$		o1,1 = j		e1,1 = a
2	2	II	8	${\displaystyle {\begin{bmatrix}1&3&1\\0&8&0\\0&4&0\end{bmatrix}}}$		o2,2 = j3		e2,2 = a3
1	2	III	5	${\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}$		o1,2 = p	e1,2 = dp = pd		p
${\displaystyle a\equiv b}$${\displaystyle \ (\mathrm {mod} \ 2)}$		I, II, or III	T even	${\displaystyle {\begin{bmatrix}1&{\frac {T}{2}}-1&1\\0&T&0\\0&{\frac {T}{2}}&0\end{bmatrix}}}$	...	oa,b		ea,b
${\displaystyle a\not \equiv b}$${\displaystyle \ (\mathrm {mod} \ 2)}$		I or III	T odd	${\displaystyle {\begin{bmatrix}1&{\frac {T-1}{2}}&0\\0&T&0\\0&{\frac {T-1}{2}}&1\end{bmatrix}}}$	...	oa,b	ea,b		oa,b

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

• Catalan
• Triakis tetrahedron
  kT
• Rhombic dodecahedron
  jC = jO = oT
• Triakis octahedron
  kO
• Tetrakis hexahedron
  kC = mT
• Deltoidal icositetrahedron
  oC = oO
• Disdyakis dodecahedron
  mC = mO
• Pentagonal icositetrahedron
  gC = gO
• Rhombic triacontahedron
  jD = jI
• Triakis icosahedron
  kI
• Pentakis dodecahedron
  kD
• Deltoidal hexecontahedron
  oD = oI
• Disdyakis triacontahedron
  mD = mI
• 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
• atI
• ttI
• ztI = ttD
• etI
• btI
• stI

• Duals
• dtI = nI = kD
• jtI
• ntI = kkD
• ktI
• otI
• mtI
• 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
• Truncated square tiling
  tQ = bQ
• Tetrakis square tiling
  kQ = mQ
• Snub square tiling
  sQ
• Cairo pentagonal tiling
  gQ

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

• Triangle tiling
  Δ = dH = kH
• Rhombille tiling
  jΔ = jH
• Triakis triangular tiling
  kΔ
• Deltoidal trihexagonal tiling
  oΔ = oH
• Kisrhombille tiling
  mΔ = mH
• 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 regular 4x4 square torus, {4,4}_4,0
• tQ24×12 projected to torus
• taQ24×12 projected to torus
• actQ24×8 projected to torus
• tH24×12 projected to torus
• taH24×8 projected to torus
• kH24×12 projected to torus