Conway polyhedron notation

Last updated November 09, 2024

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.^[1]^[2]

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, $tC$ represents a truncated cube, and $taC$ , parsed as $t (aC)$ , is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.g., a dual cube is an octahedron: $dC = O$ . Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators $a$ (ambo), $b$ (bevel), $d$ (dual), $e$ (expand), $g$ (gyro), $j$ (join), $k$ (kis), $m$ (meta), $o$ (ortho), $s$ (snub), and $t$ (truncate), while Hart added $r$ (reflect) and $p$ (propellor).^[3] Later implementations named further operators, sometimes referred to as "extended" operators.^[4]^[5] 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 ( $aa = e$ ), while a truncation after ambo produces bevel ( $ta = b$ ).

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,^[6] i.e. ⁠ $a(C)=aC$ ⁠, and a truncated cuboctahedron is ⁠ $t(a(C))=t(aC)=taC$ ⁠. 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 $D n$ where n is the number of sides of a face, while chiral operators correspond to cyclic groups $C n$ 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.^[7]^[8]^[9] 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)

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.^[6] 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	${\begin{bmatrix}a&b&c\\0&d&0\\a'&b'&c'\end{bmatrix}}=\mathbf {M} _{x}$
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	${\begin{bmatrix}c'&b'&a'\\0&d&0\\c&b&a\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}\mathbf {M} _{d}$

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.^[7]

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 $2,2,4,6,4,20,20,54,64,\cdots$ .^[8]

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	${\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$	Seed: S	Dual: d		Seed: dd = S	Dual replaces each face with a vertex, and each vertex with a face.
2	${\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}$	Join: j		Ambo: a		Join creates quadrilateral faces. Ambo creates degree-4 vertices, and is also called rectification, or the medial graph in graph theory.^[10]
3	${\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}$	Kis: k	Needle: n	Zip: z	Truncate: t	Kis raises a pyramid on each face, and is also called akisation, Kleetope, cumulation,^[11] accretion, or pyramid-augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges.^[12] Zip is also called bitruncation.
4	${\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}$	Ortho: o = jj		Expand: e = aa
5	${\begin{bmatrix}1&2&1\\0&5&0\\0&2&0\end{bmatrix}}$	Gyro: g	gd = rgr	sd = rsr	Snub: s	Chiral operators. See Snub (geometry). Contrary to Hart,^[3]gd is not the same as g: it is its chiral pair.^[13]
6	${\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}$	Meta: m = kj		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:^[14]

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	${\begin{bmatrix}1&2&0\\0&4&0\\0&1&1\end{bmatrix}}$	Chamfer: c	cd = du	dc = ud	Subdivide: u	Chamfer is the join-form of l. See Chamfer (geometry).
5	${\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}$	Propeller: p	dp = pd		dpd = p	Chiral operators. The propeller operator was developed by George Hart.^[15]
5	${\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}$	Loft: l	ld	dl	dld
6	${\begin{bmatrix}1&3&0\\0&6&0\\0&2&1\end{bmatrix}}$	Quinto: q	qd	dq	dqd
6	${\begin{bmatrix}1&2&0\\0&6&0\\0&3&1\end{bmatrix}}$	Join-lace: L₀	L₀d	dL₀	dL₀d	See below for explanation of join notation.
7	${\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}$	Lace: L	Ld	dL	dLd
7	${\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}$	Stake: K	Kd	dK	dKd
7	${\begin{bmatrix}1&4&0\\0&7&0\\0&2&1\end{bmatrix}}$	Whirl: w	wd = dv	vd = dw	Volute: v	Chiral operators.
8	${\begin{bmatrix}1&2&1\\0&8&0\\0&5&0\end{bmatrix}}$	Join-kis-kis: ⁠ $(kk)_{0}$ ⁠	⁠ $(kk)_{0}d$ ⁠	⁠ $d(kk)_{0}$ ⁠	⁠ $d(kk)_{0}d$ ⁠	Sometimes named J.^[4] See below for explanation of join notation. The non-join-form, kk, is not irreducible.
10	${\begin{bmatrix}1&3&1\\0&10&0\\0&6&0\end{bmatrix}}$	Cross: X	Xd	dX	dXd

Indexed extended operations

A number of operators can be grouped together by some criteria, or have their behavior modified by an index.^[4] 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	k	l	L	K	(kk)
Corresponding join-form operator	x₀	k₀ = j	l₀ = c	L₀	K₀ = jk	(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.^[4]

Meta/Bevel operators
n	Edge factor	Matrix $\mathbf {M} _{x}$	x	xd	dx	dxd
0	3	${\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}$	k = m₀	n	z = b₀	t
1	6	${\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}$	m = m₁ = kj		b = b₁ = ta
2	9	${\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}$	m₂	m₂d	b₂	b₂d
3	12	${\begin{bmatrix}1&3&1\\0&12&0\\0&8&0\end{bmatrix}}$	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.^[4]

Medial operators
n	Edge factor	Matrix $\mathbf {M} _{x}$	x	xd	dx	dxd
1	4	${\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}$	M₁ = o = jj		e = aa
2	7	${\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}$	Medial: M = M₂	Md	dM	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.^[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 $a\geq 1$ and $b\geq 0$ . 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: ⁠ $b=0$ ⁠. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. c_a,0 = c_a.
Class II: ⁠ $a=b$ ⁠. 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

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	${\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$		u₁ = S	d		c₁ = S
2	0	I	4	${\begin{bmatrix}1&1&0\\0&4&0\\0&2&1\end{bmatrix}}$		u₂ = u	dc	du	c₂ = c
3	0	I	9	${\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}$		u₃ = nn	nk	zt	c₃ = zz
4	0	I	16	${\begin{bmatrix}1&5&0\\0&16&0\\0&10&1\end{bmatrix}}$		u₄ = uu	uud = dcc	duu = ccd	c₄ = cc
5	0	I	25	${\begin{bmatrix}1&8&0\\0&25&0\\0&16&1\end{bmatrix}}$		u₅	u₅d = dc₅	du₅ = c₅d	c₅
6	0	I	36	${\begin{bmatrix}1&11&1\\0&36&0\\0&24&0\end{bmatrix}}$		u₆ = unn	unk	czt	u₆ = czz
7	0	I	49	${\begin{bmatrix}1&16&0\\0&49&0\\0&32&1\end{bmatrix}}$		u₇ = u_2,1u_1,2 = vrv	vrvd = dwrw	dvrv = wrwd	c₇ = c_2,1c_1,2 = wrw
8	0	I	64	${\begin{bmatrix}1&21&0\\0&64&0\\0&42&1\end{bmatrix}}$		u₈ = u³	u³d = dc³	du³ = c³d	c₈ = c³
9	0	I	81	${\begin{bmatrix}1&26&1\\0&81&0\\0&54&0\end{bmatrix}}$		u₉ = n⁴	n³k = kz³	tn³ = z³t	c₉ = z⁴
1	1	II	3	${\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}$		u_1,1 = n	k	t	c_1,1 = z
2	1	III	7	${\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}$		v = u_2,1	vd = dw	dv = wd	w = c_2,1
3	1	III	13	${\begin{bmatrix}1&4&0\\0&13&0\\0&8&1\end{bmatrix}}$		u_3,1	u_3,1d = dc_3,1	du_3,1 = c_3,1d	c_3,1
3	2	III	19	${\begin{bmatrix}1&6&0\\0&19&0\\0&12&1\end{bmatrix}}$		u_3,2	u_3,2d = dc_3,2	du_3,2 = c_3,2d	c_3,2
4	3	III	37	${\begin{bmatrix}1&12&0\\0&37&0\\0&24&1\end{bmatrix}}$		u_4,3	u_4,3d = dc_4,3	du_4,3 = c_4,3d	c_4,3
5	4	III	61	${\begin{bmatrix}1&20&0\\0&61&0\\0&40&1\end{bmatrix}}$		u_5,4	u_5,4d = dc_5,4	du_5,4 = c_5,4d	c_5,4
6	5	III	91	${\begin{bmatrix}1&30&0\\0&91&0\\0&60&1\end{bmatrix}}$		u_6,5 = u_1,2u_1,3	u_6,5d = dc_6,5	du_6,5 = c_6,5d	c_6,5=c_1,2c_1,3
7	6	III	127	${\begin{bmatrix}1&42&0\\0&127&0\\0&84&1\end{bmatrix}}$		u_7,6	u_7,6d = dc_7,6	du_7,6 = c_7,6d	c_7,6
8	7	III	169	${\begin{bmatrix}1&56&0\\0&169&0\\0&112&1\end{bmatrix}}$		u_8,7 = u_3,1²	u_8,7d = dc_8,7	du_8,7 = c_8,7d	c_8,7 = c_3,1²
9	8	III	217	${\begin{bmatrix}1&72&0\\0&217&0\\0&144&1\end{bmatrix}}$		u_9,8 = u_2,1u_5,1	u_9,8d = dc_9,8	du_9,8 = c_9,8d	c_9,8 = c_2,1c_5,1
$a\equiv b$ $\ (\mathrm {mod} \ 3)$		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
$a\not \equiv b$ $\ (\mathrm {mod} \ 3)$		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

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	${\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$		o₁ = S	e₁ = d		o₁ = dd = S
2	0	I	4	${\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}$		o₂ = o = j²		e₂ = e = a²
3	0	I	9	${\begin{bmatrix}1&4&0\\0&9&0\\0&4&1\end{bmatrix}}$		o₃	e₃		o₃
4	0	I	16	${\begin{bmatrix}1&7&1\\0&16&0\\0&8&0\end{bmatrix}}$		o₄ = oo = j⁴		e₄ = ee = a⁴
5	0	I	25	${\begin{bmatrix}1&12&0\\0&25&0\\0&12&1\end{bmatrix}}$		o₅ = o_2,1o_1,2 = prp	e₅ = e_2,1e_1,2		o₅= dprpd
6	0	I	36	${\begin{bmatrix}1&17&1\\0&36&0\\0&18&0\end{bmatrix}}$		o₆ = o₂o₃		e₆ = e₂e₃
7	0	I	49	${\begin{bmatrix}1&24&0\\0&49&0\\0&24&1\end{bmatrix}}$		o₇	e₇		o₇
8	0	I	64	${\begin{bmatrix}1&31&1\\0&64&0\\0&32&0\end{bmatrix}}$		o₈ = o³ = j⁶		e₈ = e³ = a⁶
9	0	I	81	${\begin{bmatrix}1&40&0\\0&81&0\\0&40&1\end{bmatrix}}$		o₉ = o₃²	e₉ = e₃²		o₉
10	0	I	100	${\begin{bmatrix}1&49&1\\0&100&0\\0&50&0\end{bmatrix}}$		o₁₀ = oo_2,1o_1,2		e₁₀ = ee_2,1e_1,2
1	1	II	2	${\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}$		o_1,1 = j		e_1,1 = a
2	2	II	8	${\begin{bmatrix}1&3&1\\0&8&0\\0&4&0\end{bmatrix}}$		o_2,2 = j³		e_2,2 = a³
1	2	III	5	${\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}$		o_1,2 = p	e_1,2 = dp = pd		p
$a\equiv b$ $\ (\mathrm {mod} \ 2)$		I, II, or III	T even	${\begin{bmatrix}1&{\frac {T}{2}}-1&1\\0&T&0\\0&{\frac {T}{2}}&0\end{bmatrix}}$	...	o_a,b		e_a,b
$a\not \equiv b$ $\ (\mathrm {mod} \ 2)$		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	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.)

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

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

Related Research Articles

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube or regular hexahedron is a three-dimensional solid object bounded by six congruent square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Snub cube</span> Archimedean solid with 38 faces

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol $, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol .$

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or $j$ -faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension $j \leq n$ .

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the Bilinski dodecahedron. There are some stellations of the rhombic dodecahedron, one of which is the Escher's solid. The rhombic dodecahedron may also appearances in the garnet crystal, the architectural philosophies, practical usages, and toys.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

<span class="mw-page-title-main">Tetrakis hexahedron</span> Catalan solid with 24 faces

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.

<span class="mw-page-title-main">Rectification (geometry)</span> Operation in Euclidean geometry

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

<span class="mw-page-title-main">Regular dodecahedron</span> Convex polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron.

In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. $GP(5,3)$ and $GP(3,5)$ are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra, of which all but the smallest one have mostly hexagonal faces.

References

↑ Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings". The Symmetries of Things. AK Peters. p. 288. ISBN 978-1-56881-220-5.
↑ Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld .
1 2 George W. Hart (1998). "Conway Notation for Polyhedra". Virtual Polyhedra.
1 2 3 4 5 Adrian Rossiter. "conway - Conway Notation transformations". Antiprism Polyhedron Modelling Software.
↑ Anselm Levskaya. "polyHédronisme".
1 2 Hart, George (1998). "Conway Notation for Polyhedra". Virtual Polyhedra. (See fourth row in table, "a = ambo".)
1 2 3 Brinkmann, G.; Goetschalckx, P.; Schein, S. (2017). "Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 473 (2206): 20170267. arXiv: 1705.02848 . Bibcode:2017RSPSA.47370267B. doi:10.1098/rspa.2017.0267. S2CID 119171258.
1 2 Goetschalckx, Pieter; Coolsaet, Kris; Van Cleemput, Nico (2020-04-12). "Generation of Local Symmetry-Preserving Operations". arXiv: 1908.11622 [math.CO].
↑ Goetschalckx, Pieter; Coolsaet, Kris; Van Cleemput, Nico (2020-04-11). "Local Orientation-Preserving Symmetry Preserving Operations on Polyhedra". arXiv: 2004.05501 [math.CO].
↑ Weisstein, Eric W. "Rectification". MathWorld .
↑ Weisstein, Eric W. "Cumulation". MathWorld .
↑ Weisstein, Eric W. "Truncation". MathWorld .
↑ "Antiprism - Chirality issue in conway".
↑ Livio Zefiro (2008). "Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra". Vismath.
↑ George W. Hart (August 2000). Sculpture based on Propellorized Polyhedra. Proceedings of MOSAIC 2000. Seattle, WA. pp. 61–70.
↑ Deza, M.; Dutour, M (2004). "Goldberg–Coxeter constructions for 3-and 4-valent plane graphs". The Electronic Journal of Combinatorics. 11: #R20. doi: 10.37236/1773 .
↑ Deza, M.-M.; Sikirić, M. D.; Shtogrin, M. I. (2015). "Goldberg–Coxeter Construction and Parameterization". Geometric Structure of Chemistry-Relevant Graphs: Zigzags and Central Circuits. Springer. pp. 131–148. ISBN 9788132224495.

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[SoT-1] Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings". The Symmetries of Things. AK Peters. p. 288. ISBN 978-1-56881-220-5.

[2] Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld .

[Hart-3] 1 2 George W. Hart (1998). "Conway Notation for Polyhedra". Virtual Polyhedra.

[Antiprism-4] 1 2 3 4 5 Adrian Rossiter. "conway - Conway Notation transformations". Antiprism Polyhedron Modelling Software.

[Levskaya-5] Anselm Levskaya. "polyHédronisme".

[hart-6] 1 2 Hart, George (1998). "Conway Notation for Polyhedra". Virtual Polyhedra. (See fourth row in table, "a = ambo".)

[Brinkmann-7] 1 2 3 Brinkmann, G.; Goetschalckx, P.; Schein, S. (2017). "Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 473 (2206): 20170267. arXiv: 1705.02848 . Bibcode:2017RSPSA.47370267B. doi:10.1098/rspa.2017.0267. S2CID 119171258.

[lsp-8] 1 2 Goetschalckx, Pieter; Coolsaet, Kris; Van Cleemput, Nico (2020-04-12). "Generation of Local Symmetry-Preserving Operations". arXiv: 1908.11622 [math.CO].

[lopsp-9] Goetschalckx, Pieter; Coolsaet, Kris; Van Cleemput, Nico (2020-04-11). "Local Orientation-Preserving Symmetry Preserving Operations on Polyhedra". arXiv: 2004.05501 [math.CO].

[10] Weisstein, Eric W. "Rectification". MathWorld .

[11] Weisstein, Eric W. "Cumulation". MathWorld .

[12] Weisstein, Eric W. "Truncation". MathWorld .

[13] "Antiprism - Chirality issue in conway".

[14] Livio Zefiro (2008). "Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra". Vismath.

[Hart2000-15] George W. Hart (August 2000). Sculpture based on Propellorized Polyhedra. Proceedings of MOSAIC 2000. Seattle, WA. pp. 61–70.

[Deza2004-16] Deza, M.; Dutour, M (2004). "Goldberg–Coxeter constructions for 3-and 4-valent plane graphs". The Electronic Journal of Combinatorics. 11: #R20. doi: 10.37236/1773 .

[Deza2015-17] Deza, M.-M.; Sikirić, M. D.; Shtogrin, M. I. (2015). "Goldberg–Coxeter Construction and Parameterization". Geometric Structure of Chemistry-Relevant Graphs: Zigzags and Central Circuits. Springer. pp. 131–148. ISBN 9788132224495.

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