Conway polyhedron notation

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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, 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 , 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 abdegjkmost, while Hart added r and p.[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.


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. , and a truncated cuboctahedron is . 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 Dn where n is the number of sides of a face, while chiral operators correspond to cyclic groups Cn 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 sides
3 (Triangle) 4 (Square) 5 (Pentagon) 6 (Hexagon)
Triangle chambers.svg Quadrilateral chambers.svg Pentagon chambers.svg Hexagon chambers.svg
The fundamental domains for polyhedron groups. The groups are  for achiral polyhedra, and  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[edit]


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 . When x is the operator, are the vertices, edges, and faces of the seed (respectively), and are the vertices, edges, and faces of the result, then


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:


Two dual operators cancel out; dd = S, and the square of 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[edit]

The number of LSPs for each inflation rate is 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 .[8]

Original operations[edit]

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 x xd dx dxd Notes
1 Conway C.png
Seed: S
Conway dC.png
Dual: d
Conway C.png
Seed: dd = S
Dual replaces each face with a vertex, and each vertex with a face.
2 Conway jC.png
Join: j
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.[10]
3 Conway kC.png
Kis: k
Conway kdC.png
Needle: n
Conway dkC.png
Zip: z
Conway tC.png
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 Conway oC.png
Ortho: o = jj
Conway eC.png
Expand: e = aa
5 Conway gC.png
Gyro: g
gd = rgr sd = rsr Conway sC.png
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 Conway mC.png
Meta: m = kj
Conway bC.png
Bevel: b = ta


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 (Pn) for n-gonal forms; antiprisms (An); cupolae (Un); anticupolae (Vn); and pyramids (Yn). Any Johnson solid can be referenced as Jn, for n=1..92.

All of the five Platonic solids can be generated from prismatic generators with zero to two operators:[14]

The regular Euclidean tilings can also be used as seeds:

Extended operations[edit]

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). To simplify, only irreducible operators are included in this list: others can be created by composing operators together.

Irreducible extended operators
Edge factor Matrix x xd dx dxd Notes
4 Conway cC.png
Chamfer: c
Conway duC.png
cd = du
Conway dcC.png
dc = ud
Conway uC.png
Subdivide: u
Chamfer is the join-form of l. See Chamfer (geometry).
5 Conway pC.png
Propeller: p
Conway dpC.png
dp = pd
Conway pC.png
dpd = p
Chiral operators. The propeller operator was developed by George Hart.[15]
5 Conway lC.png
Loft: l
Conway ldC.png
Conway dlC.png
Conway dldC.png
6 Conway qC.png
Quinto: q
Conway qdC.png
Conway dqC.png
Conway dqdC.png
6 Conway L0C.png
Join-lace: L0
Conway Diagram L0d.png
Conway dL0C.png
Conway dL0d.png
See below for explanation of join notation.
7 Conway LC.png
Lace: L
Conway L0dC.png
Conway dLC.png
Conway dLdC.png
7 Conway KC.png
Stake: K
Conway KdC.png
Conway dKC.png
Conway dKdC.png
7 Conway wC.png
Whirl: w
wd = dv Conway dwC.png
vd = dw
Volute: v Chiral operators.
8 Conway (kk)0C.png
Conway (kk)0dC.png
Conway d(kk)0C.png
Conway d(kk)0dC.png
Sometimes named J.[4] See below for explanation of join notation. The non-join-form, kk, is not irreducible.
10 Conway XC.png
Cross: X
Conway XdC.png
Conway dXC.png
Conway dXdC.png

Indexed extended operations[edit]

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: xn.


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.

Operator k l L K (kk)
x Conway kC.png Conway lC.png Conway LC.png Conway KC.png Conway kkC.png
x0 Conway jC.png
k0 = j
Conway cC.png
l0 = c
Conway L0C.png
Conway K0C.png
K0 = jk
Conway (kk)0C.png
Augmentation Pyramid Prism Antiprism

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

Some of the extended operators can be created in special cases with kn and tn 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 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 x xd dx dxd
0 3 Conway kC.png
k = m0
Conway kdC.png
Conway dkC.png
z = b0
Conway tC.png
1 6 Conway mC.png
m = m1 = kj
Conway bC.png
b = b1 = ta
2 9 Conway m3C.png
Conway m3dC.png
Conway b3C.png
Conway dm3dC.png
3 12 Conway m4C.png
m3d b3 b3d
n 3n+3 mn mnd bn bnd


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
Matrix x xd dx dxd
1 4 Conway oC.png
M1 = o = jj
Conway eC.png
e = aa
2 7 Conway MC.png
Medial: M = M2
Conway MdC.png
Conway dMC.png
Conway dMdC.png
n 3n+1 Mn Mnd dMn dMnd


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, ca,b and ua,b, and the quadrilateral GC family, ea,b and oa,b. Both the GC families are indexed by two integers and . 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: . Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. ca,0 = ca.
  • Class II: . Also achiral. Can be decomposed as ca,a = cac1,1
  • Class III: All other operators. These are chiral, and ca,b and cb,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 Goldberg-Coxeter operators
a b Class Edge factor
T = a2 + ab + b2
Matrix Master triangle x xd dx dxd
1 0 I 1 Subdivided triangle 01 00.svg Conway C.png
u1 = S
Conway dC.png
Conway C.png
c1 = S
2 0 I 4 Subdivided triangle 02 00.svg Conway uC.png
u2 = u
Conway dcC.png
Conway duC.png
Conway cC.png
c2 = c
3 0 I 9 Subdivided triangle 03 00.svg Conway ktC.png
u3 = nn
Conway dtkC.png
Conway dktC.png
Conway tkC.png
c3 = zz
4 0 I 16 Subdivided triangle 04 00.svg Conway u4C.png
u4 = uu
uud = dcc duu = ccd c4 = cc
5 0 I 25 Subdivided triangle 05 00.svg Conway u5C.png
u5d = dc5 du5 = c5d c5
6 0 I 36 Subdivided triangle 06 00.svg Conway u6C.png
u6 = unn
unk czt u6 = czz
7 0 I 49 Subdivided triangle 07 00.svg Conway u7.png
u7 = u2,1u1,2 = vrv
vrvd = dwrw dvrv = wrwd c7 = c2,1c1,2 = wrw
8 0 I 64 Subdivided triangle 08 00.svg Conway u8C.png
u8 = u3
u3d = dc3 du3 = c3d c8 = c3
9 0 I 81 Subdivided triangle 09 00.svg Conway u9C.png
u9 = n4
n3k = kz3 tn3 = z3t c9 = z4
1 1 II 3 Subdivided triangle 01 01.svg Conway kdC.png
u1,1 = n
Conway kC.png
Conway tC.png
Conway dkC.png
c1,1 = z
2 1 III 7 Subdivided triangle 02 01.svg v = u2,1 Conway dwC.png
vd = dw
dv = wd Conway wC.png
w = c2,1
3 1 III 13 Subdivided triangle 03 01.svg u3,1 u3,1d = dc3,1 du3,1 = c3,1d Conway w3C.png
3 2 III 19 Subdivided triangle 03 02.svg u3,2 u3,2d = dc3,2 du3,2 = c3,2d Conway w3-2.png
4 3 III 37 Subdivided triangle 04 03.svg u4,3 u4,3d = dc4,3 du4,3 = c4,3d Conway w4-3C.png
5 4 III 61 Subdivided triangle 05 04.svg u5,4 u5,4d = dc5,4 du5,4 = c5,4d Conway w5-4C.png
6 5 III 91 Subdivided triangle 06 05.svg u6,5 = u1,2u1,3 u6,5d = dc6,5 du6,5 = c6,5d Conway w6-5C.png
7 6 III 127 Subdivided triangle 07 06.svg u7,6 u7,6d = dc7,6 du7,6 = c7,6d Conway w7C.png
8 7 III 169 Subdivided triangle 08 07.svg u8,7 = u3,12 u8,7d = dc8,7 du8,7 = c8,7d Conway w8C.png
c8,7 = c3,12
9 8 III 217 Subdivided triangle 09 08.svg u9,8 = u2,1u5,1 u9,8d = dc9,8 du9,8 = c9,8d Conway w9C.png
c9,8 = c2,1c5,1
I, II, or III ... ua,b ua,bd = dca,b dua,b = ca,bd ca,b
I or III ... ua,b ua,bd = dca,b dua,b = ca,bd ca,b

By basic number theory, for any values of a and b, .


Quadrilateral Goldberg-Coxeter operators
a b Class Edge factor
T = a2 + b2
Matrix Master square x xd dx dxd
1 0 I 1 Subdivided square 01 00.svg Conway C.png
o1 = S
Conway dC.png
e1 = d
Conway C.png
o1 = dd = S
2 0 I 4 Subdivided square 02 00.svg Conway oC.png
o2 = o = j2
Conway eC.png
e2 = e = a2
3 0 I 9 Subdivided square 03 00.svg Conway o3C.png
Conway e3C.png
Conway o3C.png
4 0 I 16 Subdivided square 04 00.svg Conway deeC.png
o4 = oo = j4
Conway eeC.png
e4 = ee = a4
5 0 I 25 Subdivided square 05 00.svg Conway o5C.png
o5 = o2,1o1,2 = prp
e5 = e2,1e1,2 Conway o5C.png
o5= dprpd
6 0 I 36 Subdivided square 06 00.svg Conway o6C.png
o6 = o2o3
e6 = e2e3
7 0 I 49 Subdivided square 07 00.svg Conway o7C.png
e7 Conway o7C.png
8 0 I 64 Subdivided square 08 00.svg Conway o8C.png
o8 = o3 = j6
e8 = e3 = a6
9 0 I 81 Subdivided square 09 00.svg Conway o9C.png
o9 = o32

e9 = e32
Conway o9C.png
10 0 I 100 Subdivided square 10 00.svg Conway o10C.png
o10 = oo2,1o1,2
e10 = ee2,1e1,2
1 1 II 2 Subdivided square 01 01.svg Conway jC.png
o1,1 = j
Conway aC.png
e1,1 = a
2 2 II 8 Subdivided square 02 02.svg Conway daaaC.png
o2,2 = j3
Conway aaaC.png
e2,2 = a3
1 2 III 5 Subdivided square 01 02.svg Conway pC.png
o1,2 = p
Conway dpC.png
e1,2 = dp = pd
Conway pC.png
I, II, or III T even ... oa,b ea,b
I or III T odd ... oa,b ea,b oa,b


See also List of geodesic polyhedra and Goldberg polyhedra.

Archimedean and Catalan solids[edit]

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[edit]

The truncated icosahedron, tI = zD, can be used as a seed to create some more visually-pleasing polyhedra, although these are neither vertex nor face-transitive.

On the plane[edit]

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

On a torus[edit]

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

See also[edit]


  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.
  3. ^ a b George W. Hart (1998). "Conway Notation for Polyhedra". Virtual Polyhedra.
  4. ^ a b c d e Adrian Rossiter. "conway - Conway Notation transformations". Antiprism Polyhedron Modelling Software.
  5. ^ Anselm Levskaya. "polyHédronisme".
  6. ^ a b Hart, George (1998). "Conway Notation for Polyhedra". Virtual Polyhedra. (See fourth row in table, "a = ambo".)
  7. ^ a b c 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.
  8. ^ a b Goetschalckx, Pieter; Coolsaet, Kris; Van Cleemput, Nico (2020-04-12). "Generation of Local Symmetry-Preserving Operations". arXiv:1908.11622 [math.CO].
  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.
  15. ^ George W. Hart (August 2000). Sculpture based on Propellorized Polyhedra. Proceedings of MOSAIC 2000. Seattle, WA. pp. 61–70.
  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.
  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.

External links[edit]

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