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][3]

Conway and Hart extended the idea of using operators, like truncation 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 a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like 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.[4] 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. Later implementations named further operators, sometimes referred to as "extended" operators.[5][6]

In general, it is difficult to predict the resulting appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice is the expand operation: aa = e, while a truncation after ambo produces bevel: ta = b. Many basic questions about Conway operators remain open, for instance, how many operators of a given "size" exist.[7]

Conway operators[edit]

Conway operators are applied like functions, from right to left. For example, a cuboctahedron is an ambo cube, i.e. t(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. The resulting polyhedron has a fixed topology (vertices, edges, faces), while exact geometry is not specified: it can be thought of as one of many embeddings of a polyhedral graph on the sphere. Often the polyhedron is put into canonical form.

Individual operators can be visualized in terms of "chambers", as below. Each white chamber is a rotated version of the others. For achiral operators, the red chambers are a reflection of the white chambers. Achiral and chiral operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively, although the exact definition is a little more restrictive.[7]

x
xd
dx
dxd

The relationship between the number of vertices, edges, and faces of the seed and the resulting 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, 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 less 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.

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). r has no effect on achiral seeds, and rr returns the original seed. Alternately an overline can be used to indicate the other chiral form of an operator, like s = rs. r does not affect the matrix.

Some open questions about Conway operators include:[7]

  • Are there two non-equivalent series of operations, not related by d or r, that create the same polyhedron from the same seed?
  • How many Conway operators exist for a given inflation rate?
  • Can an algorithm be developed to generate all the Conway operators for a given inflation rate?
  • Can an algorithm be developed to decompose a given polyhedron into a series of operations on a smaller seed?

Seeds[edit]

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 regular polyhedra can be generated from prismatic generators with zero to two operators:[8]

The regular Euclidean tilings can also be used as seeds:

Primitive operations[edit]

Primitive operations cannot be expressed as a composition of operators aside from d and r. The majority of Conway's original operators are primitive: the exceptions are e, b, o, and m.

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. Note that many more operators 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).

Basic[edit]

Primitive basic 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 valence-4 vertices, and is also called rectification, or the medial graph in graph theory.[9]
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,[10] accretion, or pyramid-augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges.[11] Needle and zip are extended operators, but can be expressed in terms of other basic operators and dual, so are included here. Zip is also called bitruncation.
5 Conway gC.png
Gyro: g
gd = rgr sd = rsr Conway sC.png
Snub: s
Chiral operators. See Snub (geometry). Contrary to Hart[4], gd is not the same as g: it is its chiral pair.[12]
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.[13]

Extended[edit]

Primitive 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 lC.png
Loft: l
Conway ldC.png
ld
Conway dlC.png
dl
Conway dldC.png
dld
6 Conway qC.png
Quinto: q
Conway qdC.png
qd
Conway dqC.png
dq
Conway dqdC.png
dqd
6 Conway L0C.png
Join-lace: L0

L0d
Conway dL0C.png
dL0
Conway dL0d.png
dL0d
See below for explanation of join notation.
7 Conway LC.png
Lace: L
Conway L0dC.png
Ld
Conway dLC.png
dL
Conway dLdC.png
dLd
7 Conway KC.png
Stake: K
Conway KdC.png
Kd
Conway dKC.png
dK
Conway dKdC.png
dKd
7 Conway wC.png
Whirl: w
wd = dv Conway dwC.png
vd = dw
Volute: v Chiral operators.
8 Conway (kk)0C.png
Join-kis-kis: (kk)0
Conway (kk)0dC.png
(kk)0d
Conway d(kk)0C.png
d(kk)0
Conway d(kk)0dC.png
d(kk)0d
Sometimes named J.[5] See below for explanation of join notation. The non-join-form, kk, is not primitive.
10 Conway XC.png
Cross: X
Conway XdC.png
Xd
Conway dXC.png
dX
Conway dXdC.png
dXd

Indexed operations[edit]

A number of operators can be grouped together by some criteria, or have their behavior modified by an index.[5] These are written as an operator with a subscript: xn.

Augmentation[edit]

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
L0
Conway K0C.png
K0 = jk
Conway (kk)0C.png
(kk)0
Augmentation Pyramid Prism Antiprism

The truncate operator t also has an index form tn, indicating that only vertices of a certain valence 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 valence-4 vertices truncated. A lofted cube, lC is the same as t4kC. And a quinto-dodecahedron, qD can be constructed as t5daaD or t5deD or t5oD, a deltoidal hexecontahedron, deD or oD, with its valence-5 vertices truncated.

Meta/Bevel[edit]

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.[5]

Meta/Bevel operators
n Edge factor Matrix x xd dx dxd
0 3 Conway kC.png
k = m0
Conway kdC.png
n
Conway dkC.png
z = b0
Conway tC.png
t
1 6 Conway mC.png
Meta: m = m1 = kj
Conway bC.png
Bevel: b = b1 = ta
2 9 Conway m3C.png
m2
Conway m3dC.png
m2d
Conway b3C.png
b2
Conway dm3dC.png
b2d
3 12 Conway m4C.png
m3
m3d b3 b3d
n 3n+3 mn mnd bn bnd

Medial[edit]

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.[5]

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

Goldberg-Coxeter[edit]

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction.[14][15] 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: b=0. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. ca,0 = ca.
  • Class II: a=b. 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 primitive basic 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[edit]

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
d
Conway C.png
c1 = S
2 0 I 4 Subdivided triangle 02 00.svg Conway uC.png
u2 = u
Conway dcC.png
dc
Conway duC.png
du
Conway cC.png
c2 = c
3 0 I 9 Subdivided triangle 03 00.svg Conway ktC.png
u3 = nn
Conway dtkC.png
nk
Conway dktC.png
zt
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
u5
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
k
Conway tC.png
t
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
c3,1
3 2 III 19 Subdivided triangle 03 02.svg u3,2 u3,2d = dc3,2 du3,2 = c3,2d Conway w3-2.png
c3,2
4 3 III 37 Subdivided triangle 04 03.svg u4,3 u4,3d = dc4,3 du4,3 = c4,3d Conway w4-3C.png
c4,3
5 4 III 61 Subdivided triangle 05 04.svg u5,4 u5,4d = dc5,4 du5,4 = c5,4d Conway w5-4C.png
c5,4
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
c6,5=c1,2c1,3
7 6 III 127 Subdivided triangle 07 06.svg u7,6 u7,6d = dc7,6 du7,6 = c7,6d Conway w7C.png
c7,6
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[edit]

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
o3
Conway e3C.png
e3
Conway o3C.png
o3
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
o7
e7 Conway o7C.png
o7
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
o9
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
p
I, II, or III T even ... oa,b ea,b
I or III T odd ... oa,b ea,b oa,b

Examples[edit]

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.

Composite operators[edit]

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

Other surfaces[edit]

On the plane

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

On the torus

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

See also[edit]

References[edit]

  1. ^ John Horton Conway; Heidi Burgiel; Chaim Goodman-Strass (2008). "Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings". The Symmetries of Things. ISBN 978-1-56881-220-5. 
  2. ^ Hidetoshi Nonaka. "Visualization of Conway Polyhedron Notation". World Academy of Science, Engineering and Technology 50 2009. 
  3. ^ Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld. 
  4. ^ a b George W. Hart. "Conway Notation for Polyhedra". Virtual Polyhedra. 
  5. ^ a b c d e Adrian Rossiter. "conway - Conway Notation transformations". Antiprism Polyhedron Modelling Software. 
  6. ^ Anselm Levskaya. "polyHédronisme". 
  7. ^ a b c d e 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:1705.02848Freely accessible. doi:10.1098/rspa.2017.0267. 
  8. ^ Livio Zefiro (2008). "Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra". Vismath. 
  9. ^ Weisstein, Eric W. "Rectification". MathWorld. 
  10. ^ Weisstein, Eric W. "Cumulation". MathWorld. 
  11. ^ Weisstein, Eric W. "Truncation". MathWorld. 
  12. ^ "Antiprism - Chirality issue in conway". 
  13. ^ George W. Hart (August 2000). Sculpture based on Propellorized Polyhedra. Proceedings of MOSAIC 2000. Seattle, WA. pp. 61–70. 
  14. ^ Deza, M.; Dutour, M (2004). "Goldberg–Coxeter constructions for 3-and 4-valent plane graphs". The Electronic Journal of Combinatorics. 11: #R20. 
  15. ^ 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.