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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] 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.[4][5]

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 general, it is difficult to predict the result 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.[6]

Contents

OperatorsEdit

In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a cuboctahedron is an ambo[7] cube, 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.

Groups of SymmetryEdit

Individual operators can be visualized in terms of fundamental domains, 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, and all are transitive. 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.[6]

 

These are the dihedral groups of order 3, 4, 5, 6, and are formed by the semidirect product of incremental rotations and reflections. So   by means of  .[8] Each right triangle is a fundamental domain which can be mapped to any other right triangle.

x  
xd  
dx  
dxd  

Operations as MatricesEdit

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.[6]

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.

Hart introduced the reflection operator r, that gives the mirror image of the polyhedron.[citation needed] 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. An overline can be used to indicate the other chiral form of an operator, like s = rsr. r does not affect the matrix.

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.

Some open questions about Conway operators include:[6]

  • 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?

Original operationsEdit

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    
Seed: S
 
Dual: d
 
Seed: dd = S
Dual replaces each face with a vertex, and each vertex with a face.
2    
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.[9]
3    
Kis: k
 
Needle: n
 
Zip: z
 
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] Zip is also called bitruncation.
4    
Ortho: o = jj
 
Expand: e = aa
5    
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.[12]
6    
Meta: m = kj
 
Bevel: b = ta

SeedsEdit

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:[13]

The regular Euclidean tilings can also be used as seeds:

Extended operationsEdit

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    
Chamfer: c
 
cd = du
 
dc = ud
 
Subdivide: u
Chamfer is the join-form of l. See Chamfer (geometry).
5    
Propeller: p
 
dp = pd
 
dpd = p
Chiral operators. The propeller operator was developed by George Hart.[14]
5    
Loft: l
 
ld
 
dl
 
dld
6    
Quinto: q
 
qd
 
dq
 
dqd
6    
Join-lace: L0
 
L0d
 
dL0
 
dL0d
See below for explanation of join notation.
7    
Lace: L
 
Ld
 
dL
 
dLd
7    
Stake: K
 
Kd
 
dK
 
dKd
7    
Whirl: w
wd = dv  
vd = dw
Volute: v Chiral operators.
8    
Join-kis-kis:  
 
 
 
 
 
 
Sometimes named J.[4] See below for explanation of join notation. The non-join-form, kk, is not irreducible.
10    
Cross: X
 
Xd
 
dX
 
dXd

Indexed extended operationsEdit

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.

AugmentationEdit

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          
x0  
k0 = j
 
l0 = c
 
L0
 
K0 = jk
 
 
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/BevelEdit

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    
k = m0
 
n
 
z = b0
 
t
1 6    
m = m1 = kj
 
b = b1 = ta
2 9    
m2
 
m2d
 
b2
 
b2d
3 12    
m3
m3d b3 b3d
n 3n+3   mn mnd bn bnd

MedialEdit

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   x xd dx dxd
1 4    
M1 = o = jj
 
e = aa
2 7    
Medial: M = M2
 
Md
 
dM
 
dMd
n 3n+1   Mn Mnd dMn dMnd

Goldberg-CoxeterEdit

The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction.[15][16] 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").[6] 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.

TriangularEdit

Triangular Goldberg-Coxeter operators
a b Class Edge factor
T = a2 + ab + b2
Matrix   Master triangle x xd dx dxd
1 0 I 1      
u1 = S
 
d
 
c1 = S
2 0 I 4      
u2 = u
 
dc
 
du
 
c2 = c
3 0 I 9      
u3 = nn
 
nk
 
zt
 
c3 = zz
4 0 I 16      
u4 = uu
uud = dcc duu = ccd c4 = cc
5 0 I 25      
u5
u5d = dc5 du5 = c5d c5
6 0 I 36      
u6 = unn
unk czt u6 = czz
7 0 I 49      
u7 = u2,1u1,2 = vrv
vrvd = dwrw dvrv = wrwd c7 = c2,1c1,2 = wrw
8 0 I 64      
u8 = u3
u3d = dc3 du3 = c3d c8 = c3
9 0 I 81      
u9 = n4
n3k = kz3 tn3 = z3t c9 = z4
1 1 II 3      
u1,1 = n
 
k
 
t
 
c1,1 = z
2 1 III 7     v = u2,1  
vd = dw
dv = wd  
w = c2,1
3 1 III 13     u3,1 u3,1d = dc3,1 du3,1 = c3,1d  
c3,1
3 2 III 19     u3,2 u3,2d = dc3,2 du3,2 = c3,2d  
c3,2
4 3 III 37     u4,3 u4,3d = dc4,3 du4,3 = c4,3d  
c4,3
5 4 III 61     u5,4 u5,4d = dc5,4 du5,4 = c5,4d  
c5,4
6 5 III 91     u6,5 = u1,2u1,3 u6,5d = dc6,5 du6,5 = c6,5d  
c6,5=c1,2c1,3
7 6 III 127     u7,6 u7,6d = dc7,6 du7,6 = c7,6d  
c7,6
8 7 III 169     u8,7 = u3,12 u8,7d = dc8,7 du8,7 = c8,7d  
c8,7 = c3,12
9 8 III 217     u9,8 = u2,1u5,1 u9,8d = dc9,8 du9,8 = c9,8d  
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,  .

QuadrilateralEdit

Quadrilateral Goldberg-Coxeter operators
a b Class Edge factor
T = a2 + b2
Matrix   Master square x xd dx dxd
1 0 I 1      
o1 = S
 
e1 = d
 
o1 = dd = S
2 0 I 4      
o2 = o = j2
 
e2 = e = a2
3 0 I 9      
o3
 
e3
 
o3
4 0 I 16      
o4 = oo = j4
 
e4 = ee = a4
5 0 I 25      
o5 = o2,1o1,2 = prp
e5 = e2,1e1,2  
o5= dprpd
6 0 I 36      
o6 = o2o3
e6 = e2e3
7 0 I 49      
o7
e7  
o7
8 0 I 64      
o8 = o3 = j6
e8 = e3 = a6
9 0 I 81      
o9 = o32

e9 = e32
 
o9
10 0 I 100      
o10 = oo2,1o1,2
e10 = ee2,1e1,2
1 1 II 2      
o1,1 = j
 
e1,1 = a
2 2 II 8      
o2,2 = j3
 
e2,2 = a3
1 2 III 5      
o1,2 = p
 
e1,2 = dp = pd
 
p
   I, II, or III T even   ... oa,b ea,b
   I or III T odd   ... oa,b ea,b oa,b

ExamplesEdit

See also List of geodesic polyhedra and Goldberg polyhedra.

Archimedean and Catalan solidsEdit

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 operatorsEdit

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.

Other surfacesEdit

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.

Special Tilings (Expand and Ortho)Edit

Below are the expand and ortho results of a basis of planar tilings: 11 uniform tilings, 4 semiregular tilings, one 4-uniform tiling ('gem' tiling), and one 92-uniform tiling with 14 distinct planigons ('take all' tiling). To expand a uniform tiling, take the midpoints of all regular polygons (ambo), take the midpoints of the resulting regular polygons, take the midpoints of the vertex-figure polygons (in the gaps), and alternate between shrunk regular polygons and shrunk planigons connected by vertices in checkerboard fashion (since e=aa). There will be new gap quadrilaterals (4-sided faces) in the expand tilings of lesser significance, corresponding to the edges in the original uniform tilings.[3] The expand operation heuristically tends to form rings around larger regular polygons, with smaller regular polygons acting as ring borders. To ortho a uniform tiling, merely superimpose its dual. Superimposing the dual (only for uniform tilings) is consistent with inserting a vertex in the middle of each regular polygon (centroid) and connecting the vertex to the midpoints of the regular polygons via new edges (edge midpoints);[3] an edge connecting the centers (centroids) of two regular polygon faces sharing exactly one edge must intersect perpendicularly at the midpoint of that common edge due to the reflection in dihedral symmetry (it is a perpendicular bisector), hence orthogonality. Note that '—' stands for elongation, which is not officially a Conway operation.

Expand and Ortho Versions of a Basis of Planar Tilings
eQ eH eaH etH eeH=eeΔ
           
oQ oH oaH otH oeH=oeΔ
           
ebH=etaH esQ e—Q esH etQ e[3.4.3.12; 3.122]
           
obH=otaH osQ o—Q osH otQ o[3.4.3.12; 3.122]
           
e[32.4.12; 36] e[3.42.6; (3.6)2]1 e[32.62; (3.6)2] e[32.4.12; 3.122; 32.4.3.4; 36] e[36; 33.42; 32.4.3.4; 34.6;
3.42.6; 32.4.12; 4.6.12]
e[92-uniform tiling]
           
o[32.4.12; 36] o[3.42.6; (3.6)2]1 o[32.62; (3.6)2] o[32.4.12; 3.122; 32.4.3.4; 36] o[36; 33.42; 32.4.3.4; 34.6;
3.42.6; 32.4.12; 4.6.12]
o[92-uniform tiling]
           

Note that the canonically expanded triangular and hexagonal tilings are not identical to the rhombitrihexagonal tiling (more appropriately called the rectrihexagonal tiling), but have thin rectangles of ratio   instead of (regular) squares.


See alsoEdit

ReferencesEdit

  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. ^ Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld.
  3. ^ a b c d 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 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:1705.02848. Bibcode:2017RSPSA.47370267B. doi:10.1098/rspa.2017.0267.
  7. ^ Hart, George (1998). "Conway Notation for Polyhedra". Virtual Polyhedra. (See fourth row in table, "a = ambo".)
  8. ^ Armstrong, M. A. (Mark Anthony) (1988). Groups and symmetry. New York: Springer-Verlag. ISBN 0387966757. OCLC 17354658.
  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. ^ Livio Zefiro (2008). "Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra". Vismath.
  14. ^ George W. Hart (August 2000). Sculpture based on Propellorized Polyhedra. Proceedings of MOSAIC 2000. Seattle, WA. pp. 61–70.
  15. ^ Deza, M.; Dutour, M (2004). "Goldberg–Coxeter constructions for 3-and 4-valent plane graphs". The Electronic Journal of Combinatorics. 11: #R20.
  16. ^ 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 linksEdit

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