Conway polyhedron notation
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
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
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. , and a truncated cuboctahedron is . Repeated application of an operator can be denoted with an exponent: j2 = o. In general, Conway operators are not
Individual operators can be visualized in terms of
3 (Triangle) | 4 (Square) | 5 (Pentagon) | 6 (Hexagon) |
---|---|---|---|
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
x | |
---|---|
xd | |
dx | |
dxd |
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
- ,
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
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
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.
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.[10] | |||
3 | Kis: k |
Needle: n |
Zip: z |
Truncate: t |
Kis raises a pyramid on each face, and is also called akisation, augmentation. Truncate cuts off the polyhedron at its vertices but leaves a portion of the original edges.[12] 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.[13] | |
6 | 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
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)
- 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)
- I = k5A5 (A special
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). To simplify, only irreducible operators are included in this list: others can be created by composing operators together.
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.[15] | ||
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 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: xn.
Augmentation
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
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
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 |
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]
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-Coxeter
The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the
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 Gaussian integersfor 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
By basic number theory, for any values of a and b, .
Quadrilateral
Examples
Archimedean and Catalan solids
Conway's original set of operators can create all of the
-
Cuboctahedron
aC = aO = eT -
Truncated octahedron
tO = bT -
Rhombicuboctahedron
eC = eO -
truncated cuboctahedron
bC = bO -
snub cube
sC = sO -
icosidodecahedron
aD = aI -
rhombicosidodecahedron
eD = eI -
truncated icosidodecahedron
bD = bI -
snub dodecahedron
sD = sI
-
Rhombic dodecahedron
jC = jO = oT -
Tetrakis hexahedron
kC = mT -
Deltoidal icositetrahedron
oC = oO -
Disdyakis dodecahedron
mC = mO -
Pentagonal icositetrahedron
gC = gO -
Rhombic triacontahedron
jD = jI -
Deltoidal hexecontahedron
oD = oI -
Disdyakis triacontahedron
mD = mI -
Pentagonal hexecontahedron
gD = gI
Composite operators
The
-
tI
-
atI
-
ttI
-
ztI = ttD
-
etI
-
btI
-
stI
-
dtI = nI = kD
-
jtI
-
ntI = kkD
-
ktI
-
otI
-
mtI
-
gtI
On the plane
Each of the
-
Truncated square tiling
tQ = bQ -
Tetrakis square tiling
kQ = mQ
-
Hexagonal tiling
H = dΔ = tΔ -
Trihexagonal tiling
aH = aΔ -
Rhombitrihexagonal tiling
eH = eΔ -
Snub trihexagonal tiling
sH = sΔ
-
Triangle tiling
Δ = dH = kH -
Rhombille tiling
jΔ = jH -
Triakis triangular tiling
kΔ -
Deltoidal trihexagonal tiling
oΔ = oH -
Kisrhombille tiling
mΔ = mH -
Floret pentagonal tiling
gΔ = gH
On a torus
Conway operators can also be applied to
-
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
See also
References
- ISBN 978-1-56881-220-5.
- ^ Weisstein, Eric W. "Conway Polyhedron Notation". MathWorld.
- ^ a b George W. Hart (1998). "Conway Notation for Polyhedra". Virtual Polyhedra.
- ^ a b c d e Adrian Rossiter. "conway - Conway Notation transformations". Antiprism Polyhedron Modelling Software.
- ^ Anselm Levskaya. "polyHédronisme".
- ^ a b Hart, George (1998). "Conway Notation for Polyhedra". Virtual Polyhedra. (See fourth row in table, "a = ambo".)
- ^ S2CID 119171258.
- ^ arXiv:1908.11622 [math.CO].
- 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.
- doi:10.37236/1773.
- ISBN 9788132224495.
External links
- polyHédronisme: generates polyhedra in HTML5 canvas, taking Conway notation as input