User:Tomruen/Conway polyhedron notation



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.

Conway and Hart extended the idea of using operators, like truncation defined by Kepler, to build related polyhedra of the same symmetry. The basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. 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. A resulting polyhedron will have a fixed topology (vertices, edges, faces), while exact geometry is not constrained: it can be thought of as one of many embeddings of a polyhedral graph on the sphere.

The seed polyhedra are the Platonic solids, represented by the first letter of their name (T,O,C,I,D); the prisms (Pn) for n-gonal forms, antiprisms (An), cupolae (Un), anticupolae (Vn) and pyramids (Yn). Any polyhedron can serve as a seed, as long as the operations can be executed on it. For example regular-faced Johnson solids can be referenced as Jn, for n=1..92.

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 becomes the same as the expand operation: aa=e, while a truncation after ambo produces bevel: ta=b. There has been no general theory describing what polyhedra can be generated in by any set of operators. Instead all results have been discovered empirically.

Operations on polyhedra
Elements are given from the seed (v,e,f) to the new forms, assuming seed is a convex polyhedron: (a topological sphere, Euler characteristic = 2) An example image is given for each operation, based on a cubic seed. The basic operations are sufficient to generate the reflective uniform polyhedra and their duals. Some basic operations can be made as composites of others. One way to classify operations is by the ratio of the number of edges after the operation to the number before: for a large class of operations, this is an integer value that does not vary depending on the seed.

Special forms
 * The kis operator has a variation, kn, which only adds pyramids to n-sided faces.
 * The truncate operator has a variation, tn, which only truncates order-n vertices.

The 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.

Chirality operator
 * r – "reflect" – makes the mirror image of the seed; it has no effect unless the seed was made with s or g. Alternately an overline can be used for picking the other chiral form, like $\overline{s}$ = rs.

The operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices.

Generating regular seeds
All of the five regular polyhedra can be generated from prismatic generators with zero to two operators:
 * 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

Examples
The cube can generate all the convex uniform polyhedra with octahedral symmetry. The first row generates the Archimedean solids and the second row the Catalan solids, the second row forms being duals of the first. Comparing each new polyhedron with the cube, each operation can be visually understood.

The truncated icosahedron, tI or zD, which is Goldberg polyhedron G(2,0), creates more polyhedra which are neither vertex nor face-transitive.

Geometric coordinates of derived forms
In general the seed polyhedron can be considered a tiling of a surface since the operators represent topological operations so the exact geometric positions of the vertices of the derived forms are not defined in general. A convex regular polyhedron seed can be considered a tiling on a sphere, and so the derived polyhedron can equally be assumed to be positioned on the surface of a sphere. Similar a regular tiling on a plane, such as a hexagonal tiling can be a seed tiling for derived tilings. Nonconvex polyhedra can become seeds if a related topological surface is defined to constrain the positions of the vertices. For example, toroidal polyhedra can derive other polyhedra with point on the same torus surface.

Derived operations
Mixing two or more basic operations leads to a wide variety of forms. There are many more derived operations, for example, mixing two ambo, kis, or expand, along with up to 3 interspaced duals. Using alternative operators like join, truncate, ortho, bevel and medial can simply the names and remove the dual operators. The numbers of total edges of a derived operation can be computed as the product of the number of total edges of each individual operator.

The operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that cross original vertices.

Chiral derived operations
There are more derived operators mixing at least one gyro with ambo, kis or expand, and up to 3 duals.

Extended operators
These extended operators can't be created in general from the basic operations above. Some can be created in special cases with k and t operators only applied to specific sided faces and vertices. 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.

Some further extended operators suggest a sequence and are given a following integer for higher order forms. For example, ortho divides a square face into 4 squares, and a o3 can divide into 9 squares. o3 is a unique construction while o4 can be derived as oo, ortho applied twice. The loft operator can include an index, similar to kis, to limit the effect to faces with that number of sides.

The chamfer operation creates Goldberg polyhedra G(2,0), with new hexagons between original faces. Sequential chamfers create G(2n,0).

Extended chiral operators
These operators can't be created in general from the basic operations above. Geometric artist George W. Hart created an operation he called a propellor.


 * p – "propellor" (A rotation operator that creates quadrilaterals at the vertices). This operation is self-dual: dpX=pdX.

Operations that preserve original edges
These augmentation operations retain original edges, and allowing the operator to apply to any independent subset of faces. Conway notation supports an optional index to these operators to specific how many sides affected faces will have.

Subdivision
A subdivision operation divides original edges into n new edges and face interiors into smaller triangles or other polygons.

Square subdivision
The ortho operator can be applied in series for powers of two quad divisions. Other divisions can be produced by the product of factorized divisions. The propellor operator applied in sequence, in reverse chiral directions produces a 5-ortho division. If the seed polyhedron has nonquadrilaeral faces, they will be retained as smaller copies for odd-ortho operators.

Chiral hexagonal subdivision
A whirl operation creates Goldberg polyhedra, G(2,1) with new hexagonal faces around each original vertex. Two sequential whirls create G(3,5). In general, a whirl can transform a G(a,b) into G(a + 3b,2a − b) for a>b and the same chiral direction. If chiral directions are reversed, G(a,b) becomes G(2a + 3b,a − 2b) if a ≥ 2b, and G(3a + b,2b − a) if a < 2b. Higher n-whirls can be defined as G(n,n − 1), and m,n-whirl G(m,n).

Whirl-n operators generate Goldberg polyhedra (n,n − 1) and can be defined by dividing a seed polyhedron's edges into 2n − 1 subedges as rings around brick pattern hexagons. Some can also be generated by composite operators with smaller Whirl-m,n operators.

The product of whirl-n and its reverse generates a (3n2 − 3n + 1,0) Goldberg polyhedron. wrw generates (7,0) w3rw3 generates (19,0), w4rw4 generates (37,0), w5rw5 generates (61,0), and w6rw6 generates (91,0). The product of two whirl-n is ((n − 1)(3n − 1),2n − 1) or (3n2 − 4n + 1,2n − 1). The product of wa by wb gives (3ab − 2(a + b) + 1,a + b − 1), and wa by reverse wb is (3ab − a − 2b + 1,a − b) for a ≥ b.

The product of two identical whirl-n operators generates Goldberg ((n − 1)(3n − 1),2n − 1). The product of a k-whirl and zip is (3k − 2,1).

Triangulated subdivision


An operation un divides faces into triangles with n-divisions along each edge, called an n-frequency subdivision in Buckminster Fuller's geodesic polyhedra.

Conway polyhedron operators can construct many of these subdivisions.

If the original faces are all triangles, the new polyhedra will also have all triangular faces, and create triangular tilings within each original face. If the original polyhedra has higher polygons, all new faces won't necessarily be triangles. In such cases a polyhedron can first be kised, with new vertices inserted in the center of each face.

Geodesic polyhedra
Conway operations can duplicate some of the Goldberg polyhedra and geodesic duals. The number of vertices, edges, and faces of Goldberg polyhedron G(m,n) can be computed from m and n, with T = m2 + mn + n2 = (m + n)2 − mn as the number of new triangles in each subdivided triangle. (m,0) and (m,m) constructions are listed below from Conway operators.

Class I
For Goldberg duals, an operator uk is defined here as dividing faces with k edge subdivisions, with Conway u = u2, while its conjugate operator, dud is chamfer, c. This operator is used in computer graphics, loop subdivision surface, as recursive iterations of u2, doubling each application. The operator u3 is given a Conway operator nn=kt, and its conjugate operator zz=tk. The product of two whirl operators with reverse chirality, wrw or w$\overline{g}$, produces 7 subdivisions as Goldberg polyhedron G(7,0), thus u7=vrv. Higher subdivision and whirl operations in chiral pairs can construct more class I forms. w(3,1)rw(3,1) gives Goldberg G(13,0). w(3,2)rw(3,2) gives G(19,0).

Class II
Orthogonal subdivision can also be defined, using operator n=kd. The operator transforms geodesic polyhedron (a,b) into (a+2b,a-b), for a>b. It transforms (a,0) into (a,a), and (a,a) into (3a,0). The operator z=dk does the same for the Goldberg polyhedra.

This is also called a Triacon method, dividing into subtriangles along their height, so they require an even number of triangles along each edge.

Class III
Most geodesic polyhedra and dual Goldberg polyhedra G(n,m) can't be constructed from derived Conway operators. The whirl operation creates Goldberg polyhedra, G(2,1) with new hexagonal faces around each original vertex, and n-whirl genereates G(n,n-1). On icosahedral symmetry forms, t5g is equivalent to whirl in this case. The v=volute operation represents the triangular subdivision dual of whirl. On icosahedral forms it can be made by the derived operator k5s, a pentakis snub.

Two sequential whirls create G(3,5). In general, a whirl can transform a G(a,b) into G(a+3b,2a-b) for a>b and the same chiral direction. If chiral directions are reversed, G(a,b) becomes G(2a+3b,a-2b) if a>=2b, and G(3a+b,2b-a) if a<2b.

Example polyhedra by symmetry
Iterating operators on simple forms can produce progressively larger polyhedra, maintaining the fundamental symmetry of the seed element.

Toroidal symmetry
Torioidal tilings exist on the flat torus on the surface of a duocylinder in four dimensions but can be projected down to three dimensions as an ordinary torus. These tilings are topologically similar subsets of the Euclidean plane tilings.

External links and references

 * George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input. Also includes helpful explanations of the operations.
 * Polyhedra Names
 * Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
 * polyHédronisme: generates polyhedra in HTML5 canvas, taking Conway notation as input
 * John Conway's notation Visualization of Conway Polyhedron Notation
 * (truncate)
 * (ambo)
 * (kis)
 * Conway operators, PolyGloss, Wendy Krieger
 * Derived Solids
 * Derived Solids