Conway polyhedron notation

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 theirs duals. Some basic operations can be made as composites of others.

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 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 not vertex or face uniform.

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.

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.

Operations that preserve original edges

These augmentation operations retain original edges, and allow a number to specific which sided-faces to to be applied.

Other symmetry reducing operators

These operations only work for polyhedra with even-sided faces, and due to degeneracy their operations work slightly differently on square faces.

The half operator, h, from Coxeter, reduces square faces into digons, with two coinciding edges, which may or may not be replaced by a single edge. If digons remain, subsequently truncation operations can expand digons into square faces.

The flex operator, f, adds a vertex in the center of the faces, and bisects all edges, but only connects new edges from each center to half of the edges, creating new hexagonal faces. Original square faces do not require the central vertex and need only a single edge across the face, creating pairs of pentagons. A dodecahedron (tetartoid) can be constructed as fC (flex cube), as well as gT (gyro tetrahedron).

Unlike the other operators, these operators can reduce the symmetry order of a seed polyhedron. On a cube, the half operator reduces full octahedral symmetry, order 48 to full tetrahedral symmetry, order 24. And the flex operator reduces full octahedral symmetry, order 48, to pyritohedral symmetry, order 24.

Subdivision

A subdivision operation divides original edges into n new edges.

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 producess a 5-ortho division. If the seed polyhedron has nonsquare faces, they will be retained as smaller copies for odd-ortho operators.

Triangulated subdivision

An operation u_n 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.

For Goldberg duals, an operator u_k 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 kt=x, and its conjugate operator y=dxd=tk. The product of two whirl operators with reverse chirality, wrw or ww, 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. w3rw3 gives Goldberg G(13,0). u(3,2)ru(3,2) gives G(19,0).

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.

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. 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. A generalized n-whirl is defined here to generate G(n,1).

Example polyhedra by symmetry

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

Chiral

Chiral

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.