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.
This chart adds 3 new operations: George Hart's p=propellor operator that add quadrilaterals, g=gyro operation that creates pentagons, and a c=Chamfer operation that replaces edges with hexagons

Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.

The seed polyhedra are the Platonic solids, represented by the first letter of their name (T,O,C,I,D); the prisms (Pn), antiprisms (An) and pyramids (Yn). Any convex polyhedron can serve as a seed, as long as the operations can be executed on it.

John Conway extended the idea of using operators, like truncation defined by Kepler, to build related polyhedra of the same symmetry. His descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. Applied in a series, these operators allow many higher order polyhedra to be generated.

Operations on polyhedra[edit]

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)

Operator Name Alternate
construction
vertices edges faces Description
Seed v e f Seed form
r Reflect
(Hart)
v e f Mirror image for chiral forms
d dual f e v dual of the seed polyhedron - each vertex creates a new face
a ambo e 2e 2 + e The edges are new vertices, while old vertices disappear. (rectify)
j join da e + 2 2e e The seed is augmented with pyramids at a height high enough so that 2 coplanar triangles from 2 different pyramids share an edge.
t truncate dkd 2e 3e e + 2 truncate all vertices.
-- -- dk 2e 3e e + 2 Dual of kis, (bitruncation)
-- -- kd e + 2 3e 2e Kis of dual
k kis dtd e + 2 3e 2e raises a pyramid on each face.
c chamfer e + v 4e 2e + f New hexagonal faces are added in place of edges.
- - dc 2e + f 4e e + v
e expand aa 2e 4e 2e + 2 Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate)
o ortho de 2e + 2 4e 2e Each n-gon faces are divided into n quadrilaterals.
p propellor
(Hart)
v + 2e 4e e + f A face rotation that creates quadrilaterals at vertices (self-dual)
- - dp e + f 4e v + 2e
s snub dg 2e 5e 3e + 2 "expand and twist" – each vertex creates a new face and each edge creates two new triangles
g gyro ds 3e + 2 5e 2e Each n-gon face is divided into n pentagons.
b bevel ta 4e 6e 2e + 2 New faces are added in place of edges and vertices, Omnitruncation (Known as cantitruncation in higher polytopes).
m meta db & kj 2e + 2 6e 4e n-gon faces are divided into 2n triangles

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:

All operations are symmetry-preserving except twisting ones like s and g which lose reflection symmetry.

Examples[edit]

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. (Two polyhedron forms don't have single operator names given by Conway.)

Cube
"seed"
ambo
(rectify)
truncate bitruncate expand
(cantellate)
bevel
(omnitruncate)
snub
Uniform polyhedron-43-t0.png
C
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
aC = djC
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t01.png
tC = dkdC
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t12.png
tdC = dkC
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t02.png
eC = aaC = doC
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-t012.png
bC = taC = dmC = dkjC
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-43-s012.png
sC = dgC
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
dual join kis
(vertex-bisect)
ortho
(edge-bisect)
meta
(full-bisect)
gyro
Uniform polyhedron-43-t2.png
dC
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Rhombicdodecahedron.jpg
jC = daC
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Triakisoctahedron.jpg
kdC = dtC
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Tetrakishexahedron.jpg
kC = dtdC
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Deltoidalicositetrahedron.jpg
oC = deC = daaC
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png
Disdyakisdodecahedron.jpg
mC = dbC = kjC
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Pentagonalicositetrahedronccw.jpg
gC = dsC
CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png

Generating regular seeds[edit]

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

Extensions to Conway's symbols[edit]

The above operations allow all of the semiregular polyhedrons and Catalan solids to be generated from regular polyhedrons. Combined many higher operations can be made, but many interesting higher order polyhedra require new operators to be constructed.

For example, geometric artist George W. Hart created an operation he called a propellor, and another reflect to create mirror images of the rotated forms.

  • p – "propellor" (A rotation operator that creates quadrilaterals at the vertices). This operation is self-dual: dpX=pdX.
  • r – "reflect" – makes the mirror image of the seed; it has no effect unless the seed was made with s or g.

Geometric coordinates of derived forms[edit]

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.

Example: A dodecahedron seed as a spherical tiling
Uniform tiling 532-t0.png
D
Uniform tiling 532-t01.png
tD
Uniform tiling 532-t1.png
aD
Uniform tiling 532-t12.png
tdD
Uniform tiling 532-t02.png
eD
Uniform tiling 532-t012.png
taD
Spherical snub dodecahedron.png
sD
Uniform tiling 532-t2.png
dD
Icosahedral reflection domains.png
dtaD
Example: A Euclidean hexagonal tiling seed (H)
Uniform tiling 63-t0.png
H
Uniform tiling 63-t01.png
tH
Uniform tiling 63-t1.png
aH
Uniform tiling 63-t12.png
tdH = H
Uniform tiling 63-t02.png
eH
Uniform tiling 63-t012.png
taH
Uniform tiling 63-snub.png
sH
Uniform tiling 63-t2.png
dH
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
dtH
Tiling Dual Semiregular V3-6-3-6 Quasiregular Rhombic.svg
daH
Uniform tiling 63-t2.png
dtdH = dH
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
deH
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg
dtaH
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
dsH
Example: A transparent Tetrahedron seed (T)
Tetrahedron.jpg
T
Truncatedtetrahedron.jpg
tT
Octahedron.svg
aT
Truncatedtetrahedron.jpg
tdT
Cuboctahedron.jpg
eT
Truncatedoctahedron.jpg
bT
Icosahedron.jpg
sT
Tetrahedron.jpg
dT
Triakistetrahedron.jpg
dtT
Hexahedron.jpg
jT
Triakistetrahedron.jpg
kT
Rhombicdodecahedron.jpg
oT
Tetrakishexahedron.jpg
mT
Dodecahedron.jpg
gT
Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"
truncate ambo
(rectify)
bitruncate expand
(cantellate)
bevel
(omnitruncate)
snub
Uniform tiling 73-t0.png Uniform tiling 73-t01.png Uniform tiling 73-t1.png Uniform tiling 73-t12.png Uniform tiling 73-t02.png Uniform tiling 73-t012.png Uniform tiling 73-snub.png
dual join kis
(vertex-bisect)
ortho
(edge-bisect)
meta
(full-bisect)
gyro
Uniform tiling 73-t2.png Ord7 triakis triang til.png Order73 qreg rhombic til.png Order3 heptakis heptagonal til.png Deltoidal triheptagonal til.png Order-3 heptakis heptagonal tiling.png Ord7 3 floret penta til.png

Other polyhedra[edit]

Iterating operators on simple forms can produce progressively larger polyhedra, maintaining the fundamental symmetry of the seed element. The vertices are assumed to be on the same spherical radius. Some generated forms can exist as spherical tilings, but fail to produce polyhedra with planar faces.

Tetrahedral symmetry[edit]

Octahedral symmetry[edit]

Icosahedral symmetry[edit]

Rhombic:

Triangular:

Dual triangular:

Triangular chiral:

Dual triangular chiral:

Dihedral symmetry[edit]

See also[edit]

References[edit]

  • George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70 [1]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
    • Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings

External links and references[edit]