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) An example image is given for each operation, based on a cubic seed.

Basic operations
Operator Example Name Alternate
construction
vertices edges faces Description
Conway C.png Seed v e f Seed form
d Conway dC.png dual f e v dual of the seed polyhedron - each vertex creates a new face
a Conway aC.png ambo e 2e 2 + e The edges are new vertices, while old vertices disappear. (rectify)
j Conway jC.png 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 Conway tC.png truncate dkd 2e 3e e + 2 truncate all vertices.
conjugate kis
k Conway kC.png kis dtd e + 2 3e 2e raises a pyramid on each face.
i Conway dkC.png -- dk 2e 3e e + 2 Dual of kis. (bitruncation)
n Conway kdC.png -- kd e + 2 3e 2e Kis of dual
e Conway eC.png expand aa 2e 4e 2e + 2 Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate)
o Conway oC.png ortho de = jj 2e + 2 4e 2e Each n-gon faces are divided into n quadrilaterals.
b Conway bC.png bevel ta 4e 6e 2e + 2 New faces are added in place of edges and vertices. (cantitruncation)
m Conway mC.png meta db & kj 2e + 2 6e 4e n-gon faces are divided into 2n triangles
Extended operations
Operator Example Name Alternate
construction
vertices edges faces Description
Conway C.png Seed v e f Seed form
r reflect
(Hart)
v e f Mirror image for chiral forms
h Conway hC.png half * v/2 e f+v/2 Alternation, remove half vertices,
limited to seed polyhedra with even-sided faces
c Conway cC.png chamfer v + 2e  4e f + e New hexagonal faces are added in place of edges.
- Conway dcC.png - dc f + e 4e v + 2e
p Conway pC.png propellor
(Hart)
v + 2e 4e f + e A face rotation that creates quadrilaterals at vertices (self-dual)
- Conway dpC.png - dp = pd f + e 4e v + 2e
s Conway sC.png snub dg = hta 2e 5e 3e + 2 "expand and twist" – each vertex creates a new face and each edge creates two new triangles
g Conway gC.png gyro ds 3e + 2 5e 2e Each n-gon face is divided into n pentagons.
w Conway wC.png whirl v+4e 7e f+2e Gyro followed by truncation of vertices centered on original faces.
This create 2 new hexagons for every original edge
- Conway dwC.png - dw f+2e 7e v+4e Dual of whirl

Note: * - The half operator, h, reduces square faces into digons, with two coinciding edges, which can be replaced by a single edge. Otherwise digons have a topological existence which can be subsequently truncated back into square faces.

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.

Generating regular seeds[edit]

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

The regular Euclidean tilings can also be used as seeds:

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.

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 truncate bitruncate expand bevel
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
dual join dual truncate kis ortho meta
Uniform polyhedron-43-t2.png
dC
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Rhombicdodecahedron.jpg
jC
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
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Deltoidalicositetrahedron.jpg
oC
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png
Disdyakisdodecahedron.jpg
mC
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Extended operations
snub propellor chamfer whirl
Uniform polyhedron-43-s012.png
sC
Propellor cube.png
pC
Chamfered cube.png
cC
Whirled cube.png
wC
gyro dual propeller dual chamfer dual whirl
Pentagonalicositetrahedronccw.jpg
gC = dsC
Dual propellor cube.png
dpC
Dual chamfered cube.png
dcC
Dual whirled cube.png
dwC
Tetrahedron seed (T)
Uniform polyhedron-33-t0.png
T
Uniform polyhedron-33-t01.png
tT
Uniform polyhedron-33-t1.png
aT
Uniform polyhedron-33-t12.png
tdT
Uniform polyhedron-33-t02.png
eT
Uniform polyhedron-33-t012.png
bT
Uniform polyhedron-33-s012.png
sT
Tetrahedron.jpg
dT
Triakistetrahedron.jpg
dtT
Hexahedron.jpg
jT
Triakistetrahedron.jpg
kT
Rhombicdodecahedron.jpg
oT
Tetrakishexahedron.jpg
mT
Dodecahedron.jpg
gT

The truncated icosahedron as a nonregular seed creates more polyhedra which are not vertex or face uniform.

Truncated icosahedron seed
"seed" ambo truncate bitruncate expand bevel
Uniform polyhedron-53-t12.png
tI
Rectified truncated icosahedron.png
atI
Truncated truncated icosahedron.png
ttI
Conway polyhedron Dk6k5tI.png
tdtI
Expanded truncated icosahedron.png
etI
Truncated rectified truncated icosahedron.png
btI
dual join kis ortho meta
Pentakisdodecahedron.jpg
dtI
Joined truncated icosahedron.png
jtI
Kissed kissed dodecahedron.png
kdtI
Conway polyhedron K6k5tI.png
ktI
Ortho truncated icosahedron.png
otI
Meta truncated icosahedron.png
mtI
Extended operations
snub propellor chamfer whirl
Snub rectified truncated icosahedron.png
stI
Propellor truncated icosahedron.png
ptI
Chamfered truncated icosahedron2.png
ctI
Whirled truncated icosahedron.png
wtI
gyro dual propeller dual chamfer dual whirl
Gyro truncated icosahedron.png
gtI
Dual propellor truncated icosahedron.png
dptI
Dual chamfered truncated icosahedron.png
dctI
Dual whirled truncated icosahedron.png
dwtI

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
Spherical triakis icosahedron.png
dtD
Spherical rhombic triacontahedron.png
daD = jD
Spherical pentakis dodecahedron.png
dtdD = kD
Spherical deltoidal hexecontahedron.png
deD = oD
Spherical disdyakis triacontahedron.png
dtaD = mD
Spherical pentagonal hexecontahedron.png
gD
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 = bH
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 = jH
Uniform tiling 63-t2.png
dtdH = kH
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
deH = oH
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg
dtaH = mH
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
dsH = gH
Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"
truncate ambo bitruncate expand bevel 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 dual kis join kis ortho meta 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]