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Conway polyhedron notation

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Title: Conway polyhedron notation  
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Subject: Chamfer (geometry), Deltoidal icositetrahedron, Snub (geometry), Capsid, Catalan solid
Collection: Elementary Geometry, Mathematical Notation, Polyhedra
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Conway polyhedron notation

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.

Contents

  • Operations on polyhedra 1
  • Generating regular seeds 2
  • Extensions to Conway's symbols 3
  • Examples 4
  • Geometric coordinates of derived forms 5
  • Other polyhedra 6
    • Tetrahedral symmetry 6.1
    • Octahedral symmetry 6.2
    • Icosahedral symmetry 6.3
    • Dihedral symmetry 6.4
  • See also 7
  • References 8
  • External links and references 9

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.

Basic operations
Operator Example Name Alternate
construction
vertices edges faces Description
Seed v e f Seed form
d dual f e v dual of the seed polyhedron - each vertex creates a new face
a ambo e 2e 2 + e New vertices are added mid-edges, while old vertices are removed. (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.
conjugate kis
k kis dtd e + 2 3e 2e raises a pyramid on each face.
i -- dk 2e 3e e + 2 Dual of kis. (bitruncation)
n -- kd e + 2 3e 2e Kis of dual
e expand aa = aj 2e 4e 2e + 2 Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate)
o ortho de = ja = jj 2e + 2 4e 2e Each n-gon faces are divided into n quadrilaterals.
b bevel ta 4e 6e 2e + 2 New faces are added in place of edges and vertices. (cantitruncation)
m 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
Seed v e f Seed form
r reflect
(Hart)
v e f Mirror image for chiral forms
h half * v/2 e f+v/2 Alternation, remove half vertices,
limited to seed polyhedra with even-sided faces
c chamfer v + 2e  4e f + e An edge-truncation. New hexagonal faces are added in place of edges.
- - dc f + e 4e v + 2e
p propellor
(Hart)
v + 2e 4e f + e A face rotation that creates quadrilaterals at vertices (self-dual)
- - dp = pd f + e 4e v + 2e
s snub dg = hta 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.
w 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
- - 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

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

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

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

C

aC = djC

tC = dkdC

tdC = dkC

eC = aaC = doC

bC = taC = dmC = dkjC
dual join dual truncate kis ortho meta

dC

jC

kdC = dtC

kC

oC

mC
Extended operations
snub propellor chamfer whirl

sC

pC

cC

wC
gyro dual propeller dual chamfer dual whirl

gC = dsC

dpC

dcC

dwC
Tetrahedron seed (T)

T

tT

aT

tdT

eT

bT

sT

dT

dtT

jT

kT

oT

mT

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

tI

atI

ttI

tdtI

etI

btI
dual join kis ortho meta

dtI

jtI

kdtI

ktI

otI

mtI
Extended operations
snub propellor chamfer whirl

stI

ptI

ctI

wtI
gyro dual propeller dual chamfer dual whirl

gtI

dptI

dctI

dwtI

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.

Example: A dodecahedron seed as a spherical tiling

D

tD

aD

tdD

eD

taD

sD

dD

dtD

daD = jD

dtdD = kD

deD = oD

dtaD = mD

gD
Example: A Euclidean hexagonal tiling seed (H)

H

tH

aH

tdH = H

eH

taH = bH

sH

dH

dtH

daH = jH

dtdH = kH

deH = oH

dtaH = mH

dsH = gH
Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"
truncate ambo bitruncate expand bevel snub
dual dual kis join kis ortho meta gyro

Other polyhedra

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

Octahedral symmetry

Icosahedral symmetry

Rhombic:

Triangular:

Dual triangular:

Triangular chiral:

Dual triangular chiral:

Dihedral symmetry

See also

References

  • George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70 [2]
  • 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

  • George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
    • 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
  • Weisstein, Eric W., "Conway Polyhedron Notation", MathWorld.
  • John Conway's notation
  • Weisstein, Eric W., "Truncation", MathWorld. (truncate)
  • Weisstein, Eric W., "Rectification", MathWorld. (ambo)
  • Weisstein, Eric W., "Cumulation or Apiculation", MathWorld. (kis)
  • Conway operators, PolyGloss, Wendy Krieger
  • Derived Solids
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