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.

Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operators. The seed polyhedra are the Platonic solids, represented by their 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 that can generate all the Archimedean solids and Catalan solids from regular seeds. Applied in series these operators, it allows many higher order polyhedra.

[edit] Operations on polyhedra

• d - the dual of the seed polyhedron - each vertex creates a new face
• tn - truncates all the n-fold vertices; if n is omitted, truncates all vertices
• kn - "kis" operator raises a pyramid on each n-gonal face; if n is omitted, elevates all faces
• a - "ambo" truncates to the edge midpoints, rectifying the polyhedron - each vertex creates a new face.
• e - "expand" (cantellate - each vertex creates a new face and each edge creates a new quadrilateral )
• s - "snub" (""expand and twist" - each vertex creates a new face and each edge creates two new triangles)

Some frequent combinations of operators have a shorter alternate notation:

• kn - "kis"  : knX = dtndX (Each n-gon faces are divided into n triangles)
• g - "gyro" : gX = dsX (Each n-gon face is divided into n pentagons)
• o - "ortho": oX = deX (Each n-gon faces are divided into n quadrilaterals)
• m - "meta" : mX = dbX = kjX (n-gon faces are divided into 2n triangles)
• j - "join" : jX = daX (New kite-shaped faces are created in place of each edge)
• b - "bevel": bX = taX (New faces are added in place of edges and vertices)

The operators are applied like functions from right to left. For example:

• the dual of a tetrahedron is dT;
• the truncation of a cube is t3C or tC;
• the truncation of a Cuboctahedron is t4aC or taC.

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

[edit] Examples

The cube can generate all the convex Octahedral symmetry uniform polyhedra. 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		expand (cantellate)	bevel (omnitruncate)	snub
C	aC = djC	tC = dkdC	tdC = dkC	eC = aaC = doC	bC = dmC = dkjC	sC = dgC
dual	join		kis (vertex-bisect)	ortho (edge-bisect)	meta (full-bisect)	gyro
dC	jC = daC	kdC = dtC	kC = dtdC	oC = deC	mC = dbC = kjC	gC = dsC

[edit] 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 = aY3 (Rectified tetrahedron)
  • C = daY3 (dual to rectified tetrahedron)
  • I = sY3 (snub tetrahedron)
  • D = dsY3 (dual to snub 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)

[edit] 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 create 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 p.

[edit] See also

• Uniform polyhedra
• Computer graphics algorithms:
  • Doo-Sabin subdivision surface - expand operator
  • Catmull-Clark subdivision surface - ortho operator

[edit] References

• 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 Symmetry of Things 2008, ISBN 978-1-56881-220-5
  • Chapter 24: Naming the Archimedean and Catalan polyhedra and Tilings

[edit] External links and references

• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
• Eric W. Weisstein, Conway Polyhedron Notation at MathWorld.
• John Conway's notation
• Olshevsky, George, Apiculation at Glossary for Hyperspace. (kis)
• Olshevsky, George, Truncation at Glossary for Hyperspace. (truncate)
• Olshevsky, George, Rectification at Glossary for Hyperspace. (ambo)