Conway symbol (InstanceTopic, 3)
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'''Conway polyhedron notation''' is used to describe [[polyhedron|polyhedra]] based on a seed polyhedron modified by various [[operator]]s. | '''Conway polyhedron notation''' is used to describe [[polyhedron|polyhedra]] based on a seed polyhedron modified by various [[operator]]s. | ||
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Since all seed polyhedra can be made from Y3, the symbols T, O, C, I and D will not be used in [[STS template]]s. The compound symbols j, k, o, b, m, g will also not be used. | Since all seed polyhedra can be made from Y3, the symbols T, O, C, I and D will not be used in [[STS template]]s. The compound symbols j, k, o, b, m, g will also not be used. | ||
- | + | <[#embed [hash 99AD0N31G7D3PRDBGE68TNV2PV]]> | |
== Operations on [[polyhedra]] == | == Operations on [[polyhedra]] == | ||
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![[snub]] | ![[snub]] | ||
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- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash RSBDW92GAYMGP90J7F5GG0GCZZ]]><BR>[[Cube|C]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 3W65RR0DXPD2Y8NXE8QMY47BE2]]><BR>[[Cuboctahedron|aC = djC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 9RNYG4DKQSMSPHDHMZ2XM6WYNA]]><BR>[[Truncated cube|tC = dkdC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash JEPRANJMQXYFY4V4WAP4Y410QB]]><BR>[[Truncated octahedron|tdC = dkC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash SZV9GZM043D12HYA1NHDE025ER]]><BR>[[Small rhombicuboctahedron|eC = aaC = doC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash WFB4Y2N0581CMZMK6DYV2DDPM2]]><BR>[[Great rhombicuboctahedron|bC = taC = dmC = dkjC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 095S4NKSBZ68MN094F8WR31YVK]]><BR>[[Snub cube|sC = dgC]] |
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![[Dual polyhedron|dual]] | ![[Dual polyhedron|dual]] | ||
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!gyro | !gyro | ||
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- | |align=center| | + | |align=center|<[#embed [hash 6J50TS8W96HFE4N1VP8XW6B1SQ]]><BR>[[Octahedron|dC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash TW7NCKFYHH3VG1HHS60ZTKQVZJ]]><BR>[[Rhombic dodecahedron|jC = daC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash XJ3C89WYXAAWWMA5GYJDYT7Q9E]]><BR>[[Triakis octahedron|kdC = dtC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash Z01KREF99SWWCH3JH07PTYPA1B]]><BR>[[Tetrakis hexahedron|kC = dtdC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 2BQ5M1RW82P5R6J91R0A8ZXN62]]><BR>[[Deltoidal icositetrahedron|oC = deC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 9AZ2YT0XBZWBJH07ZDFQYPT323]]><BR>[[Disdyakis dodecahedron|mC = dbC = kjC]] |
- | |align=center valign=bottom| | + | |align=center valign=bottom|<[#embed [hash 35RFPM5Z73HJJTS61RJZTMZ6NY]]><BR>[[Pentagonal icositetrahedron|gC = dsC]] |
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{| class=wikitable | {| class=wikitable | ||
|+Example: A [[dodecahedron]] seed as a spherical tiling | |+Example: A [[dodecahedron]] seed as a spherical tiling | ||
- | |align=center| | + | |align=center|<[#embed [hash J66VJVS8KN5R472ZJQSH2DD4VG]]><BR>[[Dodecahedron|D]] |
- | |align=center| | + | |align=center|<[#embed [hash FD0TJJGTZJJF2N5PWE17A0KN3E]]><BR>[[Truncated dodecahedron|tD]] |
- | |align=center| | + | |align=center|<[#embed [hash 0EWY4GKX9W0ZC5C2TSHYP729B5]]><BR>[[Icosidodecahedron|aD]] |
- | |align=center| | + | |align=center|<[#embed [hash QMXY4MJNYWV76JQGQW6TM4VC5V]]><BR>[[Truncated icosahedron|tdD]] |
- | |align=center| | + | |align=center|<[#embed [hash X4YC8AEXAXNCTJKGG0Q5R5D3NZ]]><BR>[[Icosahedron|dD]] |
- | |align=center| | + | |align=center|<[#embed [hash B2QEJX0TFZ5XMEYH923QT1HRVN]]><BR>[[Rhombicosidodecahedron|eD]] |
- | |align=center| | + | |align=center|<[#embed [hash 0CXX6Y454HAQW4N6A5N80X784G]]><BR>[[Truncated icosidodecahedron|teD]] |
|} | |} | ||
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Latest revision as of 20:50, 11 February 2014
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 a series, these operators allows many higher order polyhedra to be generated.
Since all seed polyhedra can be made from Y3, the symbols T, O, C, I and D will not be used in STS templates. The compound symbols j, k, o, b, m, g will also not be used.
Operations on polyhedra
Operator | 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 | truncates to the edge midpoints, each vertex creates a new face. (rectification) | |
j | join | da | e+2 | 2e | e | new kite-shaped faces are created in place of each edge. |
t | truncate | 2e | 3e | e+2 | truncate all vertices. | |
-- | -- | dt | e+2 | 3e | 2e | dual of truncation (bitruncation) |
-- | -- | dk | 2e | 3e | e+2 | dual of kis |
k | kis | dtd | e+2 | 3e | 2e | raises a pyramid on each face. |
e | expand | 2e | 4e | 2e+2 | Each vertex creates a new face and each edge creates a new quadrilateral. (cantellation) | |
o | ortho | de | 2e+2 | 4e | 2e | Each n-gon faces are divided into n quadrilaterals. |
b | bevel | ta | 4e | 5e | 2e+2 | New faces are added in place of edges and vertices. |
m | meta | db & kj | 2e+2 | 5e | 4e | n-gon faces are divided into 2n triangles |
s | snub | 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. |
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:
- 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.
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 (Rectification) | truncate | bitruncate | expand (cantellation) | bevel (omnitruncation) | snub |
---|---|---|---|---|---|---|
C | aC = djC | tC = dkdC | tdC = dkC | eC = aaC = doC | bC = taC = 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 |
Generating regular seeds
All of the five regular polyhedra can be generated from the tetrahedron, Y3, with zero to two operators:
- T = Y3
- O = aY3 (Rectified tetrahedron)
- C = daY3 (dual to rectified tetrahedron)
- I = sY3 (snub tetrahedron)
- D = dsY3 (dual to snub tetrahedron)
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 [1] 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.
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 torus-shaped polyhedra can derived other polyhedra with point on the same torus surface.
D | tD | aD | tdD | dD | eD | teD |