User:Tomruen/Wythoffian operators
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[edit] Wythoffian operator names
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Some example operations on a cube
These operators apply to regular polytopes in dimensions up to 5.
According to Norman Johnson the fundamental cutting operations are:
- Truncation - cuts vertices (truncating vertices) (polygons and higher)
- Cantellation - cuts edges (beveling edges) (polyhedron and higher)
- Runcination - cuts faces (planing faces) (polychoron and higher)
- Stericating - cuts cells (soliding cells) (5-polytopes and higher) (Like steradian)
See also:
- Image:Polyhedron kaleidoscope triangles4.png
- Image:Polyhedron truncation example3.png
- Image:Polychoron truncation chart.png
- Image:Uniform honeycomb truncations.png
| Step | J-abbr | Johnson | Bowers | ||
|---|---|---|---|---|---|
| 0 | - | Rectified | Rectated | ||
| 1 | t | Truncated | Truncated | ||
| 2 | c | Cantellated | Rhombated | ||
| 3 | r | Runcinated | Prismated | ||
| 4 | s | Stericated | Cellated |
This table shows the operator names used by Johnson and Bowers. Bowers notation is similar, but uses a great prefix to group a complete set of 2 or more lower operations together.
| Right->left | ||||||||
|---|---|---|---|---|---|---|---|---|
| t-notation | Rings | First | Binary | B-Ind | Op-Ind | J-ab | Johnson name | Bowers name |
| t_0 | 1 | 1 | 00001 | 1 | 1 | 0 | regular | regular |
| t_1 | 1 | 2 | 00010 | 2 | 2 | 1 | rectified | rectated |
| t_2 | 1 | 3 | 00100 | 4 | 3 | 2 | birectified | birectated |
| t_3 | 1 | 4 | 01000 | 8 | 4 | 3 | trirectified | trirectated |
| t_4 | 1 | 5 | 10000 | 16 | 5 | 4 | quadrirectified | quadrirectated |
| t_0,1 | 2 | 1 | 00011 | 3 | 7 | t1 | truncated | truncated |
| t_1,2 | 2 | 2 | 00110 | 6 | 8 | t2 | bitruncated | bitruncated |
| t_2,3 | 2 | 3 | 01100 | 12 | 9 | t3 | tritruncated | tritruncated |
| t_3,4 | 2 | 4 | 11000 | 24 | 10 | t4 | quadritruncated | quadritruncated |
| t_0,2 | 2 | 1 | 00101 | 5 | 12 | c1 | cantellated | small rhombated |
| t_1,3 | 2 | 2 | 01010 | 10 | 13 | c2 | bicantellated | small birhombated |
| t_2,4 | 2 | 3 | 10100 | 20 | 14 | c3 | tricantellated | small trirhombated |
| t_0,3 | 2 | 1 | 01001 | 9 | 16 | r1 | runcinated | small prismated |
| t_1,4 | 2 | 2 | 10010 | 18 | 17 | r2 | biruncinated | small biprismated |
| t_0,4 | 2 | 1 | 10001 | 17 | 19 | s1 | stericated | small cellated |
| t_0,1,2 | 3 | 1 | 000111 | 7 | 22 | ct | cantitruncated | great rhombated |
| t_1,2,3 | 3 | 2 | 01110 | 14 | 23 | ct2 | bicantitruncated | great birhombated |
| t_2,3,4 | 3 | 3 | 11100 | 28 | 24 | ct3 | tricantitruncated | great trirhombated |
| t_0,1,3 | 3 | 1 | 01011 | 11 | 26 | rt1 | runcitruncated | prismatotruncated |
| t_1,2,4 | 3 | 2 | 10110 | 22 | 27 | rt2 | biruncitruncated | biprismatotruncated |
| t_0,2,3 | 3 | 1 | 01101 | 13 | 29 | rc1 | runcicantellated | prismatorhombated |
| t_1,3,4 | 3 | 2 | 11010 | 26 | 30 | rc2 | biruncicantellated | biprismatorhombated |
| t_0,1,4 | 3 | 1 | 10011 | 19 | 32 | st1 | steritruncated | cellitruncated |
| t_0,2,4 | 3 | 1 | 010101 | 21 | 34 | sc1 | stericantellated | small cellirhombated |
| t_0,3,4 | 3 | 1 | 11001 | 25 | 36 | sr1 | steriruncinated | celliprismated |
| t_0,1,2,3 | 4 | 1 | 01111 | 15 | 42 | rct1 | runcicantitruncated | great prismated |
| t_1,2,3,4 | 4 | 2 | 11110 | 30 | 43 | rct2 | biruncicantitruncated | great biprismated |
| t_0,1,2,4 | 4 | 1 | 10111 | 23 | 45 | sct1 | stericantitruncated | great cellirhombated |
| t_0,1,3,4 | 4 | 1 | 11011 | 27 | 47 | srt1 | steriruncitruncated | celliprismatotruncated |
| t_0,2,3,4 | 4 | 1 | 11101 | 29 | 49 | src1 | steriruncicantellated | celliprismatorhombated |
| t_0,1,2,3,4 | 5 | 1 | 11111 | 31 | 57 | srct1 | steriruncicantitruncated | great cellated |
