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)
| Operator | Name | Alternate construction |
vertices | edges | faces | Description |
|---|---|---|---|---|---|---|
| Seed | v | e | f | Seed form | ||
| r | Reflect (Hart) |
v | e | f | Mirror image for chiral forms | |
| d | dual | f | e | v | dual of the seed polyhedron - each vertex creates a new face | |
| a | ambo | e | 2e | 2+e | The edges are new vertices, while old vertices disappear. (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. |
| -- | -- | dk | 2e | 3e | e+2 | Dual of kis, (bitruncation) |
| -- | -- | kd | e+2 | 3e | 2e | Kis of dual |
| k | kis | dtd | e+2 | 3e | 2e | raises a pyramid on each face. |
| c | chamfer | e+v | 4e | 2e+f | New hexagonal faces are added in place of edges. | |
| - | - | dc | 2e+f | 4e | e+v | |
| e | expand | aa | 2e | 4e | 2e+2 | Each vertex creates a new face and each edge creates a new quadrilateral. (cantellate) |
| o | ortho | de | 2e+2 | 4e | 2e | Each n-gon faces are divided into n quadrilaterals. |
| p | propellor (Hart) |
v+2e | 4e | e+f | A face rotation that creates quadrilaterals at vertices (self-dual) | |
| - | - | dp | e+f | 4e | v+2e | |
| s | snub | dg | 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. |
| b | bevel | ta | 4e | 6e | 2e+2 | New faces are added in place of edges and vertices, Omnitruncation (Known as cantitruncation in higher polytopes). |
| m | meta | db & kj | 2e+2 | 6e | 4e | n-gon faces are divided into 2n triangles |
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.
Read more about this topic: Conway Polyhedron Notation
Famous quotes containing the word operations:
“It may seem strange that any road through such a wilderness should be passable, even in winter, when the snow is three or four feet deep, but at that season, wherever lumbering operations are actively carried on, teams are continually passing on the single track, and it becomes as smooth almost as a railway.”
—Henry David Thoreau (1817–1862)