Notes On The Wythoff Construction For The Uniform 5-polytopes
Construction of the reflective 5-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter-Dynkin diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 5-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.
Here's the primary operators available for constructing and naming the uniform 5-polytopes.
The last operation, the snub, and more generally the alternation, are the operation that can create nonreflective forms. These are drawn with "hollow rings" at the nodes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
| Operation | Extended Schläfli symbol |
Coxeter- Dynkin diagram |
Description |
|---|---|---|---|
| Parent | t0{p,q,r,s} | Any regular 5-polytope | |
| Rectified | t1{p,q,r,s} | The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual. | |
| Birectified | t2{p,q,r,s} | Birectification reduces cells to their duals. | |
| Truncated | t0,1{p,q,r,s} | Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual. |
|
| Cantellated | t0,2{p,q,r,s} | In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms. |
|
| Runcinated | t0,3{p,q,r,s} | Runcination reduces cells and creates new cells at the vertices and edges. | |
| Stericated | t0,4{p,q,r,s} | Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for polyterons.) | |
| Omnitruncated | t0,1,2,3,4{p,q,r,s} | All four operators, truncation, cantellation, runcination, and sterication are applied. |
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