Special Cuts of 600-cell
The 600-cell is a regular 4-polytope with 120 vertices and 600 tetrahedral facets. This project explores its special cuts—sets of non-adjacent vertices that, when removed, yield unique regular-faced polytopes.
Geometric Background
- Definition: A special cut is a family S of vertices of the 600-cell such that no two vertices in S are adjacent.
- Polytope Construction: Removing vertices in S and taking the convex hull results in a regular-faced polytope, where vertices in S correspond to icosahedral cells.
- Symmetries: The symmetry group of the 600-cell is the Coxeter group H4 (14,400 elements). Equivalent cuts under this group produce isometric polytopes.
- Snub 24-cell: The maximal case of 24 vertices in S corresponds to the semiregular snub 24-cell.
Enumeration Results
Below is the number of orbits of special cuts for each size s of the set S:
| s | Orbits | Data | s | Orbits | Data |
|---|---|---|---|---|---|
| 1 | 1 | View | 13 | 74,619,659 | View |
| 2 | 7 | View | 14 | 54,482,049 | View |
| 3 | 39 | View | 15 | 26,749,384 | View |
| 4 | 436 | View | 16 | 8,690,111 | View |
| 5 | 4,776 | View | 17 | 1,856,685 | View |
| 6 | 45,775 | View | 18 | 263,268 | View |
| 7 | 334,380 | View | 19 | 25,265 | View |
| 8 | 1,826,415 | View | 20 | 1,683 | View |
| 9 | 7,355,498 | View | 21 | 86 | View |
| 10 | 21,671,527 | View | 22 | 9 | View |
| 11 | 46,176,020 | View | 23 | 1 | View |
| 12 | 70,145,269 | View | 24 | 1 | View |
Programs & Additional Data
- Adjacency Matrix: Orbit vertices are indexed relative to this adjacency matrix.
- Group Data: Full list of symmetry group elements available here.
- Enumeration Program: Download the source code here.
- Analysis: Detailed case analysis available here.
Note: The enumeration program is highly parallelizable and optimized for finding cliques up to symmetry in large graphs.