List of regular stars in 4D
Regular star polychora are analogous to regular star 3D shapes. Ludwig Schläfli and Edmund Hess figured out there are 10 in total. John Conway gave a name to each one.
We can create them by applying some operations to regular 4D shapes:
- Stellate: Replace pentagons with pentagrams which have them as core.
- Greaten: Replace dodecahedra with great dodecahedra, or icosahedra with great icosahedra, which have them as core.
- Aggrandize: Replace each cell with a larger one in the same realm (3D space).
- Facet: Connect vertices to create new faces or cells.
Small stellated 120-cell
Made by 120 Sissid's.
Icosahedral 120-cell (Faceted 600-cell)
Made by 120 Ike's. Similar to the Ex, but constructed by 120 intersecting Ike's instead of 600 Tet's. Its vertices, edges and faces are the same as those of the Ex.
Great 120-cell
Made by 120 Gad's.
Grand 120-cell
Made by 120 Doe's.
Great stellated 120-cell
Made by 120 Gissid's.
Grand stellated 120-cell
Made by 120 Sissid's.
Great grand 120-cell
Made by 120 Gad's.
Great icosahedral 120-cell (Great faceted 600-cell)
Made by 120 Gike's.
Grand 600-cell
Made by 600 Tet's.
Great grand stellated 120-cell
Made by 120 Gissid's. The only one with 600 vertices of the Hi and looks very pointy.
Entire list
| Cells | Schläfli symbol | Density | |
| Small stellated 120-cell | 120 Sissid's | 5/2,5,3 | 4 |
| Icosahedral 120-cell | 120 Ike's | 3,5,5/2 | |
| Great 120-cell | 120 Gad's | 5,5/2,5 | 6 |
| Great stellated 120-cell | 120 Gissid's | 5/2,3,5 | 20 |
| Grand 120-cell | 120 Doe's | 5,3,5/2 | |
| Grand stellated 120-cell | 120 Sissid's | 5/2,5,5/2 | 66 |
| Great icosahedral 120-cell | 120 Gike's | 3,5/2,5 | 76 |
| Great grand 120-cell | 120 Gad's | 5,5/2,3 | |
| Great grand stellated 120-cell | 120 Gissid's | 5/2,3,3 | 191 |
| Grand 600-cell | 600 Tet's | 3,3,5/2 |
Trivia
In 3D, all stellations of Doe are regular; these are 3 of 4 regular stars. However, in 4D, stellations of 120-cell are much more diverse. As shown above, these include 9 of 10 regular stars. In addition, there are more than 100 non-regulars; some of these are noble.
I expect that stellations of 600-cell are much more diverse.
Density is how many times the shape covers the center. For example, pentagram =(5/2) has density of 2. If a polyhedron or polytope has (5/2) as a face, it is counted twice. The higher the density, the more complex and intersecting the shape. Dual shapes have the same density. Great 120-cell and Grand stellated 120-cell are self-dual.
When expressed in Schläfli symbols, it is known that we can perform a "conjugate transformation," that is to leave 3 as it is and switch 5 between 5/2.