A306602 Number of elements in the ten nonconvex regular 4-polytopes (regular 4-dimensional star-polytopes), as a four-column array, read by rows, with rows ordered first by increasing density, then by increasing cell-count, then by increasing face-count, then by increasing edge-count and then by increasing vertex-count.

120, 720, 1200, 120, 120, 1200, 720, 120, 120, 720, 720, 120, 120, 720, 720, 120, 120, 720, 720, 120, 120, 720, 720, 120, 120, 720, 1200, 120, 120, 1200, 720, 120, 120, 720, 1200, 600, 600, 1200, 720, 120

Offset: 1

Author

Felix Fröhlich, Feb 27 2019

Keywords

• nonn
• tabf
• fini
• full

Comments

Only included for completeness (see other sequences in crossrefs). This is one of those entries that I think should be included since it is such an elementary sequence in geometry, but where the terms do otherwise not really make a very spectacular sequence.

The ordering of the polytopes in the enumeration of the element counts is somewhat arbitrary, though based on invariants inherently associated with these polytopes. For example, ordering first by vertex-count, then by edge-count, then by face-count and then by cell-count might have been as good a choice as the one used for the sequence.

Examples

			Polytope            | Schläfli symbol | Density | Cells | Faces | Edges | Vertices
----------------------------------------------------------------------------------
Sm st 120-cell      | {5/2,5,3}       |       4 |   120 |   720 |  1200 |      120
Ico 120-cell        | {3,5,5/2}       |       4 |   120 |  1200 |   720 |      120
Grt 120-cell        | {5,5/2,5}       |       6 |   120 |   720 |   720 |      120
Grd 120-cell        | {5,3,5/2}       |      20 |   120 |   720 |   720 |      120
Grt st 120-cell     | {5/2,3,5}       |      20 |   120 |   720 |   720 |      120
Grd st 120-cell     | {5/2,5,5/2}     |      66 |   120 |   720 |   720 |      120
Grt grd 120-cell    | {5,5/2,3}       |      76 |   120 |   720 |  1200 |      120
Grt ico 120-cell    | {3,5/2,5}       |      76 |   120 |  1200 |   720 |      120
Grt grd st 120-cell | {5/2,3,3}       |     191 |   120 |   720 |  1200 |      600
Grd 600-cell        | {3,3,5/2}       |     191 |   600 |  1200 |   720 |      120
======================
| Keys               |
======================
| Sm   | Small       |
| St   | Stellated   |
| Grt  | Great       |
| Grd  | Grand       |
| Ico  | Icosahedral |
======================
		

Links

• Wikipedia, Regular 4-polytope - Regular star (Schläfli-Hess) 4-polytopes

Crossrefs

Cf. A048789, A063924, A063925, A063926, A063927.