A359201 -id:A359201 - cp's OEIS frontend

A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

4, 6, 8, 10, 12, 20, 24, 32, 35, 56, 80, 84, 96, 120, 160, 165, 220, 240, 280, 286, 364, 448, 455, 560, 672, 680, 720, 816, 960, 969, 1140, 1200, 1320, 1330, 1540, 1760, 1771, 1792, 2024, 2288, 2300, 2600, 2912, 2925, 3276, 3640, 3654, 4060, 4480, 4495, 4608

Offset: 1

Marco Ripà, Dec 20 2022

In 3 dimensions there are five (convex) regular polytopes and they have 4, 6, 8, 12, or 20 (bidimensional) faces (A053016).
In 4 dimensions there are six regular 4-polytopes and they have 10, 24, 32, 96, 720, or 1200 faces (A063925).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of bidimensional faces in ascending order and define the present sequence.

			6 is a term since a cube has 6 faces.

Mathematics StackExchange, What are the formulas for the number of vertices, edges, faces, cells, 4-faces, ..., n-faces, of convex regular polytopes in n≥5 dimensions?
Wikipedia, List of regular polytopes and compounds

Cf. A000292, A001788, A053016, A063925, A130809.

Cf. A359201 (edges), A359662 (cells).

{a(n), n >= 1} = {{12, 96, 720, 1200} U {A000292} U {A001788} U {A130809}} \ {0, 1}.

A359662 Number of (3-dimensional) cells of regular m-polytopes for m >= 3.

Original entry on oeis.org

1, 5, 8, 15, 16, 24, 35, 40, 70, 80, 120, 126, 160, 210, 240, 330, 495, 560, 600, 715, 1001, 1120, 1365, 1792, 1820, 2016, 2380, 3060, 3360, 3876, 4845, 5280, 5376, 5985, 7315, 7920, 8855, 10626, 11440, 12650, 14950, 15360, 16016, 17550, 20475, 21840, 23751

Offset: 1

Marco Ripà, Jan 10 2023

In 3 dimensions there are five (convex) regular polytopes and each of them (trivially) consists of a single cell.
In 4 dimensions there are six regular 4-polytopes and they have 5, 8, 16, 24, 120, 600 3-dimensional cells (A063924).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of (3-dimensional) cells in ascending order and define the present sequence.

			8 is a term since the hypersurface of a tesseract consists of 8 (cubical) cells.

MathStackExchange, MathStackExchange discussion concerning the number of edges of convex regular n-polytopes
Wikipedia, List of regular polytopes and compounds

Cf. A000332, A001789, A063924, A130810.

Cf. A359201 (edges), A359202 (faces).

Equals {{24, 120, 600} U {A000332} U {A001789} U {A130810}} \ {0}.

cp's OEIS Frontend

A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

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