A063926 -id:A063926 - cp's OEIS frontend

A063924 Number of 3-dimensional cells in the regular 4-dimensional polytopes.

5, 8, 16, 24, 120, 600

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 3-dimensional cells.

Also, number of vertices of the regular 4-dimensional polyhedra. - Douglas Boffey, Aug 12 2012

			a(2) = 8 since a 4D hypercube contains eight cubes.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A053016, A060296, A063722, A063723.

a(n) = A063925(n) - A063926(n) + A063927(n).

Corrected faces to cells by Douglas Boffey, Aug 12 2012

A063927 Number of vertices in the regular 4-dimensional polytopes.

Original entry on oeis.org

5, 16, 8, 24, 600, 120

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 3-dimensional faces.

			a(2) = 16 since a 4D hypercube contains sixteen vertices.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A060296, A053016, A063722, A063723.

a(n) = A063926(n) - A063925(n) + A063924(n).

A063925 Number of 2-dimensional faces in the regular 4-dimensional polytopes.

Original entry on oeis.org

10, 24, 32, 96, 720, 1200

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 2-dimensional faces.

Also the number of edges in the regular 4-dimensional polytopes [Douglas Boffey, Aug 12 2012]

			a(2) = 24 since a 4D hypercube contains twenty-four faces.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A053016, A063722, A063723.

a(n) = A063924(n)+A063926(n)-A063927(n).

A359201 Number of edges of regular m-polytopes for m >= 3.

Original entry on oeis.org

6, 10, 12, 15, 21, 24, 28, 30, 32, 36, 40, 45, 55, 60, 66, 78, 80, 84, 91, 96, 105, 112, 120, 136, 144, 153, 171, 180, 190, 192, 210, 220, 231, 253, 264, 276, 300, 312, 325, 351, 364, 378, 406, 420, 435, 448, 465, 480, 496, 528, 544, 561, 595, 612, 630, 666

Offset: 1

Marco Ripà, Dec 20 2022

In 3 dimensions there are five (convex) regular polytopes and they have 6, 12, or 30 edges (A063722).
In 4 dimensions there are six regular 4-polytopes and they have 10, 24, 32, 96, 720, or 1200 edges (A063926).
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 edges in ascending order and define the present sequence.

			6 is a term since a tetrahedron has 6 edges.

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. A000217, A001787, A046092, A063722, A063926.

Cf. A359202 (faces), A359662 (cells).

{a(n), n >= 1} = {{30, 96, 720} U {A000217} U {A001787} U {A046092}} \ {0, 1, 3, 4}.

A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

Original entry on oeis.org

1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742

Offset: 0

Jonathan Vos Post, Jul 15 2008

Andrew Weimholt suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.

			a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.

Georg Fischer, Table of n, a(n) for n = 0..1000
Gil Kalai, Five Open Problems Regarding Convex Polytopes.
Eric W. Weisstein, Polytope
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000943, A000944, A019503, A053016, A060296, A063924, A063925, A063926, A063927, A065984, A086653, A093478, A093479, A105230, A105231.

LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)

For n > 4, a(n) = A086653(n) + 1 = 2^n + 3*n + 1.

G.f.: -(1488*x^7 - 3656*x^6 + 2794*x^5 - 569*x^4 - 58*x^3 + 3*x^2 + 2*x - 1)/((1-x)^2*(1-2*x)). [Colin Barker, Sep 05 2012]

a(14)-a(15) corrected by Georg Fischer, May 02 2019

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.

Original entry on oeis.org

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

Felix Fröhlich, Feb 27 2019

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.

			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 |
======================

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

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

cp's OEIS Frontend

A063924 Number of 3-dimensional cells in the regular 4-dimensional polytopes.

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A063927 Number of vertices in the regular 4-dimensional polytopes.

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A063925 Number of 2-dimensional faces in the regular 4-dimensional polytopes.

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A359201 Number of edges of regular m-polytopes for m >= 3.

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A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

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