cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A053016 Numbers of vertices of Platonic solids in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron. Equally, numbers of faces of Platonic solids in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron.

Original entry on oeis.org

4, 6, 8, 12, 20
Offset: 1

Views

Author

Jeffrey Keller (jeff(AT)auctionflow.com), Feb 24 2000

Keywords

Comments

It appears that the stereographic projection of the Platonic solids requires respectively 4, 6, 8, 6, 10, different colors to represent them. - Eric Desbiaux, Feb 15 2009

References

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

Crossrefs

Extensions

Definition expanded by N. J. A. Sloane, Nov 06 2020

A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

4, 8, 6, 20, 12
Offset: 1

Views

Author

Henry Bottomley, Aug 14 2001

Keywords

Comments

The preferred order for these five numbers is 4, 6, 8, 12, 20 (tetrahedron, octahedron, cube, icosahedron, dodecahedron), as in A053016. - N. J. A. Sloane, Nov 05 2020
Also number of faces of Platonic solids ordered by increasing ratios of volumes to their respective circumscribed spheres. See cross-references for actual ratios. - Rick L. Shepherd, Oct 04 2009
Also the expected lengths of nontrivial random walks along the edges of a Platonic solid from one vertex back to itself. - Jens Voß, Jan 02 2014

Examples

			a(2) = 8 since a cube has eight vertices.
		

Crossrefs

Cf. A165922 (tetrahedron), A049541 (octahedron), A165952 (cube), A165954 (icosahedron), A165953 (dodecahedron). - Rick L. Shepherd, Oct 04 2009
Cf. A234974. - Jens Voß, Jan 02 2014

Formula

a(n) = A063722(n) - A053016(n) + 2.

A338622 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, formed when the five Platonic solids, in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron, are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

1, 8, 72, 24, 2160, 360, 205320, 208680, 94800, 34200, 7920, 1560, 120
Offset: 1

Views

Author

Scott R. Shannon, Nov 04 2020

Keywords

Comments

See A338571 for further details and images of this sequence.
The author thanks Zach J. Shannon for producing the images for this sequence.

Examples

			The cube is cut with 14 internal planes defined by all 3-vertex combinations of its 8 vertices. This leads to the creation of 72 4-faced polyhedra and 24 5-faced polyhedra, 96 pieces in all. See A338571 and A333539.
The table is:
1;
8;
72, 24;
2160, 360;
205320, 208680, 94800, 34200, 7920, 1560, 120;
		

Crossrefs

Cf. A338571 (total number of polyhedra), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427, A333543.

Formula

Sum of row n = A338571(n).

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

Original entry on oeis.org

5, 8, 16, 24, 120, 600
Offset: 1

Views

Author

Henry Bottomley, Aug 15 2001

Keywords

Comments

Sorted by number of 3-dimensional cells.
Also, number of vertices of the regular 4-dimensional polyhedra. - Douglas Boffey, Aug 12 2012

Examples

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

References

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

Crossrefs

Formula

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

Extensions

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

Views

Author

Henry Bottomley, Aug 15 2001

Keywords

Comments

Sorted by number of 3-dimensional faces.

Examples

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

References

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

Crossrefs

Formula

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

A338571 Number of polyhedra formed when the five Platonic solids, in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron, are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

1, 8, 96, 2520, 552600
Offset: 1

Views

Author

Scott R. Shannon, Nov 03 2020

Keywords

Comments

For a Platonic solid create all possible planes defined by connecting any three of its vertices. For example, in the case of a cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for the Platonic solids, ordered by number of vertices: tetrahedron, octahedron, cube, icosahedron, dodecahedron.
See A338622 for the number and images of the k-faced polyhedra in each dissection for each of the five solids.
The author thanks Zach J. Shannon for producing the images for this sequence.

Examples

			a(1) = 1. The tetrahedron has no internal cutting planes so the single polyhedron after cutting is the tetrahedron itself.
a(2) = 8. The octahedron has 3 internal cutting planes resulting in 8 polyhedra.
a(3) = 96. The cube has 14 internal cutting planes resulting in 96 polyhedra. See also A333539.
a(4) = 2520. The icosahedron has 47 cutting planes resulting in 2520 polyhedra.
See A338622 for a breakdown of the above totals into the corresponding number of k-faced polyhedra.
a(5) = 552600. The dodecahedron has 307 internal cutting planes resulting in 552600 polyhedra. It is the only Platonic solid which produces polyhedra with 6 or more faces.
		

Crossrefs

Cf. A338622 (number of k-faced polyhedra in each dissection), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427.

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

Views

Author

Henry Bottomley, Aug 15 2001

Keywords

Comments

Sorted by number of 2-dimensional faces.
Also the number of edges in the regular 4-dimensional polytopes [Douglas Boffey, Aug 12 2012]

Examples

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

References

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

Crossrefs

Formula

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

A063926 Number of edges in the six regular 4-dimensional polytopes.

Original entry on oeis.org

10, 32, 24, 96, 1200, 720
Offset: 1

Views

Author

Henry Bottomley, Aug 15 2001

Keywords

Comments

Sorted by number of 3-dimensional faces.

Examples

			a(2) = 32 since a 4D hypercube contains thirty-two edges.
		

References

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

Crossrefs

Formula

a(n) = A063927(n) + A063925(n) - A063924(n).

A198861 The number of ways to paint the faces of the five Platonic solids using exactly n colors where n is the number of faces of each solid.

Original entry on oeis.org

2, 30, 1680, 7983360, 40548366802944000
Offset: 1

Views

Author

Geoffrey Critzer, Oct 30 2011

Keywords

Comments

Let G, the group of rotations in 3 dimensional space act on the set of n! paintings of each Platonic solid having n faces. There are n! fixed points in the action table since the only element in G that leaves a painting fixed is the identity element. The order of G is A098427/2. So by Burnside's Lemma a(n)=n!/|G|.

Crossrefs

Cf. A053016 (number of faces), A063722 (number of edges).

Programs

  • PARI
    lista() = {ve = [6, 12, 12, 30, 30 ]; vf = [4, 6, 8, 12, 20 ]; for (i=1, 5, nb = vf[i]!/(2*ve[i]); print1(nb, ", "););} \\ Michel Marcus, Aug 25 2014

Formula

a(n) = A053016(n)!/(2*A063722(n)) (see link). - Michel Marcus, Aug 24 2014

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

Views

Author

Marco Ripà, Dec 20 2022

Keywords

Comments

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.

Examples

			6 is a term since a tetrahedron has 6 edges.
		

Crossrefs

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

Formula

{a(n), n >= 1} = {{30, 96, 720} U {A000217} U {A001787} U {A046092}} \ {0, 1, 3, 4}.
Showing 1-10 of 12 results. Next