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Regular polychoron numbers

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There are six regular convex polychora (4-dimensional hyper-solids,) which, except for the 24-cell, are the analogues of the Platonic solids. Listed by increasing number of vertices, the six regular convex polychora are:

  • the 5 vertices (self dual) 5-cell (pentachoron) or 4-simplex (a hyper-tetrahedron,)
  • the 8 vertices 16-cell or 4-cross polytope or 4-orthoplex (a hyper-octahedron,)
  • the 16 vertices 8-cell or 4-cube or 4-orthotope or tesseract (a hyper-cube,)
  • the 24 vertices (self dual) 24-cell, which has no perfect analogy in higher or lower dimensional spaces,
  • the 120 vertices 600-cell (a hyper-icosahedron,)
  • and the 600 vertices 120-cell (a hyper-dodecahedron.)

The 5-cell and 24-cell are self-dual, the 16-cell is the dual of the 8-cell, and the 600- and 120-cells are dual to each other.

The number of cells, faces, edges and vertices, for each of the six regular convex polychora are give the sequences:

  • A063924 Number of cells (3-dimensional elements) in the regular 4-dimensional polytopes.
  • A063925 Number of faces (2-dimensional elements) in the regular 4-dimensional polytopes.
  • A063926 Number of edges (1-dimensional elements) in the six regular 4-dimensional polytopes.
  • A063927 Number of vertices (0-dimensional elements) in the regular 4-dimensional polytopes.

The polychoron numbers are the numbers of dots in a layered geometric arrangement into one of the 6 regular convex polychoron hyper-solids.[1] The polychoron numbers start with one initial dot (n = 1,) then with one dot at each vertex of a given polychoron hyper-solid (n = 2,) with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots are counted only once.

The 6 types of regular convex polychoron numbers are:

  • A000332(n+3) Pentachoron numbers, binomial(n+3,4).
  • A000583 The tesseract numbers, fourth powers: n^4.
  • A092182 Figurate numbers based on the 600-cell (4-D polytope with Schlaefli symbol {3,3,5}).
  • A092183 Figurate numbers based on the 120-cell (4-D polytope with Schlaefli symbol {5,3,3}).
  • A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).


All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The nth 4-dimensional N3-cell regular polytope (having N0 vertices) number is given by the formula:[2]

For the 5-cell numbers:

For the 16-cell and 8-cell numbers:

For the 24-cell numbers:

For the 600-cell and 120-cell numbers:

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[3]

For 4-dimensional (d = 4) regular convex polytopes:

where N0 is the number of 0-dimensional elements (vertices V), N1 is the number of 1-dimensional elements (edges V), N2 is the number of 2-dimensional elements (faces F) and N3 is the number of 3-dimensional elements (cells C) of the regular convex polytope.

Recurrence equation

The recurrence equation seems to be the same for all polychoron numbers (TO BE VERIFIED for 5-cell, 16-cell and 8-cell numbers):

with initial conditions

For the 5-cell numbers:

For the 16-cell and 8-cell numbers:

For the 24-cell numbers:

For the 600-cell and 120-cell numbers:

Ordinary generating function

For the 5-cell numbers:

For the 16-cell and 8-cell numbers:

For the 24-cell numbers:

For the 600-cell and 120-cell numbers:

Procedure to obtain the generating functions

where (Cf. Generating function for regular orthotopic numbers)

Exponential generating function

For the 5-cell numbers:

For the 16-cell and 8-cell numbers:

For the 24-cell numbers:

For the 600-cell and 120-cell numbers:

Dirichlet generating function

For the 5-cell numbers:

For the 16-cell and 8-cell numbers:

[4]

For the 24-cell numbers:

For the 600-cell and 120-cell numbers:

Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[5] Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)

A nonempty subset of nonnegative integers is called a basis of order if is the minimum number with the property that every nonnegative integer can be written as a sum of elements in . Lagrange’s sum of four squares can be restated as the set of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every , the set of k-gon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of k-gon numbers.

We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number such that every nonnegative integer is a sum of th powers, i.e. the set of th powers forms a basis of order . The Hilbert-Waring problem is concerned with the study of for . This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

Differences

Partial sums

The partial sums correspond to 5-dimensional polychoron hyperpyramidal numbers.

Partial sums of reciprocals

Sum of reciprocals

Table of formulae and values

N0, N1, N2 and N3 are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N0 of vertices.

Regular polychoron numbers formulae and values
Rank


r

N0 Name

(N0, N1, N2, N3)

Schläfli symbol[6]

Formulae


Generating

function


n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

0 5 Pentachoron

5 cell

(5, 10, 10, 5)

{3, 3, 3}


[7]


0 1 5 15 35 70 126 210 330 495 715 1001 1365 A000332(n+3)
1 8 16 cell

(8, 24, 32, 16)

{3, 3, 4}



0 1 8 33 96 225 456 833 1408 2241 3400 4961 7008 A014820(n-1)
2 16 Tesseract

8 cell

(16, 32, 24, 8)

{4, 3, 3}


0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 A000583
3 24 24 cell

(24, 96, 96, 24)

{3, 4, 3}


0 1 24 153 544 1425 3096 5929 10368 16929 26200 38841 55584 A092181
4 120 600 cell

(120, 720, 1200, 600)

{3, 3, 5}


0 1 120 947 3652 9985 22276 43435 76952 126897 197920 295251 424700 A092182
5 600 120 cell

(600, 1200, 720, 120)

{5, 3, 3}


0 1 600 4983 19468 53505 119676 233695 414408 683793 1066960 1592151 2290740 A092183


Table of related formulae and values

N0, N1, N2 and N3 are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N0 of vertices.

Regular polychoron numbers related formulae and values
Rank


r

N0 Name

(N0, N1, N2, N3)

Schläfli symbol[6]

Order

of basis


Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals[8][9]

0 5


Pentachoron

5 cell

(5, 10, 10, 5)

{3, 3, 3}


[10]


Tetrahedral numbers


(A000292(n))


[7]


[11]



A000389

1 8


16 cell

(8, 24, 32, 16)

{3, 3, 4}


[10]


Centered octahedral numbers


(A001845(n-1))




A061927(n-1)?

2 16


Tesseract

8 cell

(16, 32, 24, 8)

{4, 3, 3}


[12]


Centered cubic(n) +
6 Square pyramidal(n-1)


Rhombic dodecahedral numbers [13]


(A005917(n-1))




A000538

[4]
3 24



24 cell

(24, 96, 96, 24)

{3, 4, 3}


[10]


4 120



600 cell

(120, 720, 1200, 600)

{3, 3, 5}


[10]


5 600


120 cell

(600, 1200, 720, 120)

{5, 3, 3}


[10]



Table of sequences

Regular polychoron numbers sequences
N0 sequences
5 {0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, ...}
8 {0, 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, 7008, 9633, 12936, 17025, 22016, 28033, 35208, 43681, 53600, 65121, 78408, 93633, 110976, 130625, 152776, 177633, ...}
16 {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, ...}
24 {0, 1, 24, 153, 544, 1425, 3096, 5929, 10368, 16929, 26200, 38841, 55584, 77233, 104664, 138825, 180736, 231489, 292248, 364249, 448800, 547281, 661144, 791913, 941184, ...}
120 {0, 1, 120, 947, 3652, 9985, 22276, 43435, 76952, 126897, 197920, 295251, 424700, 592657, 806092, 1072555, 1400176, 1797665, 2274312, 2839987, 3505140, 4280801, ...}
600 {0, 1, 600, 4983, 19468, 53505, 119676, 233695, 414408, 683793, 1066960, 1592151, 2290740, 3197233, 4349268, 5787615, 7556176, 9701985, 12275208, 15329143, 18920220, ...}


See also

Centered regular polychoron numbers

Notes

  1. Weisstein, Eric W., Regular Polychoron, From MathWorld--A Wolfram Web Resource.
  2. Where is the d-dimensional regular convex polytope number with N0 0-dimensional elements (vertices V.)
  3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  4. 4.0 4.1 Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  6. 6.0 6.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  7. 7.0 7.1 Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
  8. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  9. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
  10. 10.0 10.1 10.2 10.3 10.4 HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
  11. Weisstein, Eric W., Multichoose, From MathWorld--A Wolfram Web Resource.
  12. Where , k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (k+2)-gonal base (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the polygonal base (hyper)pyramid.
  13. Weisstein, Eric W., Rhombic Dodecahedral Number, From MathWorld--A Wolfram Web Resource.

External links