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Simplicial polytopic numbers
The simplicial polytopic numbers are a family of sequences of figurate numbers corresponding to the d-dimensional simplex for each dimension d, where d is a nonnegative integer.
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 Minimal nondegenerate polytopes in a d-dimensional Euclidean space, d ≥ 0
- 2 Formulae
- 3 Recurrence relation
- 4 Generating function
- 5 Simplicial polytopic numbers and Pascal's (rectangular) triangle columns
- 6 Order of basis
- 7 Differences
- 8 Partial sums
- 9 Partial sums of reciprocals
- 10 Sum of reciprocals
- 11 Number of j-dimensional "vertices"
- 12 Table of formulae and values
- 13 Table of related formulae and values
- 14 Table of sequences
- 15 See also
- 16 Notes
- 17 External links
Minimal nondegenerate polytopes in a d-dimensional Euclidean space, d ≥ 0
In a d-dimensional Euclidean space , d ≥ 0, the minimal number of vertices d + 1 gives the simplest d-polytope (the d-simplex,) i.e.:
- d = 0: the 0-simplex (having 1 vertex) is the point (the 1 (-1)-cell, with 1 null polytope as facet)
- d = 1: the 1-simplex (having 2 vertices) is the triangular gnomon (the 2 0-cell, with 2 points as facets)
- d = 2: the 2-simplex (having 3 vertices) is the trigon (triangle) (the 3 1-cell, with 3 segments as facets)
- d = 3: the 3-simplex (having 4 vertices) is the tetrahedron (the 4 2-cell, with 4 faces as facets)
- d = 4: the 4-simplex (having 5 vertices) is the pentachoron (the 5 3-cell, with 5 rooms as facets)
- d = 5: the 5-simplex (having 6 vertices) is the hexateron (the 6 4-cell, with 6 4-cells as facets)
- d = 6: the 6-simplex (having 7 vertices) is the heptapeton (the 7 5-cell, with 7 5-cells as facets)
- d = 7: the 7-simplex (having 8 vertices) is the octahexon (the 8 6-cell, with 8 6-cells as facets)
- d = 8: the 8-simplex (having 9 vertices) is the enneahepton (the 9 7-cell, with 9 7-cells as facets)
- ...
- d = d: the d-simplex (having d+1 vertices) is the d+1 (d-1)-cell, with d+1 (d-1)-cells as facets
Formulae
The nth simplicial d-polytopic numbers are given by the formulae [1] [2][3]:
- , or
where d ≥ 0 is the dimension and n-1 ≥ 0 is the number of nondegenerate layered simplices (n-1 = 0 giving a single dot, a degenerate simplex) of the d-dimensional regular convex simplicial polytope number (d-simplex number.)
Recurrence relation
Generating function
Simplicial polytopic numbers and Pascal's (rectangular) triangle columns
n = 0 | 1 | ||||||||||||
1 | 1 | 1 | |||||||||||
2 | 1 | 2 | 1 | ||||||||||
3 | 1 | 3 | 3 | 1 | |||||||||
4 | 1 | 4 | 6 | 4 | 1 | ||||||||
5 | 1 | 5 | 10 | 10 | 5 | 1 | |||||||
6 | 1 | 6 | 15 | 20 | 15 | 6 | 1 | ||||||
7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | |||||
8 | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | ||||
9 | 1 | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | |||
10 | 1 | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | ||
11 | 1 | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | |
12 | 1 | 12 | 66 | 220 | 495 | 792 | 924 | 792 | 495 | 220 | 66 | 12 | 1 |
d = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
d = 0 | 0-simplicial numbers | Point numbers | Form point | (1 (-1)-cells "faces") | (0-simplex) |
d = 1 | 1-simplicial numbers | Linear numbers | Form segments | (2 0-cells "faces") | (1-simplex) |
d = 2 | 2-simplicial numbers | Triangular numbers | Form triangles | (3 1-cells "faces") | (2-simplex) |
d = 3 | 3-simplicial numbers | Tetrahedral numbers | Form tetrahedrons | (4 2-cells "faces") | (3-simplex) |
d = 4 | 4-simplicial numbers | Pentachoron numbers | Form pentachorons | (5 3-cells "faces") | (4-simplex) |
d = 5 | 5-simplicial numbers | Hexateron numbers | Form hexaterons | (6 4-cells "faces") | (5-simplex) |
d = 6 | 6-simplicial numbers | Heptapeton numbers | Form heptapetons | (7 5-cells "faces") | (6-simplex) |
d = 7 | 7-simplicial numbers | Octahexon numbers | Form octahexons | (8 6-cells "faces") | (7-simplex) |
d = 8 | 8-simplicial numbers | Nonahepton numbers | Form nonaheptons | (9 7-cells "faces") | (8-simplex) |
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
Partial sums of reciprocals
Sum of reciprocals
The sum of reciprocals can be interpreted as , where is the probability that does not divide a random integer or the probability that two random integers and have different residues modulo .
The reciprocal of the sum of reciprocals thus gives:
Graphically, the second interpretation is the probability that a 2-D integer lattice point does not lay on any straight line of the family , for integer values of . This family of functions consists of the bisection of the first and third quadrants and all its parallels at distances . When this includes all lattice points, hence .
Number of j-dimensional "vertices"
Table of formulae and values
N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The simplicial polytopic numbers are listed by increasing number N0 of vertices.
d | Name
d-simplex d+1 (d-1)-cell (N0, N1, N2, ...) Schläfli symbol[6] |
Formulae
|
n = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | OEIS
number |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | Point
0-simplex hena-(-1)-cell () {} |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | (NOT A057427) [7] | |
1 | Triangular gnomon
1-simplex di-0-cell (2) {} |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | A000027(n) | |
2 | Triangular
2-simplex tri-1-cell (3, 3) {3} |
0 | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | A000217(n) | |
3 | Tetrahedral
3-simplex tetra-2-cell (4, 6, 4) {3, 3} |
0 | 1 | 4 | 10 | 20 | 35 | 56 | 84 | 120 | 165 | 220 | 286 | 364 | A000292(n) | |
4 | Pentachoron
4-simplex penta-3-cell (5, 10, 10, 5) {3, 3, 3} |
0 | 1 | 5 | 15 | 35 | 70 | 126 | 210 | 330 | 495 | 715 | 1001 | 1365 | A000332(n+3) | |
5 | Hexateron
5-simplex hexa-4-cell (6, 15, 20, 15, 6) {3, 3, 3, 3} |
0 | 1 | 6 | 21 | 56 | 126 | 252 | 462 | 792 | 1287 | 2002 | 3003 | 4368 | A000389(n+4) | |
6 | Heptapeton
6-simplex hepta-5-cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} |
0 | 1 | 7 | 28 | 84 | 210 | 462 | 924 | 1716 | 3003 | 5005 | 8008 | 12376 | A000579(n+5) | |
7 | Octahexon
7-simplex octa-6-cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} |
0 | 1 | 8 | 36 | 120 | 330 | 792 | 1716 | 3432 | 6435 | 11440 | 19448 | 31824 | A000580(n+6) | |
8 | Enneahepton
8-simplex nona-7-cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} |
0 | 1 | 9 | 45 | 165 | 495 | 1287 | 3003 | 6435 | 12870 | 24310 | 43758 | 75582 | A000581(n+7) | |
9 | Decaocton
9-simplex deca-8-cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} |
0 | 1 | 10 | 55 | 220 | 715 | 2002 | 5005 | 11440 | 24310 | 48620 | 92378 | 167960 | A000582(n+8) | |
10 | Hendecaenneon
10-simplex hendeca-9-cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} |
0 | 1 | 11 | 66 | 286 | 1001 | 3003 | 8008 | 19448 | 43758 | 92378 | 184756 | 352716 | A001287(n+9) | |
11 | Dodecadecon
11-simplex dodeca-10-cell (12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
0 | 1 | 12 | 78 | 364 | 1365 | 4368 | 12376 | 31824 | 75582 | 167960 | 352716 | 705432 | A001288(n+10) | |
12 | Tridecahendecon
12-simplex trideca-11-cell (13, ... Pascal's triangle 13th row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
0 | 1 | 13 | 91 | 455 | 1820 | 6188 | 18564 | 50388 | 125970 | 293930 | 646646 | 1352078 | A010965(n+11) |
N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The simplicial polytopic numbers are listed by increasing number N0 of vertices.
d | Name
d-simplex d+1 (d-1)-cell (N0, N1, N2, ...) Schläfli symbol[6] |
Generating
function
|
Order
of basis
|
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of reciprocals[11]
|
---|---|---|---|---|---|---|---|
1 | Triangular gnomon
1-simplex bi-0-cell (2) {} |
|
[12] [1] | ||||
2 | Triangular
2-simplex tri-1-cell (3, 3) {3} |
|
[2] =
|
||||
3 | Tetrahedral
3-simplex tetra-2-cell (4, 6, 4) {3, 3} |
|
[3] =
|
[4] | |||
4 | Pentachoron
4-simplex penta-3-cell (5, 10, 10, 5) {3, 3, 3} |
|
[5] =
|
||||
5 | Hexateron
5-simplex hexa-4-cell (6, 15, 20, 15, 6) {3, 3, 3, 3} |
|
[6] | ||||
6 | Heptapeton
6-simplex hepta-5-cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} |
|
[7] | ||||
7 | Octahexon
7-simplex octa-6-cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} |
|
[8] | ||||
8 | Enneahepton
8-simplex nona-7-cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} |
[9] | |||||
9 | Decaocton
9-simplex deca-8-cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} |
[10] | |||||
10 | Hendecaenneon
10-simplex hendeca-9-cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} |
[11] | |||||
11 | Dodecadecon
11-simplex dodeca-10-cell (12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
[12] | |||||
12 | Tridecahendecon
12-simplex trideca-11-cell (13, ... Pascal's triangle 13th row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
[13] |
Table of sequences
d | sequences |
---|---|
1 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ...} |
2 | {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, ...} |
3 | {0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, ...} |
4 | {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, ...} |
5 | {0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, ...} |
6 | {0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, ...} |
7 | {0, 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, ...} |
8 | {0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, ...} |
9 | {0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, ...} |
10 | {0, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, ... } |
11 | {0, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, ...} |
12 | {0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, ...} |
See also
Centered simplicial polytopic numbers
Notes
- ↑ Where is the d-dimensional regular convex polytope number with N0 0-dimensional elements (vertices V.)
- ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Multichoose, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Simplex, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ 6.0 6.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
- ↑ A057427 is the sign function (-1 for n < 0, 0 for n = 0, +1 for n > 0,) while what we get here is the characteristic function of positive integers (0 for n ≤ 0, +1 for n ≥ 1.)
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
- ↑ Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
- ↑ Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.
External links
- S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
- S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
- Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.