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Platonic numbers
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There are five regular convex polyhedra (3dimensional regular convex solids,) know as the 5 Platonic solids:
 the tetrahedron (4 vertices, 6 edges and 4 faces)
 the octahedron (6 vertices, 12 edges and 8 faces)
 the cube or hexahedron (8 vertices, 12 edges and 6 faces)
 the icosahedron (12 vertices, 30 edges and 20 faces)
 the dodecahedron (20 vertices, 30 edges and 12 faces)
The tetrahedron is selfdual, the cube and the octahedron are duals, and the dodecahedron and icosahedron are duals. (Dual pairs have same number of edges and have vertices corresponding to faces of each other.)
Number of vertices, edges and faces of the 5 Platonic solids:
 A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
 A063722 Number of edges in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
 A053016 Number of faces of Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).
The Platonic numbers (Cf. A053012) are the numbers of dots in a layered geometric arrangement into one of the 5 Platonic solids.^{[1]} The platonic numbers start with one initial dot (for n = 1,) then with one dot at each vertex of a given Platonic 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 (the dot at the initial vertex and the dots on all the edges sharing that initial vertex) are counted only once.
The 5 types of Platonic numbers (by increasing number of vertices) are:
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Formulae
The n^{th} Platonic N_{2}hedral number (having N_{0} vertices) is given by the formulae:^{[2]}
 for the (self dual) tetrahedral (N_{0} = 4) numbers;
 for the (dual pair) octahedral (N_{0} = 6) and hexahedral (cubic) (N_{0} = 8) numbers;
 for the (dual pair) icosahedral (N_{0} = 12) and dodecahedral (N_{0} = 20) numbers.
where N_{0} is the number of 0dimensional elements (vertices V,) N_{1} is the number of 1dimensional elements (edges E,) N_{2} is the number of 2dimensional elements (faces F) of the polyhedron.
is the number of vertices of the Platonic solid, where is the rank of the Platonic solid (by increasing number of vertices.)
DescartesEuler (convex) polyhedral formula
DescartesEuler (convex) polyhedral formula:^{[3]}
where N_{0} is the number of 0dimensional elements (vertices V,) N_{1} is the number of 1dimensional elements (edges E,) N_{2} is the number of 2dimensional elements (faces F) of the polyhedron.
Recurrence relation
with initial conditions
Ordinary generating function
for the (self dual) tetrahedral (rank r = 0, N_{0} = 4 vertices) numbers;
for the (dual pair) octahedral (rank r = 1, N_{0} = 6 vertices) and hexahedral (cubic) (rank r = 2, N_{0} = 8 vertices) numbers;
for the (dual pair) icosahedral (rank r = 3, N_{0} = 12 vertices) and dodecahedral (rank r = 4, N_{0} = 20 vertices) numbers.
is the number of vertices of the Platonic solid, where is the rank of the Platonic solid (by increasing number of vertices.)
Exponential generating function
For the 4face numbers:
For the 8face and 6face numbers:
For the 20face and 12face numbers:
Dirichlet generating function
For the 4face numbers:
For the 8face and 6face numbers:
For the 20face and 12face 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 kpolygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.^{[4]} 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 kgon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the HilbertWaring 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 kgon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of kgon 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 HilbertWaring 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 4dimensional Platonic hyperpyramidal numbers.
Partial sums of reciprocals
Sum of reciprocals
The sum of reciprocals can be interpreted as , where is the probability that three random integers , and are coprime.^{[5]}
Table of formulae and values
N_{0}, N_{1} and N_{2} are the number of vertices (0dimensional), edges (1dimensional) and faces (2dimensional) respectively, where the faces are the actual facets. The regular Platonic numbers are listed by increasing number N_{0} of vertices.
Rank
 N_{0}
 Name
(N_{0}, N_{1}, N_{2}) Schläfli symbol^{[6]}  Formulae
 Generating
function
 n = 0  1  2  3  4  5  6  7  8  9  10  11  12  OEIS
number 

0  4  Tetrahedral
(4, 6, 4) {3, 3} 

 0  1  4  10  20  35  56  84  120  165  220  286  364  A000292 
1  6  Octahedral
(6, 12, 8) {3, 4} 

 0  1  6  19  44  85  146  231  344  489  670  891  1156  A005900 
2  8  Cube
(8, 12, 6) {4, 3} 
 0  1  8  27  64  125  216  343  512  729  1000  1331  1728  A000578  
3  12  Icosahedral
(12, 30, 20) {3, 5} 
 0  1  12  48  124  255  456  742  1128  1629  2260  3036  3972  A006564  
4  20  Dodecahedral
(20, 30, 12) {5, 3} 
 0  1  20  84  220  455  816  1330  2024  2925  4060  5456  7140  A006566 
Table of related formulae and values
N_{0}, N_{1} and N_{2} are the number of vertices (0dimensional), edges (1dimensional) and faces (2dimensional) respectively, where the faces are the actual facets. The regular Platonic numbers are listed by increasing number N_{0} of vertices.
Rank
 N_{0}
 Name
(N_{0}, N_{1}, N_{2}) Schläfli symbol^{[6]}  Order
of basis^{[9]}^{[10]}
 Differences
 Partial sums
 Partial sums of reciprocals
 Sum of Reciprocals^{[11]}^{[12]}


0  4  Tetrahedral
(4, 6, 4) {3, 3} 
 [1]  
1  6  Octahedral
(6, 12, 8) {3, 4} 

 ^{[13]} ^{[14]}
 
2  8  Cube
(8, 12, 6) {4, 3} 

 ^{[15]}^{[16]}
 
3  12  Icosahedral
(12, 30, 20) {3, 5} 

 ^{[13]} ^{[14]}
 
4  20  Dodecahedral
(20, 30, 12) {5, 3} 



Table of sequences
N_{0}  sequences 

4  {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, ...} 
6  {0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, ...} 
8  {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, ...} 
12  {0, 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, 6384, 7890, 9616, 11577, 13788, 16264, 19020, 22071, 25432, 29118, 33144, 37525, 42276, 47412, ...} 
20  {0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, ...} 
See also
Notes
 ↑ Weisstein, Eric W., Platonic Solid, From MathWorldA Wolfram Web Resource.
 ↑ Where is the ddimensional regular convex polytope number with N_{0} vertices.
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Relatively Prime, From MathWorldA Wolfram Web Resource.
 ↑ ^{6.0} ^{6.1} Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Rising Factorial, From MathWorldA Wolfram Web Resource.
 ↑ Where , k ≥ 1, n ≥ 0, is the ddimensional, 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.
 ↑ 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. 922924.
 ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
 ↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
 ↑ ^{13.0} ^{13.1} Weisstein, Eric W., EulerMascheroni Constant, From MathWorldA Wolfram Web Resource.
 ↑ ^{14.0} ^{14.1} Weisstein, Eric W., Digamma Function, From MathWorldA Wolfram Web Resource.
 ↑ Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Apéry's Constant, From MathWorldA 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, 2^{nd} ed., 1994.