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Platonic numbers

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There are five regular convex polyhedra (3-dimensional 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 self-dual, 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

Contents

Formulae

The nth Platonic N2-hedral number (having N0 vertices) is given by the formulae:[2]

P^{(3)}_{N_0}(n) = {\binom{n+2}{3} + (N_0-4) \binom{n+1}{3} + 0 \binom{n}{3}},\, for the (self dual) tetrahedral (N0 = 4) numbers;
P^{(3)}_{N_0}(n) = {\binom{n+2}{3} + (N_0-4) \binom{n+1}{3} + 1 \binom{n}{3}},\, for the (dual pair) octahedral (N0 = 6) and hexahedral (cubic) (N0 = 8) numbers;
P^{(3)}_{N_0}(n) = {\binom{n+2}{3} + (N_0-4) \binom{n+1}{3} + \bigg(\frac{N_0}{2}\bigg) \binom{n}{3}},\, for the (dual pair) icosahedral (N0 = 12) and dodecahedral (N0 = 20) numbers.

where N0 is the number of 0-dimensional elements (vertices V,) N1 is the number of 1-dimensional elements (edges E,) N2 is the number of 2-dimensional elements (faces F) of the polyhedron.

\scriptstyle N_0 = (2^r - 0^r) + 4\, is the number of vertices of the Platonic solid, where \scriptstyle r \in \{0, 1, 2, 3, 4\}\, is the rank of the Platonic solid (by increasing number of vertices.)

Descartes-Euler (convex) polyhedral formula

Descartes-Euler (convex) polyhedral formula:[3]

{\sum_{i=0}^2 (-1)^i N_i} = N_0-N_1+N_2 = V-E+F = 2,\,

where N0 is the number of 0-dimensional elements (vertices V,) N1 is the number of 1-dimensional elements (edges E,) N2 is the number of 2-dimensional elements (faces F) of the polyhedron.

Recurrence relation

P^{(3)}_{N_0}(n) = 4P^{(3)}_{N_0}(n-1) - 6P^{(3)}_{N_0}(n-2) + 4P^{(3)}_{N_0}(n-3) - P^{(3)}_{N_0}(n-4),\ n>3,\,

with initial conditions

P^{(3)}_{N_0}(0) = 0,\ P^{(3)}_{N_0}(1) = 1,\ P^{(3)}_{N_0}(2) = N_0,\ P^{(3)}_{N_0}(3) = ?\,

Ordinary generating function

G_{\{P^{(3)}_{N_0}(n)\}}(x) = {{x(1+(N_0-4)x+0x^2)}\over{(1-x)^4}} = {{x(1+(2^r-0^r)x+0x^2)}\over{(1-x)^4}},\,

for the (self dual) tetrahedral (rank r = 0, N0 = 4 vertices) numbers;

G_{\{P^{(3)}_{N_0}(n)\}}(x) = {{x(1+(N_0-4)x+1x^2)}\over{(1-x)^4}} = {{x(1+(2^r-0^r)x+1x^2)}\over{(1-x)^4}},\,

for the (dual pair) octahedral (rank r = 1, N0 = 6 vertices) and hexahedral (cubic) (rank r = 2, N0 = 8 vertices) numbers;

G_{\{P^{(3)}_{N_0}(n)\}}(x) = {{x(1+(N_0-4)x+({N_0\over2})x^2)}\over{(1-x)^4}} = {{x(1+(2^r-0^r)x+({N_0\over2})x^2)}\over{(1-x)^4}},\,

for the (dual pair) icosahedral (rank r = 3, N0 = 12 vertices) and dodecahedral (rank r = 4, N0 = 20 vertices) numbers.

\scriptstyle N_0 = (2^r - 0^r) + 4\, is the number of vertices of the Platonic solid, where \scriptstyle r \in \{0, 1, 2, 3, 4\}\, is the rank of the Platonic solid (by increasing number of vertices.)

Exponential generating function

E_{\{P^{(3)}_{N_0}(n)\}}(x) = ?\,

For the 4-face numbers:

E_{\{P^{(3)}_{4}(n)\}}(x) = ,\,

For the 8-face and 6-face numbers:

E_{\{P^{(3)}_{6}(n)\}}(x) = ,\,
E_{\{P^{(3)}_{8}(n)\}}(x) = ,\,

For the 20-face and 12-face numbers:

E_{\{P^{(3)}_{12}(n)\}}(x) = ,\,
E_{\{P^{(3)}_{20}(n)\}}(x) = .\,

Dirichlet generating function

D_{\{P^{(3)}_{N_0}(n)\}}(x) = ?\,

For the 4-face numbers:

D_{\{P^{(3)}_{4}(n)\}}(x) = ,\,

For the 8-face and 6-face numbers:

D_{\{P^{(3)}_{6}(n)\}}(x) = ,\,
D_{\{P^{(3)}_{8}(n)\}}(x) = ,\,

For the 20-face and 12-face numbers:

D_{\{P^{(3)}_{12}(n)\}}(x) = ,\,
D_{\{P^{(3)}_{20}(n)\}}(x) = .\,

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.[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 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 \scriptstyle A\, of nonnegative integers is called a basis of order \scriptstyle g\, if \scriptstyle g\, is the minimum number with the property that every nonnegative integer can be written as a sum of \scriptstyle g\, elements in \scriptstyle A\,. Lagrange’s sum of four squares can be restated as the set \scriptstyle \{n^2|n = 0, 1, 2, \ldots\}\, of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every \scriptstyle k \ge 3, the set \scriptstyle \{P(k, n)|n = 0, 1, 2, \ldots\}\, of k-gon numbers forms a basis of order \scriptstyle k\,, i.e. every nonnegative integer can be written as a sum of \scriptstyle k\, 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 \scriptstyle g(d)\, such that every nonnegative integer is a sum of \scriptstyle g(d)\, \scriptstyle d\,th powers, i.e. the set \scriptstyle \{n^d|n = 0, 1, 2, \ldots\}\, of \scriptstyle d\,th powers forms a basis of order \scriptstyle g(d)\,. The Hilbert-Waring problem is concerned with the study of \scriptstyle g(d)\, for \scriptstyle d \ge 2\,. 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

P^{(3)}_{N_0}(n) - P^{(3)}_{N_0}(n-1) = ?\,

Partial sums

The partial sums correspond to 4-dimensional Platonic hyperpyramidal numbers.

\sum_{n=0}^{m} P^{(3)}_{N_0}(n) = ?\,

Partial sums of reciprocals

\sum_{n=0}^{m} \frac{1}{P^{(3)}_{N_0}(n)} = ?\,

Sum of reciprocals

\sum_{n=0}^{\infty} \frac{1}{P^{(3)}_{N_0}(n)} = ?\,

The sum of reciprocals {\sum_{n=1}^\infty {1\over{P^{(3)}_6(n)}}} = \zeta(3) can be interpreted as \frac{1}{p}, where p = {\frac{1}{\zeta(3)}} is the probability that three random integers \scriptstyle x\,, \scriptstyle y\, and \scriptstyle z\, are coprime.[5]

Table of formulae and values

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

Platonic numbers formulae and values
Rank


r

N0


\scriptstyle (2^r - 0^r) + 4\,

Name

(N0, N1, N2)

Schläfli symbol[6]

Formulae

P^{(3)}_{N_0}(n)\,

Generating

function

G_{\{P^{(3)}_{N_0}(n)\}}(x)\,

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

number

0 4 Tetrahedral

(4, 6, 4)

{3, 3}

\binom{n+2}{3}


\frac{n^{(3)}}{3!}[7]


{n(n+1)(n+2)}\over{6}\,

{x (1+0x+0x^2)}\over{(1-x)^4}\,


{x}\over{(1-x)^4}\,

0 1 4 10 20 35 56 84 120 165 220 286 364 A000292
1 6 Octahedral

(6, 12, 8)

{3, 4}

\scriptstyle \binom{n+2}{3} + 2 \binom{n+1}{3} + \binom{n}{3}\,


\scriptstyle P^{(3)}_4(n) + 2\ P^{(3)}_4(n-1) + P^{(3)}_4(n-2)\,


Y^{(3)}_5(n) + Y^{(3)}_5(n-1)\, [8]


(Square dipyramidal numbers)


{n(2n^2 + 1)}\over{3}\,

{x(1+2x+x^2)}\over{(1-x)^4}\,


{x(1+x)^2}\over{(1-x)^4}\,

0 1 6 19 44 85 146 231 344 489 670 891 1156 A005900
2 8 Cube

(8, 12, 6)

{4, 3}

\scriptstyle \binom{n+2}{3} + 4 \binom{n+1}{3} + \binom{n}{3}\,


\scriptstyle P^{(3)}_4(n) + 4\ P^{(3)}_4(n-1) + P^{(3)}_4(n-2)\,


n^3\,

{x(1+4x+x^2)}\over{(1-x)^4}\, 0 1 8 27 64 125 216 343 512 729 1000 1331 1728 A000578
3 12 Icosahedral

(12, 30, 20)

{3, 5}

\scriptstyle \binom{n+2}{3} + 8 \binom{n+1}{3} + 6 \binom{n}{3}\,


\scriptstyle P^{(3)}_4(n) + 8\ P^{(3)}_4(n-1) + 6\ P^{(3)}_4(n-2)\,


n(5P^{(2)}_3(n-1)+1)\,


{n(5n^2-5n+2)}\over{2}\,

{x(1+8x+6x^2)}\over{(1-x)^4}\, 0 1 12 48 124 255 456 742 1128 1629 2260 3036 3972 A006564
4 20 Dodecahedral

(20, 30, 12)

{5, 3}

\scriptstyle \binom{n+2}{3} + 16 \binom{n+1}{3} + 10 \binom{n}{3}\,


\scriptstyle P^{(3)}_4(n) + 16\ P^{(3)}_4(n-1) + 10\ P^{(3)}_4(n-2)\,


n(9P^{(2)}_3(n-1)+1)\,


{n(9n^2-9n+2)}\over{2}\,


{n(3n-1)(3n-2)}\over{2}\,


\binom{3n}{3}


P^{(3)}_4(3n-2)\,


n\ \,_cP^{(2)}_9(n)\,


n\ P^{(2)}_3(3n-2)\,

{x(1+16x+10x^2)}\over{(1-x)^4}\, 0 1 20 84 220 455 816 1330 2024 2925 4060 5456 7140 A006566


Table of related formulae and values

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

Platonic numbers related formulae and values
Rank


r

N0


\scriptstyle (2^r - 0^r) + 4\,

Name

(N0, N1, N2)

Schläfli symbol[6]

Order

of basis[9][10]

g_{\{P^{(3)}_{N_0}\}} =\,


N_0 + ?\,

Differences

P^{(3)}_{N_0}(n) - \,

P^{(3)}_{N_0}(n-1) =\,

Partial sums

\sum_{n=1}^m {P^{(3)}_{N_0}(n)} =

Partial sums of reciprocals

\sum_{n=1}^m {1\over{P^{(3)}_{N_0}(n)}} =

Sum of Reciprocals[11][12]

\sum_{n=1}^\infty{1\over{P^{(3)}_{N_0}(n)}} =

0 4 Tetrahedral

(4, 6, 4)

{3, 3}

5?\,


(N_0 + 1?)\,

\frac{n(n+1)}{2}\, \, \, \frac{3}{2} [1]
1 6 Octahedral

(6, 12, 8)

{3, 4}

7?\,


(N_0 + 1?)\,

2n^2-2n+1\,


\frac{4n(n-1)+2}{2}\,

\, \, 3\bigg[\gamma+\Re \psi\bigg(\frac{i}{\surd{2}}\bigg)\bigg] [13] [14]


Base 10: A175577

2 8 Cube

(8, 12, 6)

{4, 3}

9\,


(N_0 + 1)\,

3n^2-3n+1\,


\frac{6n(n-1)+2}{2}\,

\, \, \zeta(3)\, [15][16]


Base 10: A002117

3 12 Icosahedral

(12, 30, 20)

{3, 5}

15?\,


(N_0 + 3?)\,

\frac{15n^2-25n+12}{2}\,


\frac{5n(3n-5)+12}{2}\,

\, \, \scriptstyle \gamma + \Re \psi\big(\frac{1}{2} + \frac{i\sqrt{15}}{10}\big) + \frac{\sqrt{15}\pi}{6} \tanh\big(\frac{\sqrt{15}\pi}{10}\big) [13] [14]


Base 10: A175578

4 20 Dodecahedral

(20, 30, 12)

{5, 3}

22?\,


(N_0 + 2?)\,

\frac{27n^2-45n+20}{2}\,


\frac{9n(3n-5)+20}{2}\,

\, \, \frac{\pi\sqrt{3}-3\log{3}}{2}


\approx 1.0727806\ldots\,


Table of sequences

Platonic numbers sequences
N0 P^{(3)}_{N_0}(n),\ n \ge 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

Centered Platonic numbers

Notes

  1. Weisstein, Eric W., Platonic Solid, From MathWorld--A Wolfram Web Resource.
  2. Where \scriptstyle P^{(d)}_{N_0}(n)\, is the d-dimensional regular convex polytope number with N0 vertices.
  3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  4. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W., Relatively Prime, From MathWorld--A Wolfram Web Resource.
  6. 6.0 6.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  7. Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
  8. Where \scriptstyle Y^{(d)}_{[(k+2)+(d-2)]}(n) = Y^{(d)}_{k+d}(n)\,, k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (k+2)-gonal base (hyper)pyramidal number where, for d ≥ 2, \scriptstyle N_0 = [(k+2)+(d-2)]\, is the number of vertices (including the \scriptstyle d-2\, apex vertices) of the polygonal base (hyper)pyramid.
  9. HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
  10. 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.
  11. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  12. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
  13. 13.0 13.1 Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.
  14. 14.0 14.1 Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.
  15. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.
  16. Weisstein, Eric W., Apéry's Constant, From MathWorld--A Wolfram Web Resource.

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