The five regular convex polyhedra (3-dimensional regular convex solids, known as the 5 Platonic solids), are
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 (
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
), then with one dot at each vertex of a given Platonic solid (for
), 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
th Platonic
-hedral number (having
vertices) is given by the formulae:
[2]
-
P (3) N0(n) = ( n + 23 ) + (N0 − 4) ( n + 13 ) + 0 ( n3 ) |
, for the (self dual) tetrahedral ( ) numbers;
-
P (3) N0(n) = ( n + 23 ) + (N0 − 4) ( n + 13 ) + 1 ( n3 ) |
for the (dual pair) octahedral ( ) and hexahedral (cubic) ( ) numbers;
-
P (3) N0(n) = ( n + 23 ) + (N0 − 4) ( n + 13 ) + () ( n3 ) |
for the (dual pair) icosahedral ( ) and dodecahedral ( ) numbers.
where
is the number of 0-dimensional elements (vertices
),
is the number of 1-dimensional elements (edges
),
is the number of 2-dimensional elements (faces
) 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).
Platonic roots
Tetrahedral roots
The
tetrahedral roots of
are defined as the roots
of the
cubic equation
hence
Cube roots
The
cube roots of
are defined as the roots
of the
cubic equation
yielding
Octahedral roots
The octahedral roots of
are defined as the roots
of the
cubic equation
hence
Dodecahedral roots
The dodecahedral roots of
are defined as the roots
of the
cubic equation
hence
Icosahedral roots
The icosahedral roots of
are defined as the roots
of the
cubic equation
hence
Descartes–Euler (convex) polyhedral formula
Descartes–Euler (convex) polyhedral formula:[3]
where
is the number of 0-dimensional elements (vertices
),
is the number of 1-dimensional elements (edges
),
is the number of 2-dimensional elements (faces
) of the polyhedron.
Recurrence relation
with initial conditions
Ordinary generating function
-
for the (self dual) tetrahedral (rank
vertices) numbers;
for the (dual pair) octahedral (rank
vertices) and hexahedral (cubic) (rank
vertices) numbers;
-
for the (dual pair) icosahedral (rank
vertices) and dodecahedral (rank
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 4-face numbers:
For the 8-face and 6-face numbers:
-
For the 20-face and 12-face numbers:
Dirichlet generating function
For the 4-face numbers:
For the 8-face and 6-face numbers:
-
For the 20-face and 12-face 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
-gon 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
-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 {P (k, n) | n = 0, 1, 2, …} |
of -gon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of -gon numbers.
Proof. PROOF GOES HERE. □ (Provide proof: PROOF GOES HERE. □) [5]
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 4-dimensional 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.
[6]
Table of formulae and values
and
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
of vertices.
Platonic numbers formulae and values
Rank
|
|
Name
Schläfli symbol[7]
|
Formulae
|
Generating function
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
A-number
|
0
|
4
|
Tetrahedral
(4, 6, 4)
{3, 3}
|
[8]
|
|
0
|
1
|
4
|
10
|
20
|
35
|
56
|
84
|
120
|
165
|
220
|
286
|
364
|
A000292
|
1
|
6
|
Octahedral
(6, 12, 8)
{3, 4}
|
[9]
(Square dipyramidal numbers)
|
|
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
and
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
of vertices.
Platonic numbers related formulae and values
Rank
|
|
Name
Schläfli symbol[7]
|
Order of basis[10][11]
|
Differences
P (3) N0(n) − P (3) N0(n − 1) |
|
Partial sums
|
Partial sums of reciprocals
|
Sum of Reciprocals[12][13]
|
0
|
4
|
Tetrahedral
(4, 6, 4) {3, 3}
|
|
|
|
|
[14]
|
1
|
6
|
Octahedral
(6, 12, 8) {3, 4}
|
|
|
|
|
[15] [16]
Base 10: A175577
|
2
|
8
|
Cube
(8, 12, 6) {4, 3}
|
|
|
|
|
[17][18]
Base 10: A002117
|
3
|
12
|
Icosahedral
(12, 30, 20) {3, 5}
|
|
|
|
|
[15][16]
Base 10: A175578
|
4
|
20
|
Dodecahedral
(20, 30, 12) {5, 3}
|
|
|
|
|
|
Table of sequences
Platonic numbers sequences
|
|
A-number
|
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, ...}
|
A000292
|
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, ...}
|
A005900
|
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, ...}
|
A000578
|
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, ...}
|
A006564
|
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, ...}
|
A006566
|
See also
Notes
- ↑ Weisstein, Eric W., Platonic Solid, from MathWorld—A Wolfram Web Resource.
- ↑ Where is the -dimensional regular convex polytope number with vertices.
- ↑ Weisstein, Eric W., Polyhedral Formula, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Needs proof.
- ↑ Weisstein, Eric W., Relatively Prime, from MathWorld—A Wolfram Web Resource.
- ↑ 7.0 7.1 Weisstein, Eric W., Schläfli Symbol, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource.
- ↑ Where
Y (d ) [(k + 2) + (d − 2)] (n) = Y (d ) k + d (n) |
, is the -dimensional, , -gonal base (hyper)pyramidal number where, for , 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. 922-924.
- ↑ 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 Nunbers Series.
- ↑ User:Jaume Oliver Lafont/Sum of Reciprocals of Tetrahedral Numbers
- ↑ 15.0 15.1 Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource.
- ↑ 16.0 16.1 Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource.
- ↑ Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, from MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.
- ↑ Weisstein, Eric W., Apéry's Constant, 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, 2 nd ed., 1994.