In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The th octahedral number
On
| 2 | |
| O | |
| n={n(2n |
+1)\over3}.
The first few octahedral numbers are:
1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 .
The octahedral numbers have a generating function
| z(z+1)2 | |
| (z-1)4 |
=
| infty | |
| \sum | |
| n=1 |
Onzn=z+6z2+19z3+ … .
Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers.[2] This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers.
In chemistry, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers.[3] [4]
An octahedral packing of spheres may be partitioned into two square pyramids, one upside-down underneath the other, by splitting it along a square cross-section. Therefore,the octahedral number
On
On=Pn-1+Pn.
If
On
Tn
On+4Tn-1=T2n-1.
Another relation between octahedral numbers and tetrahedral numbers is also possible, based on the fact that an octahedron may be divided into four tetrahedra each having two adjacent original faces (or alternatively, based on the fact that each square pyramidal number is the sum of two tetrahedral numbers):
On=Tn+2Tn-1+Tn-2.
If two tetrahedra are attached to opposite faces of an octahedron, the result is a rhombohedron.[5] The number of close-packed spheres in the rhombohedron is a cube, justifying the equation
On+2Tn-1=n3.
The difference between two consecutive octahedral numbers is a centered square number:[1]
On-On-1=C4,n=n2+(n-1)2.
The number of cubes in an octahedron formed by stacking centered squares is a centered octahedral number, the sum of two consecutive octahedral numbers. These numbers are
1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, ... given by the formula
On+On-1=
| (2n-1)(2n2-2n+3) | |
| 3 |
The first study of octahedral numbers appears to have been by René Descartes, around 1630, in his De solidorum elementis. Prior to Descartes, figurate numbers had been studied by the ancient Greeks and by Johann Faulhaber, but only for polygonal numbers, pyramidal numbers, and cubes. Descartes introduced the study of figurate numbers based on the Platonic solids and some of the semiregular polyhedra; his work included the octahedral numbers. However, De solidorum elementis was lost, and not rediscovered until 1860. In the meantime, octahedral numbers had been studied again by other mathematicians, including Friedrich Wilhelm Marpurg in 1774, Georg Simon Klügel in 1808, and Sir Frederick Pollock in 1850.