Octahedral number
Encyclopedia
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 nth octahedral number
can be obtained by the formula:
The first few octahedral numbers are:
Sir Frederick Pollock
conjectured in 1850 that every number is the sum of at most 7 octahedral numbers: see Pollock octahedral numbers conjecture
.
In chemistry
, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers
.
s, one upside-down underneath the other, by splitting it along a square cross-section. Therefore,
the nth octahedral number
can be obtained by adding two consecutive square pyramidal number
s together:
is the nth octahedral number and 
is the nth tetrahedral number
then
This represents the geometric fact that gluing a tetrahedron onto each of four non-adjacent faces of an octahedron produces a tetrahedron of twice the size. 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):
. The number of close-packed spheres in the rhombohedron is a cube, justifying the equation
:
Therefore, an octahedral number also represents the number of points in a square pyramid
formed by stacking centered squares; for this reason, in his book Arithmeticorum libri duo (1575), Francesco Maurolico
called these numbers "pyramides quadratae secundae".
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
given by the formula
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, an octahedral number is a figurate number
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...
that represents the number of spheres in an octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
formed from close-packed spheres. The nth octahedral number
can be obtained by the formula:The first few octahedral numbers are:
- 1, 6, 1919 (number)19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
, 4444 (number)44 is the natural number following 43 and preceding 45.- In mathematics :Forty-four is a tribonacci number, a happy number, an octahedral number and a palindromic number....
, 8585 (number)85 is the natural number following 84 and preceding 86.-In mathematics:85 is an octahedral number, a centered triangular number, a centered square number, a decagonal number, and a Smith number....
, 146, 231, 344, 489, 670, 891 .
Properties and applications
The octahedral numbers have a generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
Sir Frederick Pollock
Sir Frederick Pollock, 1st Baronet
Sir Frederick Pollock, 1st Baronet PC , was a British lawyer and Tory politician.-Background and education:...
conjectured in 1850 that every number is the sum of at most 7 octahedral numbers: see Pollock octahedral numbers conjecture
Pollock octahedral numbers conjecture
In additive number theory, the Pollock octahedral numbers conjecture is an unproven conjecture that every positive integer is the sum of at most seven octahedral numbers. It was first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician but also a contributor of papers...
.
In chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers
Magic number (chemistry)
In case a gas condenses into clusters of atoms, the number of atoms in these clusters varies between a few and hundreds. However, there are peaks at specific sizes, usually one at lower and one at larger numbers. Often, specific numbers dominate...
.
Square pyramids
An octahedral packing of spheres may be partitioned into two square pyramidSquare pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
s, one upside-down underneath the other, by splitting it along a square cross-section. Therefore,
the nth octahedral number
can be obtained by adding two consecutive square pyramidal numberSquare pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base...
s together:
Tetrahedra
If is the nth octahedral number and is the nth tetrahedral numberTetrahedral number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron...
then
This represents the geometric fact that gluing a tetrahedron onto each of four non-adjacent faces of an octahedron produces a tetrahedron of twice the size. 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):
Cubes
If two tetrahedra are attached to opposite faces of an octahedron, the result is a rhombohedronRhombohedron
In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length....
. The number of close-packed spheres in the rhombohedron is a cube, justifying the equation
Centered squares
The difference between two consecutive octahedral numbers is a centered square numberCentered square number
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a...
:
Therefore, an octahedral number also represents the number of points in a square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
formed by stacking centered squares; for this reason, in his book Arithmeticorum libri duo (1575), Francesco Maurolico
Francesco Maurolico
Francesco Maurolico was a Greek mathematician and astronomer of Sicily. Throughout his lifetime, he made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy...
called these numbers "pyramides quadratae secundae".
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