Octahedral number
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, an octahedral number is a
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 (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
that represents the number of spheres in an
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahedral numbers are: : 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 .


Properties and applications

The octahedral numbers have a
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
: \frac = \sum_^ O_n z^n = z +6z^2 + 19z^3 + \cdots . Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers. This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers. In
chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers..


Relation to other figurate numbers


Square pyramids

An octahedral packing of spheres may be partitioned into two
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 is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...
s, one upside-down underneath the other, by splitting it along a square cross-section. Therefore, the ''n''th octahedral number O_n can be obtained by adding two consecutive
square pyramidal number In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broa ...
s together: :O_n = P_ + P_n.


Tetrahedra

If O_n is the ''n''th octahedral number and T_n is the ''n''th
tetrahedral 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. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
then :O_n+4T_=T_. 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): :O_n = T_n + 2T_ + T_.


Cubes

If two tetrahedra are attached to opposite faces of an octahedron, the result is a
rhombohedron In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right ang ...
. The number of close-packed spheres in the rhombohedron is a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
, justifying the equation :O_n+2T_=n^3.


Centered squares

The difference between two consecutive octahedral numbers is a
centered 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 cen ...
: :O_n - O_ = C_ = n^2 + (n-1)^2. Therefore, an octahedral number also represents the number of points in a
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 is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...
formed by stacking centered squares; for this reason, in his book ''Arithmeticorum libri duo'' (1575),
Francesco Maurolico Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions t ...
called these numbers "pyramides quadratae secundae". The number of cubes in an octahedron formed by stacking centered squares is a
centered octahedral number A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the ...
, 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 :O_n+O_=\frac for ''n'' = 1, 2, 3, ...


History

The first study of octahedral numbers appears to have been by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
, around 1630, in his ''De solidorum elementis''. Prior to Descartes, figurate numbers had been studied by the ancient Greeks and by
Johann Faulhaber Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician. Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Bes ...
, but only for
polygonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples T ...
s,
pyramidal number A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. ...
s, and
cubes In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
. Descartes introduced the study of figurate numbers based on the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s and some of the
semiregular polyhedra In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...
; 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 Friedrich Wilhelm Marpurg (21 November 1718 – 22 May 1795) was a German music critic, music theorist and composer. He was friendly and active with many figures of the Enlightenment of the 18th century. Life Little is known of Marpurg's earl ...
in 1774,
Georg Simon Klügel Georg Simon Klügel (August 19, 1739 – August 4, 1812) was a German mathematician and physicist. He was born in Hamburg, and in 1760 went to the University of Göttingen where he initially studied theology before switching to mathematics. ...
in 1808, and Sir Frederick Pollock in 1850.


References


External links

* {{Classes of natural numbers Figurate numbers