Editing Octahedral number

{{short description|Number of close-packed spheres in an octahedron}}
[[File:Octahedral number.jpg|thumb|146 [[Neodymium magnet toys|magnetic balls]], packed in the form of an octahedron]]

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 {{mvar|n}}th octahedral number <math>O_n</math> can be obtained by the formula:<ref name="bon">{{citation|title=The Book of Numbers|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Richard K.|last2=Guy|author2-link=Richard K. Guy|publisher=Springer-Verlag|year=1996|isbn=978-0-387-97993-9|page=50}}.</ref>

:<math>O_n={n(2n^2 + 1) \over 3}.</math>

The first few octahedral numbers are: 

:[[1 (number)|1]], [[6 (number)|6]], [[19 (number)|19]], [[44 (number)|44]], [[85 (number)|85]], 146, 231, 344, 489, 670, 891 {{OEIS|id=A005900}}.

==Properties and applications==
The octahedral numbers have a [[generating function]]

:<math> \frac{z(z+1)^2}{(z-1)^4} = \sum_{n=1}^{\infty} O_n z^n = z +6z^2 + 19z^3 + \cdots .</math>

[[Sir Frederick Pollock, 1st Baronet|Sir Frederick Pollock]] conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers.<ref>{{citation|author-link=L. E. Dickson|last=Dickson|first=L. E.|series=[[History of the Theory of Numbers]]|volume=2|title=Diophantine Analysis|location=New York|publisher=Dover|year=2005|pages=22–23|isbn=9780821819357 |url=https://books.google.com/books?id=eNjKEBLt_tQC&pg=PA22}}.</ref> This statement, the [[Pollock octahedral numbers conjecture]], has been proven true for all but finitely many numbers.<ref>{{citation|last=Elessar Brady|first=Zarathustra|arxiv=1509.04316|doi=10.1112/jlms/jdv061|issue=1|journal=Journal of the London Mathematical Society|mr=3455791|pages=244–272|series=Second Series|title=Sums of seven octahedral numbers|volume=93|year=2016|s2cid=206364502 }}</ref>

In [[chemistry]], octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called [[Magic number (chemistry)|magic numbers]].<ref>{{citation|title=Magic numbers in polygonal and polyhedral clusters|first1=Boon K.|last1=Teo|first2=N. J. A.|last2=Sloane|author2-link=Neil Sloane|journal=Inorganic Chemistry|year=1985|volume=24|issue=26|pages=4545–4558|doi=10.1021/ic00220a025|url=http://www2.research.att.com/~njas/doc/magic1/magic1.pdf|access-date=2011-04-08|archive-url=https://web.archive.org/web/20120313220128/http://www2.research.att.com/~njas/doc/magic1/magic1.pdf|archive-date=2012-03-13|url-status=dead}}.</ref><ref name="nano">{{citation|title=Metal nanoparticles: synthesis, characterization, and applications|first1=Daniel L.|last1=Feldheim|first2=Colby A.|last2=Foss|publisher=CRC Press|year=2002|isbn=978-0-8247-0604-3|page=76|url=https://books.google.com/books?id=-u9tVYWfRcMC&pg=PA76}}.</ref>

==Relation to other figurate numbers==

===Square pyramids===
An octahedral packing of spheres may be partitioned into two [[square pyramid]]s, one upside-down underneath the other, by splitting it along a square cross-section. Therefore,
the {{nowrap|<math>n</math>th}} octahedral number <math>O_n</math> can be obtained by adding two consecutive [[square pyramidal number]]s together:<ref name="bon"/>
:<math>O_n = P_{n-1} + P_n.</math>

===Tetrahedra===
If <math>O_n</math> is the {{nowrap|<math>n</math>th}} octahedral number and <math>T_n</math> is the {{nowrap|<math>n</math>th}} [[tetrahedral number]] then
:<math>O_n+4T_{n-1}=T_{2n-1}.</math>
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):
:<math>O_n = T_n + 2T_{n-1} + T_{n-2}.</math>

===Cubes===
If two tetrahedra are attached to opposite faces of an octahedron, the result is a [[rhombohedron]].<ref>{{citation|first=John G.|last=Burke|title=Origins of the science of crystals|publisher=University of California Press|year=1966|page=88|url=https://books.google.com/books?id=qvxPbZtJu8QC&pg=PA88}}.</ref> The number of close-packed spheres in the rhombohedron is a [[Cube (algebra)|cube]], justifying the equation
:<math>O_n+2T_{n-1}=n^3.</math>

===Centered squares===
[[File:Pyramides quadratae secundae.svg|thumb|upright=1.35|Square pyramids in which each layer has a [[centered square number]] of cubes. The total number of cubes in each pyramid is an octahedral number.]]
The difference between two consecutive octahedral numbers is a [[centered square number]]:<ref name="bon"/>
:<math>O_n - O_{n-1} = C_{4,n} = n^2 + (n-1)^2.</math>
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".<ref>[http://www.maurolico.unipi.it/edizioni/arithmet/ariduo/ari1/ari1-2.htm Tables of integer sequences] {{Webarchive|url=https://archive.today/20120907110907/http://www.maurolico.unipi.it/edizioni/arithmet/ariduo/ari1/ari1-2.htm |date=2012-09-07 }} from ''Arithmeticorum libri duo'', retrieved 2011-04-07.</ref>

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, ... {{OEIS|A001845}}
given by the formula 
:<math>O_n+O_{n-1}=\frac{(2n-1)(2n^2-2n+3)}{3}</math> for ''n'' = 1, 2, 3, ...

==History==
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 number]]s, [[pyramidal number]]s, and [[Cube (algebra)|cubes]]. Descartes introduced the study of figurate numbers based on the [[Platonic solid]]s and some of the [[Semiregular polyhedron|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, 1st Baronet|Sir Frederick Pollock]] in 1850.<ref>{{citation|title=Descartes on Polyhedra: A Study of the "De solidorum elementis"|title-link=Descartes on Polyhedra|first=Pasquale Joseph|last=Federico|author-link=Pasquale Joseph Federico|series= Sources in the History of Mathematics and Physical Sciences|volume=4|publisher=Springer|year=1982|page=118}}</ref>

==References==
{{reflist}}

==External links==
* {{mathworld|title=Octahedral Number|urlname=OctahedralNumber|mode=cs2}}

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Figurate numbers]]