Simplicial approximation theorem - Revision history

en>Tonyxty: /* Formal statement of the theorem */ Add link

2015-01-06T05:13:34Z

Formal statement of the theorem: Add link

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	~~[[File:Octahedral number.jpg\|thumb\|146 [[Neodymium magnet toys\|magnetic balls]]~~, ~~packed~~ in the ~~form~~ of ~~an octahedron]]~~		<br><br>Blogs, unlike websites, display articles (called posts) in a reverse chronological order. What makes it unique? This can be a major drain on the planet's electricity resources. Each of these packages is designed to give you the maximum amount of flexibility you need for a price you can afford. So I can create all the sites I want to. Why decide on HostGator and what it is.It's known as one of the most excellent webhosting services; it's an internet service that aids you to make your own website online whether it is for your new-found business or just for fun or as a hobby. Customer Service The support offered by Hostgator Brazil is one of the best there, because they take your questions in a live chat if you are not up to the client company. I've paid more then that an hour with previous help when I was using another hosting service. With its simple to recognize interface and the broad array of templates ranging from dating themes to [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=business+enterprise&btnI=lucky business enterprise] layouts, that you can opt for from, you will be ready to produce a complete working web site in just a couple of hrs. If not, write a path to a new blog directory, and it will be created for you.<br><br>If you cherished this short article and you would like to obtain more data with regards to Hostgator Coupons - [http://dawonls.dothome.co.kr/db/?document_srl=284755 dawonls.dothome.co.kr] - kindly go to our own page. <br><br>We restored our new computers were purchased and offsite backup, and by early afternoon we were back in business. It's not a full time job but it certainly helps if your saving for a particular thing. There is a noble cause why HostGator have won so many awards as well as why they are so generally chosen by online marketers as well as entrepreneurs all round the globe. I've found the people on this talk window to be slow and unhelpful. That said, when I decided to pursue publication in the Internet age, I made sure to have a Web site ready and a blog to track my progress. It virtually stands out from all the other hosting providers. The packages on offer for such an affordable price is the reason why over 200,000 webmasters have decided that this [http://Photobucket.com/images/company company] is ideal for their online business. I have been getting more heavily involved in my web endeavors and had a recent inclination to do some serious research on hosting and domain companies.
	~~In [[number theory]], an '''octahedral number'''~~ is ~~a [[figurate number]] that represents~~ the ~~number~~ of ~~spheres in an [[octahedron]] formed from [[Close-packing of spheres\|close-packed spheres]]~~. ~~The ''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 number is~~ the ~~sum~~ of ~~at most 7 octahedral numbers: see [~~[Pollock octahedral numbers conjecture]].<ref>{{citation\|authorlink=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\|url=http://~~books~~.google.com/~~books~~?id=~~eNjKEBLt_tQC~~&pg=~~PA22}}.</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}}.<~~/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=~~http://~~books~~.~~google~~.~~com~~/~~books~~?~~id=-u9tVYWfRcMC&pg=PA76}}.</ref>~~

	~~==Relation to other figurate numbers==~~

	~~===Square pyramids==~~=
	~~[[File:Pyramides quadratae secundae~~.~~svg\|thumb\|300px\|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~~.]]
	~~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 ''n''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~~ '~~'n''th octahedral number and <math>T_n</math> is the ''n''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=http://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===~~
	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~~] ~~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>~~

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*{{mathworld\|title=Octahedral Number\|urlname=OctahedralNumber}}

	~~{{Classes of natural numbers}}~~
	~~[[Category:Figurate numbers]]~~

en>Rjwilmsi: Typo fixing, typo(s) fixed: continous → continuous (3) using AWB (9888)

2014-01-26T10:45:57Z

Typo fixing, typo(s) fixed: continous → continuous (3) using AWB (9888)

New page

[[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-packing of spheres|close-packed spheres]]. The ''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 number is the sum of at most 7 octahedral numbers: see [[Pollock octahedral numbers conjecture]].<ref>{{citation|authorlink=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|url=http://books.google.com/books?id=eNjKEBLt_tQC&pg=PA22}}.</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}}.</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=http://books.google.com/books?id=-u9tVYWfRcMC&pg=PA76}}.</ref>

==Relation to other figurate numbers==

===Square pyramids===
[[File:Pyramides quadratae secundae.svg|thumb|300px|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.]]
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 ''n''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 ''n''th octahedral number and <math>T_n</math> is the ''n''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=http://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===
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] 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>

==References==
{{reflist}}

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

{{Classes of natural numbers}}
[[Category:Figurate numbers]]