Pythagorean expectation - Revision history

en>Runner1928: /* External links */ navbox

2014-11-20T02:54:01Z

External links: navbox

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en>StatsGeek2009: /* Use in hockey */

2014-03-04T21:38:03Z

Use in hockey

← Older revision		Revision as of 23:38, 4 March 2014
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	~~{{Use dmy dates\|date=August 2012}}~~		The person who wrote the article is called Eusebio. South Carolina is his birth place. The favorite hobby for him and as well , his kids is so that you can fish and he's been for a while doing it for a very long time. Filing has been his profession for some time. Go to his website to find out more: http://circuspartypanama.com<br><br>Feel free to surf to my site - how to hack clash of clans - [http://circuspartypanama.com her response] -
	The ~~'''icosian calculus'''~~ is ~~a non-commutative [[algebraic structure]] discovered by the Irish mathematician [[William Rowan Hamilton]] in 1856~~.~~<ref>{{Cite journal~~
	~~\|title=Memorandum respecting a new System of Roots of Unity~~
	~~\|author=Sir William Rowan Hamilton~~
	~~\|author-link=William Rowan Hamilton~~
	~~\|url=http://www.maths.tcd~~.~~ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf~~
	~~\|journal=[[Philosophical Magazine]]~~
	~~\|volume=12~~
	~~\|year=1856~~
	~~\|page=446~~
	~~}}</ref><ref>{{cite book \|author=Thomas L. Hankins \|title=Sir William Rowan Hamilton \|publisher=~~The ~~Johns Hopkins University Press \|location=Baltimore \|year=1980 \|page=474 \|isbn=0-8018-6973-0 \|oclc= \|doi=}}</ref>~~
	~~In modern terms~~, ~~he gave a [[group presentation]] of the [[icosahedral group\|icosahedral rotation group]] by [[Generating set of a group\|generators]] and relations.~~

	~~Hamilton’s discovery derived from~~ his ~~attempts to find an algebra of [[tuple\|"triplets" or 3-tuples]]~~ that ~~he believed would reflect the three [[Cartesian coordinate system#Cartesian coordinates in three dimensions\|Cartesian axes]]. The symbols of the icosian calculus~~ can ~~be equated to moves between vertices on a [[dodecahedron]]. Hamilton’s work in this area resulted indirectly in the terms [[Hamiltonian circuit]]~~ and [[Hamiltonian path]] in graph theory.<ref name="biggs">{{cite book \|author=Norman L. Biggs, E. Keith Lloyd, Robin J. Wilson \|title=Graph theory 1736–1936 \|publisher=Clarendon Press \|location=Oxford \|year=1976 \|page=239 \|isbn=0-19-853901-0 \|oclc= \|doi=}}</ref> He also invented the [[icosian game]] as a means of illustrating and popularising his discovery.

	~~== Informal definition ==~~
	~~[[File:Icosian_grid_small_with_labels2.svg\|thumb\|right\|250px\|[[Stereographic projection]] of dodecahedron used for Hamilton~~'s ~~[[icosian game]]]]~~

	~~The algebra is based on three symbols that are each [[roots of unity]], in that repeated application of any of them yields the value 1 after~~ a ~~particular number of steps. They are:~~

	~~:<math>~~
	~~\begin{align}~~
	~~\iota^2 & = 1, \\~~
	~~\kappa^3 & = 1, \\~~
	~~\lambda^5 & = 1.~~
	~~\end{align}~~
	~~</math>~~

	~~Hamilton also gives one other relation between the symbols:~~

	~~:<math>\lambda = \iota\kappa.\,\!</math>~~
	~~(In modern terms this is the (2,3,5) [[triangle group]].)~~

	The operation is [[associative]] but not [[commutative]]. They generate a group of order 60, isomorphic to the [[group (mathematics)\|group]] of rotations of a regular [[icosahedron]] or [[dodecahedron]], and therefore to the [[alternating group]] of degree five.

	~~Although the algebra exists as a purely abstract construction,~~ it ~~can be most easily visualised in terms of operations on the edges and vertices of~~ a ~~dodecahedron~~. ~~Hamilton himself used a flattened dodecahedron as the basis~~ for his ~~instructional game.~~

	~~Imagine an insect crawling along a particular edge of Hamilton's labelled dodecahedron in a certain direction, say from <math>B</math>~~ to ~~<math>C</math>. We can represent this [[Directed graph\|directed edge]] by <math>BC</math>.~~

	~~[[File~~:~~Icosian_calculus_iota2.svg\|thumb\|right\|400px\|Geometrical illustration of operation iota in icosian calculus]]~~

	*The icosian symbol <math>\iota</math> equates to changing direction on any edge, so the insect crawls from <math>C</math> to <math>B</~~math> (following the directed edge <math>CB<~~/~~math>)~~.

	*The icosian symbol <~~math~~>~~\kappa~~<~~/math~~> ~~equates~~ to ~~rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction <math>BC</math> to become <math>DC</math>.~~

	*The icosian symbol <math>\lambda</math> equates to ~~making a right~~-~~turn at the end point, moving from <math>BC</math>~~ to ~~<math>CD</math>.~~

	~~== Legacy ==~~
	~~The icosian calculus is one~~ of ~~the earliest examples of many mathematical ideas, including:~~
	* presenting and studying a group by [[Presentation of a group\|generators and relations]];
	* a [[triangle group]], later generalized to [[Coxeter group]]s;
	* visualization of a group by a graph, which led to [[combinatorial group theory]] and later [[geometric group theory]];
	* [[Hamiltonian circuit]]s and [[Hamiltonian path]]s in graph theory;<ref name="biggs" />
	* [[dessin d'enfant]]<ref>{{Cite journal \| title = Dessins d'enfants: bipartite maps and Galois groups \| first = Gareth \| last = Jones \| journal = [[Séminaire Lotharingien de Combinatoire]] \| volume = B35d \| year = 1995 \| pages = 4 \| url = http://radon.mat.univie.ac.at/~slc/s/s35jones.html \| postscript =, [http://~~www.emis.de/journals/SLC/wpapers/s35jones~~.pdf PDF] }}</ref><ref>W. R. Hamilton, Letter to John T. Graves "On the Icosian" (17 October 1856), ''Mathematical papers, Vol. III, Algebra,'' eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967, pp. 612–625.</ref> – see [[dessin d'enfant#History\|dessin d'enfant: history]] for details.

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

	~~[[Category:Graph theory]]~~
	~~[[Category:Abstract algebra]]~~
	~~[[Category:Binary operations]]~~
	~~[[Category:Rotational symmetry]~~]

en>Monkbot: /* Use in basketball */Fix CS1 deprecated date parameter errors

2014-01-29T00:36:10Z

Use in basketball: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 02:36, 29 January 2014
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	~~Nothing~~ to ~~tell about myself I think~~.<br>~~Feels good~~ to be a ~~member~~ of this ~~site~~.<br>~~I just hope Im useful in one way~~ .<br><br>~~Here~~ is ~~my website~~; [http://www.~~ggtrust~~.~~com~~/~~forums~~/~~member~~.~~php?48470-IJAA assistance dog training~~]		{{Use dmy dates\|date=August 2012}}
			The '''icosian calculus''' is a non-commutative [[algebraic structure]] discovered by the Irish mathematician [[William Rowan Hamilton]] in 1856.<ref>{{Cite journal
			\|title=Memorandum respecting a new System of Roots of Unity
			\|author=Sir William Rowan Hamilton
			\|author-link=William Rowan Hamilton
			\|url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
			\|journal=[[Philosophical Magazine]]
			\|volume=12
			\|year=1856
			\|page=446
			}}</ref><ref>{{cite book \|author=Thomas L. Hankins \|title=Sir William Rowan Hamilton \|publisher=The Johns Hopkins University Press \|location=Baltimore \|year=1980 \|page=474 \|isbn=0-8018-6973-0 \|oclc= \|doi=}}</ref>
			In modern terms, he gave a [[group presentation]] of the [[icosahedral group\|icosahedral rotation group]] by [[Generating set of a group\|generators]] and relations.

			Hamilton’s discovery derived from his attempts to find an algebra of [[tuple\|"triplets" or 3-tuples]] that he believed would reflect the three [[Cartesian coordinate system#Cartesian coordinates in three dimensions\|Cartesian axes]]. The symbols of the icosian calculus can be equated to moves between vertices on a [[dodecahedron]]. Hamilton’s work in this area resulted indirectly in the terms [[Hamiltonian circuit]] and [[Hamiltonian path]] in graph theory.<ref name="biggs">{{cite book \|author=Norman L. Biggs, E. Keith Lloyd, Robin J. Wilson \|title=Graph theory 1736–1936 \|publisher=Clarendon Press \|location=Oxford \|year=1976 \|page=239 \|isbn=0-19-853901-0 \|oclc= \|doi=}}</ref> He also invented the [[icosian game]] as a means of illustrating and popularising his discovery.

			== Informal definition ==
			[[File:Icosian_grid_small_with_labels2.svg\|thumb\|right\|250px\|[[Stereographic projection]] of dodecahedron used for Hamilton's [[icosian game]]]]

			The algebra is based on three symbols that are each [[roots of unity]], in that repeated application of any of them yields the value 1 after a particular number of steps. They are:

			:<math>
			\begin{align}
			\iota^2 & = 1, \\
			\kappa^3 & = 1, \\
			\lambda^5 & = 1.
			\end{align}
			</math>

			Hamilton also gives one other relation between the symbols:

			:<math>\lambda = \iota\kappa.\,\!</math>
			(In modern terms this is the (2,3,5) [[triangle group]].)

			The operation is [[associative]] but not [[commutative]]. They generate a group of order 60, isomorphic to the [[group (mathematics)\|group]] of rotations of a regular [[icosahedron]] or [[dodecahedron]], and therefore to the [[alternating group]] of degree five.

			Although the algebra exists as a purely abstract construction, it can be most easily visualised in terms of operations on the edges and vertices of a dodecahedron. Hamilton himself used a flattened dodecahedron as the basis for his instructional game.

			Imagine an insect crawling along a particular edge of Hamilton's labelled dodecahedron in a certain direction, say from <math>B</math> to <math>C</math>. We can represent this [[Directed graph\|directed edge]] by <math>BC</math>.

			[[File:Icosian_calculus_iota2.svg\|thumb\|right\|400px\|Geometrical illustration of operation iota in icosian calculus]]

			*The icosian symbol <math>\iota</math> equates to changing direction on any edge, so the insect crawls from <math>C</math> to <math>B</math> (following the directed edge <math>CB</math>).

			*The icosian symbol <math>\kappa</math> equates to rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction <math>BC</math> to become <math>DC</math>.

			*The icosian symbol <math>\lambda</math> equates to making a right-turn at the end point, moving from <math>BC</math> to <math>CD</math>.

			== Legacy ==
			The icosian calculus is one of the earliest examples of many mathematical ideas, including:
			* presenting and studying a group by [[Presentation of a group\|generators and relations]];
			* a [[triangle group]], later generalized to [[Coxeter group]]s;
			* visualization of a group by a graph, which led to [[combinatorial group theory]] and later [[geometric group theory]];
			* [[Hamiltonian circuit]]s and [[Hamiltonian path]]s in graph theory;<ref name="biggs" />
			* [[dessin d'enfant]]<ref>{{Cite journal \| title = Dessins d'enfants: bipartite maps and Galois groups \| first = Gareth \| last = Jones \| journal = [[Séminaire Lotharingien de Combinatoire]] \| volume = B35d \| year = 1995 \| pages = 4 \| url = http://radon.mat.univie.ac.at/~slc/s/s35jones.html \| postscript =, [http://www.emis.de/journals/SLC/wpapers/s35jones.pdf PDF] }}</ref><ref>W. R. Hamilton, Letter to John T. Graves "On the Icosian" (17 October 1856), ''Mathematical papers, Vol. III, Algebra,'' eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967, pp. 612–625.</ref> – see [[dessin d'enfant#History\|dessin d'enfant: history]] for details.

			== References ==
			{{reflist}}

			[[Category:Graph theory]]
			[[Category:Abstract algebra]]
			[[Category:Binary operations]]
			[[Category:Rotational symmetry]]

en>Ale And Quail: a little more plain English for this phrase

2012-09-02T04:21:15Z

a little more plain English for this phrase

New page

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