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	~~Hello~~ from ~~Germany~~. I'~~m glad~~ to ~~came across you~~. My first ~~name~~ is ~~Miranda~~. <br>I ~~live~~ in a ~~city~~ called ~~Eisdorf~~ in ~~east Germany~~.<br>I was ~~also born~~ in ~~Eisdorf 29 years ago~~. ~~Married~~ in ~~March 2001~~. I'~~m working~~ at the ~~post office~~.<br><br>~~Also visit my web blog~~ [http://www.~~charlie911~~.~~com~~/~~uggboots~~.~~html~~ http://www.~~charlie911~~.~~com~~/~~uggboots~~.~~html~~]		[[Image:William_Rowan_Hamilton_Plaque_-_geograph.org.uk_-_347941.jpg\|right\|thumb\|Quaternion plaque on [[Broom Bridge\|Brougham (Broom) Bridge]], [[Dublin]], which says:<blockquote style="margin:0px;"><div>Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication<div style="white-space:nowrap;">''i''² = ''j''² = ''k''² = <i style="letter-spacing:2px;">ijk</i> = −1</div>& cut it on a stone of this bridge.</div></blockquote>]]

			In [[mathematics]], [[quaternion]]s are a non-[[Commutativity\|commutative]] number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by [[Olinde Rodrigues]] in all but name in 1840, but independently discovered by Irish mathematician Sir [[William Rowan Hamilton]] in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. This article describes the original invention and subsequent development of quaternions.

			==Hamilton's discovery==
			In 1843, Hamilton knew that the [[complex number]]s could be viewed as [[point (geometry)\|point]]s in a [[plane (mathematics)\|plane]] and that they could be added and multiplied together using certain geometric operations. Hamilton sought to find a way to do the same for points in [[space]]. Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had been stuck on defining the appropriate multiplication.

			According to a letter Hamilton wrote later to his son Archibald:
			<blockquote>Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edward and yourself used to ask me: "Well, Papa, can you multiply triples?" Whereto I was always obliged to reply, with a sad shake of the head, "No, I can only add and subtract them."</blockquote>

			On October 16, 1843, Hamilton and his wife took a walk along the [[Royal Canal]] in [[Dublin]]. While they walked across Brougham Bridge (now [[Broom Bridge]]), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for ''quadruples''. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:
			:<math>i^2 = j^2 = k^2 = ijk = -1.\,</math>

			Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted the remainder of his life to studying and teaching them. He founded a school of "quaternionists" and popularized them in several books. The last and longest, ''Elements of Quaternions'', had 800 pages and was published shortly after his death. See "''[[classical Hamiltonian quaternions]]''" for a summary of Hamilton's work.

			==Precursors==

			Hamilton's innovation consisted of expressing quaternions ''as an [[algebra (mathematics)\|algebra]]''. The formulae for the multiplication of quaternions are implicit in the [[four squares formula]] devised by [[Leonhard Euler]] in 1748; [[Olinde Rodrigues]] applied this formula to representing rotations in 1840.<ref>John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters, 2003, ISBN 1-56881-134-9, p. 9</ref>

			==After Hamilton==
			After Hamilton's death, his pupil [[Peter Guthrie Tait\|Peter Tait]], as well as [[Benjamin Peirce]], continued advocating the use of quaternions. Topics in physics and geometry that would now be described using vectors, such as [[kinematics]] in space and [[Maxwell's equations]], were described entirely in terms of quaternions. There was a professional research association which existed from 1899 to 1913, the [[Quaternion Society]], exclusively devoted to the study of quaternions.

			From the mid-1880s, quaternions began to be displaced by [[vector analysis]], which had been developed by [[Josiah Willard Gibbs]] and [[Oliver Heaviside]].<ref>Baez, p. 147</ref> Both were inspired by the quaternions as used in Maxwell's ''[[A Treatise on Electricity and Magnetism]]'', but — according to Gibbs — found that "… the idea of the quaternion was quite foreign to the subject."<ref>{{citation \| title=A history of vector analysis: The evolution of the idea of a vectorial system \| first=Michael J. \| last=Crowe \| publisher=Dover \| year=1994 \| isbn=0-486-67910-1 \| pages=152–154 }}. A reprint of the corrected edition from 1985; originally published in 1967.</ref> Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side effect of this transition is that works on [[classical Hamiltonian quaternions]] are difficult to comprehend for many modern readers because they use familiar terms from vector analysis in unfamiliar and fundamentally different ways.
			==Principal publications==
			* 1853 ''Lectures on Quaternions''<ref>[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 Lectures on Quaternions], Royal Irish Academy, weblink from [[Cornell University]] ''Historical Math Monographs''</ref>
			* 1866 ''Elements of Quaternions''<ref>[http://books.google.com/books?id=fIRAAAAAIAAJ Elements of Quaternions], [[University of Dublin]] Press. Edited by William Edwin Hamilton, son of the deceased author</ref>
			* 1873 ''Elementary Treatise'' by [[Peter Guthrie Tait]]<ref>[http://books.google.com/books?id=TwkXAAAAYAAJ Elementary Treatise on Quaternions]</ref>
			* 1878 [[Abbott Lawrence Lowell]]: Quadrics: Harvard dissertation:<ref>[[Abbott Lawrence Lowell]] (1878) [http://www.biodiversitylibrary.org/item/35671#page/248/mode/1up Surfaces of the second order, as treated by quaternions], [[Proceedings of the American Academy of Arts and Sciences]] 13:222–50, from [[Biodiversity Heritage Library]]</ref>
			* 1882 Tait and [[Philip Kelland]]: ''Introduction with Examples''<ref>[http://books.google.com/books?id=xxsFAAAAMAAJ Introduction to Quaternions with Numerous Examples]</ref>
			* 1885 Arthur Buchheim: Biquaternions<ref> [http://www.jstor.org/stable/2369176 "A Memoir on biquaternions"], [[American Journal of Mathematics]] 7(4):293 to 326 from [[Jstor]] early content</ref>
			* 1899 [[Charles Jasper Joly]]: ''Elements'' vol 1, vol 2 1901<ref>Hamilton (1899) ''Elements of Quaternions'' volume I, (1901) volume II. Edited by [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]]</ref>
			* 1901 [[Vector Analysis]] by [[Willard Gibbs]] and [[Edwin Bidwell Wilson]] (quaternion ideas without quaternions)
			* 1904 [[Cargill Gilston Knott]]: third edition of Kelland and Tait's textbook<ref>[http://books.google.com/books?id=Y-18PgAACAAJ Introduction to Quaternions]</ref>
			* 1904 ''Bibliography'' prepared for the [[Quaternion Society]] by [[Alexander Macfarlane]]<ref>[[Alexander Macfarlane]] (1904) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03030001 Bibliography of Quaternions and Allied Systems of Mathematics], weblink from Cornell University ''Historical Math Monographs''</ref>
			* 1905 C.J. Joly's ''Manual''<ref>[[Charles Jasper Joly]] (1905) ''A Manual of Quaternions'', London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84</ref>
			* 1940 [[Julian Coolidge]] in ''A History of Geometrical Methods'', page 261, uses the coordinate-free methods of Hamilton's operators and cites A. L. Lawrence's work at Harvard. Coolidge uses these operators on [[dual quaternion]]s to describe screw displacement in [[kinematics]].

			==Octonions==
			[[Octonion]]s were developed independently by [[Arthur Cayley]] in 1845 <ref>Penrose 2004 pg 202</ref>and [[John T. Graves]], a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?"<ref>Baez, p.146</ref>

			Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternion<ref>See Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...'</ref> that is nowadays often called an octonion, and showing that they were what we now call [[norm (mathematics)\|norm]]ed [[division algebra]]{{Fact\|date=March 2009}}; Graves called them ''octaves''. Hamilton needed a way to distinguish between two different types of double quaternions, the associative bi-quaternions and the octaves. He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion.<ref>[http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=octaves+hamilton+thinks&as_brr=1#PRA1-PA730,M1 Hamilton 1853 pg 740See a hard copy of Lectures on quaternions, appendix B, half of the hyphenated word double quaternion has been cut off in the online Edition ]</ref><ref>See Hamilton's talk to the Royal Irish Academy on the subject</ref> observed in reply that they were not [[associativity\|associative]], which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it; Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – as an appendix to a paper on a different subject. Hamilton was stung into protesting Graves' priority in discovery, if not publication; nevertheless, octonions are known by the name Cayley gave them – or as ''Cayley numbers''.

			The major deduction from the existence of octonions was the [[Degen's eight-square identity\|eight squares theorem]], which follows directly from the product rule from octonions, had also been previously discovered as a purely algebraic identity, by [[Ferdinand Degen]] in 1818.<ref>section from Baez, pp. 146–147</ref>

			==Mathematical uses==
			Quaternions continued to be a well-studied ''mathematical'' structure in the twentieth century, as the third term in the [[Cayley–Dickson construction]] of [[hypercomplex number]] systems over the reals, followed by the [[octonion]]s and the [[sedenion]]s; they are also useful tool in [[number theory]], particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the [[quaternion group]], is also the simplest non-commutative [[Sylow group]].

			The study of [[integral quaternion]]s began with [[Rudolf Lipschitz]] in 1886, whose system was later simplified by [[Leonard Eugene Dickson]]; but the modern system was published by [[Adolf Hurwitz]] in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions ''all four'' of whose coordinates are [[half-integers]]. Both systems are closed under subtraction and multiplication, and are therefore [[ring (mathematics)\|ring]]s, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.<ref>Hardy and Wright, ''Introduction to Number Theory'', §20.6-10''n'' (pp. 315–316, 1968 ed.)</ref>

			==Quaternions as rotations==
			Quaternions are a concise method of representing the [[automorphism]]s of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the [[simply connected]] cover of the space of three-dimensional rotations.<ref>John H. Conway, Derek A. Smith, ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry''. AK Peters, 2003, ISBN 1-56881-134-9, chapter 2.</ref>

			For this reason, quaternions are used in [[computer graphics]],<ref>[[Ken Shoemake]] (1985), [http://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf Animating Rotation with Quaternion Curves], ''[[Computer Graphics (publication)\|Computer Graphics]]'', '''19'''(3), 245–254. Presented at [[SIGGRAPH]] '85.</ref> [[control theory]], [[signal processing]], [[attitude control]], [[physics]], [[bioinformatics]], and [[orbital mechanics]]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. ''[[Tomb Raider]]'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation.<ref>Nick Bobick, "[http://www.gamasutra.com/features/19980703/quaternions_01.htm Rotating Objects Using Quaternions]", ''Game Developer'' magazine, February 1998</ref> Quaternions have received another boost from [[number theory]] because of their relation to [[quadratic form]]s.

			==Memorial==
			Since 1989, the Department of Mathematics of the [[National University of Ireland, Maynooth]] has organized a pilgrimage, where scientists (including physicists [[Murray Gell-Mann]] in 2002, [[Steven Weinberg]] in 2005, [[Frank Wilczek]] in 2007, and mathematician [[Andrew Wiles]] in 2003) take a walk from [[Dunsink Observatory]] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.<ref>[http://www.maths.nuim.ie/links/hamilton.shtml Hamilton walk] at the [[National University of Ireland, Maynooth]].]</ref>

			==References==
			*[[John C. Baez]], "[http://www.math.uchicago.edu/~mcgerty/BaezOctonions.pdf The Octonions]", ''Bulletin of the American Mathematical Society'', New Series, volume 39, number 2, pages 145–205. [[PDF]] link.
			*[[John H. Conway]], [[Derek A. Smith]], ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry''. AK Peters, 2003, ISBN 1-56881-134-9.
			*[[G. H. Hardy]] and [[E. M. Wright]], ''Introduction to Number Theory''. Many editions.

			==Notes==
			{{reflist\|2}}

			[[Category:Historical treatment of quaternions]]

69.245.250.6: small typo fix

2012-06-12T00:24:06Z

small typo fix

New page

Hello from Germany. I'm glad to came across you. My first name is Miranda. <br>I live in a city called Eisdorf in east Germany.<br>I was also born in Eisdorf 29 years ago. Married in March 2001. I'm working at the post office.<br><br>Also visit my web blog [http://www.charlie911.com/uggboots.html http://www.charlie911.com/uggboots.html]

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88.130.32.27: weird linebreaks

en>Eptified at 01:32, 15 June 2013

69.245.250.6: small typo fix