Editing Quaternion (section)

== History ==
<!-- {{refimprove section|date=October 2015}} -->
{{main|History of quaternions}}

[[File:Inscription on Broom Bridge (Dublin) regarding the discovery of Quaternions multiplication by Sir William Rowan Hamilton.jpg|right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge]], [[Dublin]], which reads:

{{bi|1= <poem>
     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
     {{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''i{{thinsp}}j{{thinsp}}k'' = −1}}
& cut it on a stone of this bridge
</poem> }} ]]

Quaternions were introduced by Hamilton in 1843.<ref name="SeeHazewinkel">See {{harvnb|Hazewinkel|Gubareni|Kirichenko|2004|p=[https://books.google.com/books?id=AibpdVNkFDYC&pg=PA12 12]}}</ref> Important precursors to this work included [[Euler's four-square identity]] (1748) and [[Olinde Rodrigues]]' [[Euler–Rodrigues parameters|parameterization of general rotations by four parameters]] (1840), but neither of these writers treated the four-parameter rotations as an algebra.<ref>{{harvnb|Conway|Smith|2003|p=[https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA9 9]}}</ref><ref>{{cite book |first1=Robert E. |last1=Bradley |first2=Charles Edward |last2=Sandifer |title=Leonhard Euler: life, work and legacy |year=2007 |isbn=978-0-444-52728-8 |url=https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA193 |page=193|publisher=Elsevier }} They mention [[Wilhelm Blaschke]]'s claim in 1959 that "the quaternions were first identified by L.&nbsp;Euler in a letter to Goldbach written on 4&nbsp;May 1748," and they comment that "it makes no sense whatsoever to say that Euler "identified" the quaternions in this letter ... this claim is absurd."</ref> [[Carl Friedrich Gauss]] had discovered quaternions in 1819, but this work was not published until 1900.<ref>Pujol, J., "[https://projecteuclid.org/journals/communications-in-mathematical-analysis/volume-13/issue-2/Hamilton-Rodrigues-Gauss-Quaternions-and-Rotations-a-Historical-Reassessment/cma/1349803591.full Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: A Historical Reassessment]"  ''Communications in Mathematical Analysis'' (2012), 13(2), 1–14</ref><ref>{{cite book |first=C.F. |last=Gauss |article=Mutationen des Raumes [Transformations of space] (c.&nbsp;1819) |others=article edited by Prof.&nbsp;Stäckel of Kiel, Germany |editor=Martin Brendel |title=Carl Friedrich Gauss Werke |trans-title=The works of Carl Friedrich Gauss |year=1900 |volume=8 |pages=357–361 |location=Göttingen, DE |publisher=Königlichen Gesellschaft der Wissenschaften [Royal Society of Sciences] |url=https://books.google.com/books?id=aecGAAAAYAAJ&pg=PA357}}</ref>

Hamilton knew that the complex numbers could be interpreted as [[point (geometry)|points]] in a [[plane (mathematics)|plane]], and he was looking for a way to do the same for points in three-dimensional [[space]]. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact, [[Ferdinand Georg Frobenius]] later [[Frobenius theorem (real division algebras)|proved]] in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: <math>\mathbb {R, C}</math> (complex numbers) and <math>\mathbb H</math> (quaternions) which have  dimension 1, 2, and 4 respectively.

The great breakthrough in quaternions finally came on Monday 16&nbsp;October 1843 in [[Dublin]], when Hamilton was on his way to the [[Royal Irish Academy]] to preside at a council meeting. As he walked along the towpath of the [[Royal Canal]] with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

<math display=block>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i\,j\,k} = -1</math>

into the stone of [[Broom Bridge|Brougham Bridge]] as he paused on it. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the [[Hamilton Walk]] for scientists and mathematicians who walk from [[Dunsink Observatory]] to the Royal Canal bridge in remembrance of Hamilton's discovery.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician, [[John T. Graves]], describing the train of thought that led to his discovery. This letter was later published in a letter to the ''[[London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science]]'';<ref name=letter1844>{{cite magazine |last=Hamilton |first=W.R. |title=Letter |magazine=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=xxv |year=1844 |pages=489–495}}</ref> Hamilton states:
{{blockquote|And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.<ref name=letter1844/>}}

Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted most of the remainder of his life to studying and teaching them. [[Classical Hamiltonian quaternions|Hamilton's treatment]] is more [[Geometry|geometric]] than the modern approach, which emphasizes quaternions' [[algebra]]ic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, ''Elements of Quaternions'',<ref name=HamiltonElements>{{cite book |title=Elements of Quaternions |author=Hamilton, Sir W.R. |author-link=William Rowan Hamilton |editor=Hamilton, W.E. |editor-link=William Edwin Hamilton |year=1866 |url=https://archive.org/details/elementsofquater00hamiuoft |location=London |publisher=Longmans, Green, & Co.}}</ref> was 800&nbsp;pages long; it was edited by [[William Edwin Hamilton|his son]] and published shortly after his death.

After Hamilton's death, the Scottish mathematical physicist  [[Peter Guthrie Tait|Peter Tait]] became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. 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 even a professional research association, the [[Quaternion Society]], devoted to the study of quaternions and other [[hypercomplex number]] systems.

From the mid-1880s, quaternions began to be displaced by [[vector analysis]], which had been developed by [[Josiah Willard Gibbs]], [[Oliver Heaviside]], and [[Hermann von Helmholtz]]. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. 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 [[classical Hamiltonian quaternions|Hamilton's work]] is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.

However, quaternions have had a revival since the late 20th&nbsp;century, primarily due to their utility in [[Quaternions and spatial rotation|describing spatial rotations]]. The representations of rotations by quaternions are more compact and quicker to compute than the representations by [[matrix (mathematics)|matrices]]. In addition, unlike Euler angles, they are not susceptible to "[[gimbal lock]]". For this reason, quaternions are used in [[computer graphics]],<ref name="Shoemake">{{cite journal |doi=10.1145/325165.325242 |first=Ken |last=Shoemake |author-link=Ken Shoemake |year=1985 |url=https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf |title=Animating Rotation with Quaternion Curves |journal=Computer Graphics |volume=19 |issue=3 |pages=245–254}} Presented at [[SIGGRAPH]] '85.</ref><ref>''[[Tomb Raider (1996 video game)|Tomb Raider]]'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth three-dimensional rotations. See, for example {{cite magazine |author=Nick Bobick |url=https://www.gamedeveloper.com/programming/rotating-objects-using-quaternions |title=Rotating objects using quaternions |magazine=Game Developer |date=July 1998}}</ref> [[computer vision]], [[robotics]],<ref>{{cite book |first=J.M. |last=McCarthy |title=An Introduction to Theoretical Kinematics |url=https://books.google.com/books?id=glOqQgAACAAJ |year=1990 |publisher=MIT Press |isbn=978-0-262-13252-7}}</ref> [[nuclear magnetic resonance]] image sampling,<ref name=Mamone /> [[control theory]], [[signal processing]], [[Spacecraft attitude control|attitude control]], [[physics]], [[bioinformatics]], [[molecular dynamics]], [[computer simulation]]s, and [[orbital mechanics]]. For example, it is common for the [[Spacecraft attitude control|attitude control]] systems of spacecraft to be commanded in terms of quaternions.  Quaternions have received another boost from [[number theory]] because of their relationships with the [[quadratic form]]s.<ref>{{citation |last=Hurwitz |first=A.
|title=Vorlesungen über die Zahlentheorie der Quaternionen |jfm=47.0106.01
|place=Berlin |publisher=J. Springer |year=1919}}, concerning [[Hurwitz quaternion]]s</ref>

=== Quaternions in physics ===

The finding of 1924 that in [[quantum mechanics]] the [[Spin (physics)|spin]] of an electron and other matter particles (known as [[spinors]]) can be described using quaternions (in the form of the famous [[Pauli matrices|Pauli spin matrices]]) furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "[[Plate trick]]").<ref>{{cite web |last1=Huerta |first1=John |title=Introducing The Quaternions |url=http://math.ucr.edu/~huerta/introquaternions.pdf |access-date=8 June 2018 |archive-url=https://web.archive.org/web/20141021093534/http://math.ucr.edu/~huerta/introquaternions.pdf |archive-date=2014-10-21 |url-status=live |date=27 September 2010}}</ref><ref>{{cite web |first=Charlie |last=Wood |title=The Strange Numbers That Birthed Modern Algebra |date=6 September 2018 |work=Abstractions blog |publisher=Quanta Magazine |url=https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/}}</ref> {{As of|2018}}, their use has not overtaken [[orthogonal group|rotation groups]].<ref group=lower-alpha>A more personal view of quaternions was written by [[Joachim Lambek]] in 1995. He wrote in his essay ''If Hamilton had prevailed: quaternions in physics'': "My own interest as a graduate student was raised by the inspiring book by Silberstein". He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics." {{cite magazine |author=Lambek, J. |title=If Hamilton had prevailed: Quaternions in physics |year=1995 |magazine=Math. Intelligencer |volume=17 |issue=4 |pages=7–15 |doi=10.1007/BF03024783}}</ref>