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{{Short description|none}}
{{Copyedit|date=September 2008}}
], ], 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;">{{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = <span style="font-style:italic;letter-spacing:2px;">ijk</span> = −1}}</div>& cut it on a stone of this bridge.</div></blockquote>]]
{{Confusing|date=January 2008}}
{{intro-tooshort}}


In ], ]s are a non-] number system that extends the ]s. Quaternions and their applications to rotations were first described in print by ] in all but name in 1840,<ref>{{cite magazine | author = Simon L. Altmann | title = Hamilton, Rodrigues and the quaternion scandal |magazine=] | year = 1989 | volume = 62 | pages = 291–308 | doi = 10.2307/2689481 | jstor = 2689481 | issue = 5 }}</ref> but independently discovered by Irish mathematician Sir ] 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 is an in-depth story of the '''history of ]'''. It tells the story of who and when.


== The golden age == == Hamilton's discovery ==
In 1843, Hamilton knew that the ]s could be viewed as ]s in a ] 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 ]. Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had difficulty defining the appropriate multiplication.
{{seealso|Classical Hamiltonian Quaternions}}


According to a letter Hamilton wrote later to his son Archibald:
Quaternions were introduced by ] mathematician ] in 1843. On October 16, he was out walking along the ] in ] with his wife when the solution in the form of the equation
<blockquote>Every morning in the early part of October 1843, on my coming down to breakfast, your brother ] 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 ] in ]. While they walked across Brougham Bridge (now ]), 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>
:{{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1}}


Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 '']'' communicated Hamilton's exposition of quaternions.<ref>W.R. Hamilton(1844 to 1850) , ], link to David R. Wilkins collection at ]</ref> In 1853 he issued ''Lectures on Quaternions'', a comprehensive treatise that also described ]s. The facility of the algebra in expressing geometric relationships led to broad acceptance of the method, several compositions by other authors, and stimulation of applied algebra generally. As mathematical terminology has grown since that time, and usage of some terms has changed, the traditional expressions are referred to as ]s.
], which says: <br><small>Here as he walked by<br> on the 16th of October 1843<br> Sir William Rowan Hamilton<br> in a flash of genius discovered<br> the fundamental formula for quaternion multiplication<br> i² = j² = k² = i j k = &minus;1<br> & cut it on a stone of this bridge.</small>]]
occurred to him; Hamilton carved this equation into the side of the nearby Brougham Bridge (now called ]). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices had yet to be developed.


== Precursors ==
Hamilton popularized quaternions with several books, the last of which, ''Elements of Quaternions'', had 800 pages and was published shortly after his death.
Hamilton's innovation consisted of expressing quaternions as an ]. The formulae for the multiplication of quaternions are implicit in the ] devised by ] in 1748. In 1840, ] used spherical trigonometry and developed a formula closely related to quaternion multiplication in order to describe the new axis and angle of two combined rotations.<ref>{{cite arxiv | arxiv=2211.07787 | title=Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un système solide...", translation and commentary | first=Richard M. | last=Friedberg | authorlink = Richard M. Friedberg | year=2022}}</ref><ref name=CS>] & Derek A. Smith (2003) ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry'', ], {{ISBN|1-56881-134-9}}</ref>{{rp|9}}


== Response ==
Reading works written before 1900 on the subject of ] is difficult for modern readers because the notation used by early writers on the subject of quaternions is different from what is used today.
The special claims of quaternions as the algebra of ] were challenged by ] with his exhibits in 1848 and 1849 of ]s and ]s as alternatives. Nevertheless, these new algebras from Cockle were, in fact, to be found inside Hamilton's ]s. From Italy, in 1858 ] responded<ref>] ( 1858) , link from HathiTrust</ref> to connect Hamilton's vector theory with his theory of ]s of directed line segments.


] led the response from France in 1874 with a textbook on the elements of quaternions. To ease the study of ]s, he introduced "biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for ] introduced in chapter 9. Hoüel replaced Hamilton's basis vectors {{math|'''i'''}}, {{math|'''j'''}}, {{math|'''k'''}} with {{math|''i''<sub>1</sub>}}, {{math|''i''<sub>2</sub>}}, and {{math|''i''<sub>3</sub>}}.
'See main article:']


The variety of fonts available led Hoüel to another notational innovation: {{math|''A''}} designates a point, {{math|''a''}} and {{math|a}} are algebraic quantities, and in the equation for a quaternion
== Turn of the century triumph of real Euclidean 3 space ==
:<math>\mathcal{ A} = \cos \alpha + \mathbf{A} \sin \alpha ,</math>
Unfortunately some of Hamilton's supporters, like ], vociferously opposed the growing fields of ] and ] (developed by ] and ], among others), maintaining that quaternions provided a superior notation.
{{math|'''A'''}} is a vector and {{math|''α''}} is an angle. This style of quaternion exposition was perpetuated by ]<ref>] (1881) , link from ]</ref> and ].<ref>A. Macfarlane (1894) '''', B. Westerman, New York, weblink from ]</ref>


] expanded the types of biquaternions, and explored ], a geometry in which the points can be viewed as versors. Fascination with quaternions began before the language of ] and ]s was available. In fact, there was little ] before the ]. The quaternions stimulated these advances: For example, the idea of a ] borrowed Hamilton's term but changed its meaning. Under the modern understanding, any quaternion is a vector in four-dimensional space. (Hamilton's vectors lie in the subspace with scalar part zero.)
The 19th century Darwinist mentality of the time, allowed the respective champion of Quaternion notation and modern vector notation to allow their pet notations to become embroiled in a battle to the death, with the intent that only the strongest notation would be 'fittest' and survive,<ref>Infamous remark by Gibbs, see crow</ref> with the weaker notation left to become extinct.{{Fact|date=January 2008}} Modern notation won the day.


Since quaternions demand their readers to imagine four dimensions, there is a metaphysical aspect to their invocation. Quaternions are a ]. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of ]s and ]s in ], for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus ] and ] made this accommodation, for pragmatism, to avoid the distracting superstructure.<ref>] (1967) ], ]</ref>
Gibbs and Wilson's advocacy of Cartesian coordinates lead them to expropriate i, j, and k, along with the term vector first introduced by Hamilton into their own notational system. The new vector was different from the vector of a quaternion.


For mathematicians the quaternion structure became familiar and lost its status as something mathematically interesting. Thus in England, when ] prepared a paper on biquaternions, it was published in the ] since some novelty in the subject lingered there. Research turned to ]s more generally. For instance, ] and ] considered the number of equations between basis vectors which would be necessary to determine a unique system. The wide interest that quaternions aroused around the world resulted in the ]. In contemporary mathematics, the ] of quaternions exemplifies an ].
As the computational power of quaternions was incorporated into the real three dimensional space, the modern notation grew more powerful, and quaternions lost favor. While ] was alive, quaternions had Tait and his school to develop and champion them, but with his death this trend reversed and other systems began to catch up and eventually surpass his quaternion idea. The book ] written by Gibbs' student ] in 1901 was an important early work that attempted to show that early modern vector notation which included ] could do everything that Hamilton's quaternions could.{{Fact|date=January 2008}} Gibbs was working too hard on statistical mechanics to help with the manuscript. His student, Wilson, based the book on his mentor's lectures.


== Principal publications ==
The dyad product and the dyadics it generated also eventually fell out of favor their functionality being replaced by the matrix.
* 1853 ''Lectures on Quaternions''<ref>, Royal Irish Academy, weblink from ] ''Historical Math Monographs''</ref>
* 1866 ''Elements of Quaternions''<ref>, ] Press. Edited by ], son of the deceased author</ref>
* 1873 ''Elementary Treatise'' by ]<ref></ref>
* 1874 ]: ''Éléments de la Théorie des Quaternions''<ref>J. Hoüel (1874) , Gauthier-Villars publisher, link from ]</ref>
* 1878 ]: Quadrics: Harvard dissertation:<ref>] (1878) , ] 13:222–50, from ]</ref>
* 1882 Tait and ]: ''Introduction with Examples''<ref></ref>
* 1885 ]: Biquaternions<ref>, ] 7(4):293 to 326 from ] early content</ref>
* 1887 Valentin Balbin: (Spanish) ''Elementos de Calculo de los Cuaterniones'', Buenos Aires<ref>Gustav Plarr (1887) in ]</ref>
* 1899 ]: ''Elements'' vol 1, vol 2 1901<ref>Hamilton (1899) ''Elements of Quaternions'' , (1901) . Edited by ]; published by ], now in ]</ref>
* 1901 ] by ] and ] (quaternion ideas without quaternions)
* 1904 ]: third edition of Kelland and Tait's textbook<ref>] (editor) (1904) via ]</ref>
* 1904 ''Bibliography'' prepared for the ] by ]<ref>] (1904) , weblink from Cornell University ''Historical Math Monographs''</ref>
* 1905 C.J. Joly's ''Manual for Quaternions''<ref>] (1905) (1905), originally published by ], now from Cornell University Historical Math Monographs</ref>
* 1940 ] 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 ]s to describe screw displacement in ].


== Octonions ==
An example of the debate at the time over quadrantal versor appears in the quaternion section of the Misplaced Pages biography of the life and thinking of ] who was an avid early participant in these debates.
{{main|Octonion}}


]s were developed independently by ] in 1845 <ref>Penrose 2004 pg 202</ref> and ], 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 , why should you stop there?"{{Sfn|Baez|2002|p=146}}
Some early formulations of ] used a quaternion-based notation (] paired his formulation in 20 equations in 20 variables with a quaternion representation<ref>Maxwell 1873</ref>), but it proved unpopular compared to the ]-based notation of Heaviside. The various notations were, of course, computationally equivalent, the difference being a matter of aesthetics and convenience.


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> which he called ''octaves'', and showed that they were what we now call a ]ed ].<ref>{{citation
The classical vector of a quaternion along with its computational power was ripped out of the classical quaternion multiplied by the square root of minus one and installed into vector analysis. The computational power of the tensor of a quaternion and the versor of a quaternion became the dyadic. The scalar and the three vector went their separate ways.
| last1 = Brown | first1 = Ezra
| last2 = Rice | first2 = Adrian
| doi = 10.1080/0025570X.2022.2125254
| issue = 5
| journal = Mathematics Magazine
| mr = 4522169
| pages = 422–436
| title = An accessible proof of Hurwitz's sums of squares theorem
| volume = 95
| year = 2022}}</ref> Hamilton observed in reply that they were not ], which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it. Hamilton needed a way to distinguish between two different types of double quaternions, the associative ]s 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></ref><ref>See Hamilton's talk to the Royal Irish Academy on the subject</ref>


Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845&nbsp;– 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&nbsp;– or as ''Cayley numbers''.
The 3 &times; 3 matrix the took over the functionality of the dyadic which also fell into obscurity.


The major deduction from the existence of octonions was the ], which follows directly from the product rule from octonions. It had also been previously discovered as a purely algebraic identity by ] in 1818.{{Sfn|Baez|2002|p=146-7}} This sum-of-squares identity is characteristic of ], a feature of complex numbers, quaternions, and octonions.
The scalar-time, 3-vector-space, and matrix-transform had emerged from the quaternion and could now march forward as three different mathematical entities, taking with them the functionality of the 19th century classical quaternion. The old notation was left behind as a relic of the Victorian era.


== Mathematical uses ==
Vector and matrix and modern tensor notation had nearly universally replaced Hamilton's quaternion notation in ] and real Euclidean three space was the mathematical model of choice in ] by the mid-20th century.
{{main|Quaternions}}


Quaternions continued to be a well-studied ''mathematical'' structure in the twentieth century, as the third term in the ] of ] systems over the reals, followed by the ]s, the ]s, the ]s; they are also a useful tool in ], particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the ], is also the simplest non-commutative ].
== Historical metaphysical 19th-century controversy ==


The study of ]s began with ] in 1886, whose system was later simplified by ]; but the modern system was published by ] 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 ]. Both systems are closed under subtraction and multiplication, and are therefore ]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>
The controversy over quaternions was more than a controversy over the best notation. It was a controversy over the nature of space and time. It was a controversy over which of two systems best represented the true nature of space time.


== Quaternions as rotations ==
=== Sign of distance squared ===
{{main| Quaternions and spatial rotation}}


Quaternions are a concise method of representing the ]s of three- and four-dimensional spaces. They have the technical advantage that ]s form the ] cover of the space of three-dimensional rotations.<ref name=CS/>{{rp|ch 2}}
This controversy involves meditating on the question, how much is one unit of distance squared? Hamilton postulated that it was a ].


For this reason, quaternions are used in ],<ref>Ken Shoemake (1985), , '']'', '''19'''(3), 245–254. Presented at ] '85.</ref> ], ],<ref>J. M. McCarthy, 1990, , MIT Press</ref> ], ], ], ], and ]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. '']'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation.<ref>Nick Bobick (February 1998) "", ]</ref> Quaternions have received another boost from ] because of their relation to ]s.
In 1833 before Hamilton invented quaternions he wrote an essay calling real number Algebra the Science of Pure Time.<ref></ref> Classical Quaternions used what we today call ]s to represent distance and real numbers to represent time. To put it in classical quaternion terminology the SQUARE of EVERY VECTOR is a NEGATIVE SCALER<ref></ref> In other words in the classical quaternion system a quantity of distance was a different kind of number from a quantity of time.


== Memorial ==
] did not like the idea of minus one having a square root. Descartes called it an ]<ref>]</ref> number. Hamilton objected to calling the square root of minus one an imaginary number. In Descartes day complex number was a polite term for imaginary number, but they meant the same thing.<ref>]</ref>{{Fact|date=January 2008}}<!--The side note on imaginary being offensive in the 19th century is not all that relevant. The citation only proves that they meant the same thing, but does not tell about the conotation of each -->
Since 1989, the Department of Mathematics of the ] has organized a pilgrimage, where scientists (including physicists ] in 2002, ] in 2005, ] in 2007, and mathematician ] in 2003) take a walk from ] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.<ref> at the ].</ref>


== References ==
When Hamilton speculated that there was not just one, but an infinite number of square roots of minus one, and took three of them to use as a bases for a model of three dimensional space, rivaling ] there immediately arose a controversy about the use of quaternions that escalated after Hamilton's death.{{Fact|date=January 2008}}
* {{citation|first=John C.|last= Baez|title= The Octonions|journal=Bulletin of the American Mathematical Society |series=New Series|volume= 39|issue= 2|pages= 145–205|doi=10.1090/S0273-0979-01-00934-X |year=2002|mr=1886087 |arxiv=math/0105155|s2cid= 586512}}
* ] and ], ''Introduction to Number Theory''. Many editions.
* Johannes C. Familton (2015) , Ph.D. thesis in ] Department of Mathematics Education.
* {{cite journal |last=Whittaker |first=E.T. | authorlink=E. T. Whittaker |title=The sequence of ideas in the discovery of quaternions |journal=Proceedings of the Royal Irish Academy A |year=1944 |volume=50 |pages=93–98 |jstor=20520633 |url=https://www.jstor.org/stable/20520633}}


== Notes ==
== Nature of space and time controversy ==
{{reflist|2}}

Hamilton also on a philosophical level believed space to be of a four dimensional or quaternion nature, with time being the fourth dimension. His quaternions importantly embodied this philosophy. On this last count of the 19 century debate Hamilton in the 21st century has been declared with winner. To an extent any model of space and time as a four dimensional entity on a metaphysical level, can be thought of as type of "quaternion" space, even if on a notational and computational level Hamilton's original four space has continued to evolve.

An element on the other side opposing Hamilton's camp in the 19th century debate believed that real ] three space was the one and only true model mathematical model of the universe in which we live.{{Fact|date=January 2008}} The 19th century advocates of Euclidean three space, have by the 20th century been proven wrong. Obviously in the 21st century the final chapter on the nature of space time has yet to be written. Hamilton was correct in suggesting that the Euclidean real 3-space, universally accepted at the time, might not be the one and only true model of space and time.

== Comparison with modern vector notation ==

Around the turn of the 19th into the 20th century early text books on modern vector analysis<ref>]</ref> did much to move standard notation away from that classical quaternion notation, in favor of modern vector notation based on real Euclidean three space.
=== Reinterpretation of ''i'', ''j'', ''k'' ===

Cartesian Coordinates represented three space with an ordered triplet of real numbers, (x,y,z). Quaternion notation introduced a different representation for the ]:

<center>'''V'''q = xi + yj + zk<ref></ref>
</center>

Cartesian coordinates were separated by commas; but classical quaternions were separated by plus or minus signs, and considered the summation of numbers of different types.

Vector analysis appropriated this form, and indeed the expression above from 1887 looks a lot like a modern vector. But there was an important difference. In the quaternion system each of the terms ''i'',&nbsp;''j'',&nbsp;''k'' is a square root of minus one. But the vector system rejected this. In the vector system ''i'',&nbsp;''j'',&nbsp;and&nbsp;''k'' remained to indicate different orthogonal unit basis vectors. But, unlike the quaternion basis, these were considered to be derived from real Cartesian coordinates, made up of only real numbers. The idea that in any sense ''i'' times ''i'' could equal minus one was rejected.

In vector notation, the ''i'',&nbsp;''j'',&nbsp;and&nbsp;''k'' had come to mean something rather different from what they had in quaternion notation.

=== Four new multiplications ===

The classical quaternion notational system had only one kind of multiplication. But in that system the product of a pure vector of the form 0 + xi +yj + kz with another pure vector produced a quaternion.

To add the functionality of classical quaternions to the real three space early modern vector analysis required four different kinds of multiplication.<ref>]</ref> In addition to regular multiplication which got the name '']'' to distinguish it from the three new kinds, it required ''two'' different kinds of ''vector products''. The fourth product in the new system was called the '']''.

==== Dot product ====

The first new product was called the "scalar product"<ref>Vector Analysis Gibbs-Wilson 1901 pg 55</ref> of two vectors, and was represented with a dot. Computationally it was equivalent to the operation of taking the negative of the scalar part of the quaternion product of the vector parts of two quaternions, so the new a · b corresponded to the old quaternion operation &minus;'''S'''(VA &times; VB).

This meant that ''i''&nbsp;·&nbsp;''i'' in the new system was +1. And the type of "vector" in the modern system was different as well; the new "vector" was not the vector of the classical quaternion system, because it did not consist of a triplet of imaginary components. Rather it was a "modern vector" which had been striped of the classical property that the product was ''ii''&nbsp;=&nbsp;&minus;1.

====Cross product ====

The second new product in the new system was the cross product, that was computationally similar to '''V'''(VA &times; VB) or taking the vector part of the product of the vector part of two classical quaternions.

In the new notational system it was still true that (''i''&nbsp;&times;&nbsp;''j'') = ''k'', however, unlike (''i''&nbsp;&nbsp;''i'')&nbsp;=&nbsp;+1 in the second new type of multiplication (''i''&nbsp;&times;&nbsp;''i'')&nbsp;=&nbsp;0.

====Dyadic product ====

The third new product, in the early modern system was the ] product. It was needed to perform some of the linear vector functions,<ref>]</ref> that quaternions multiplied into vectors had performed. A dyad was written in some early text books as AB<ref>]</ref> without a dot or cross in the middle. Three dyads made up a ]. This vector product took over the quaternion operations of version and tension. This early aspect of the Gibbs/Wilson system has become more obscure over time.

=== New system questioned ===

In the 19th century supporters of classical quaternion notation and modern vector notation debated over which was best notational system, as described above.

To provide a vastly oversimplified, short introduction to what motivated these debates consider that in the new notation that i · i =+1, j · j =+1 and k · k =+1. So apparently i,j,k in the modern vector notational system represent three new square roots of positive one.

In the new notational system i, j, and k also apparently represented square roots of zero, since i &times; i = 0 , j &times; j = 0, k &times; k = 0. The new notation system was then based on numbers that were the square root of both zero and positive one. Advocates of the classical quaternion system liked the older idea of a single vector product with a unit vector multiplied by itself being negative one better.

=== The quadrantal versor argument ===

An important argument in favor of classical quaternion notation was that i, j, and k doubled as quadrantal ]s. '''i &times; (i &times; j) = &minus;j''' and '''(i &times; i) &times; j = &minus;j''' This was not the case in the new notational system of modern vector analysis because their cross product was not associative. In the new notation (i &times; i) &times; j = 0, and however i &times; (i &times; j) = &minus;j.

=== Turn of the century triumph of modern vector notation===

Modern vector notation eventually replaced the classical concept of the vector of a quaternion.

Advocates of Cartesian coordinates expropriated i, j, and k, along with the term vector into the modern notational system. The new modern vector was different from the vector of a quaternion.

As the computational power of quaternions was incorporated into modern vector notation,<ref>]</ref> classical quaternion notation lost favor.

The classical vector of a quaternion was multiplied by the square root of minus one and then again by negative one, and installed into modern vector analysis. The computational power of the classical quaternion vector product was exported into the new notation as the new cross and dot products. The computational power of the tensor of a quaternion and the versor of a quaternion became the dyadic, and then the matrix. The scalar and the three vector went their separate ways.

The 3 &times; 3 matrix rotation matrix took over the functionality of the dyadic which also fell into obscurity.

The scalar-time, 3-vector-space, and rotation matrix-transform had emerged from the classical quaternion and could now march forward as three different mathematical entities, taking with them the functionality of the 19th century classical quaternion. The old notation was left behind as a relic of the Victorian era.

Modern vector and matrix and modern tensor notation had nearly universally replaced Hamilton's quaternion notation in ] and real Euclidean three space was the mathematical model of choice in ] by the mid-20th century.

== 20th-century extensions ==

In the early 20th century, there has been considerable effort with quaternions and other hypercomplex numbers, due to their apparent relation with space-time geometry. ], ], or ], just to mention a few concepts that were looked at.

Descriptions of physics using quaternions turned out to either not work, or to not yield "new" physics (i.e. one might just as well continue to not use quaternions).

The conclusion is that if quaternions are not required, they are a "nice-to-have", a mathematical curiosity, at least from the viewpoint of physics.

The historical development went to ] for multi-dimensional analysis,{{Dubious|date=March 2009}} ] for description of gravity, and ] for describing internal (non-spacetime) symmetries.{{Dubious|date=March 2009}} All three approaches (Clifford, Lie, tensors) include quaternions, so in that respect they've become quite "mainstream", so to speak.

== Modern synthesis ==
{{Unreferenced|date=March 2009}}
{{Merge|date=March 2009}}
Quaternions have had a revival in the late 20th century, primarily due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. For this reason, quaternions are used in ], ], ], ], ], ], and ]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions.

An important conceptual step forward has been the wider realisation{{Who|date=March 2009}} that the "vector part" of quaternions most naturally represents not vectors in 3D but ]s, the ]s of vectors. Pseudovectors in 3D (also known as ]s) are associated with the direction of oriented 2D planes. A difference with vectors is that whereas vectors change sign under co-ordinate inversion, '''a''' → &minus;'''a''', '''b''' → −'''b''', pseudovectors in 3D (bivectors) remain unchanged, '''a''' ^ '''b''' → &minus;'''a''' ^ &minus;'''b''' = '''a''' ^ '''b'''.

The two systems, vector and quaternion, are combined by grafting a new element onto the quaternion system, the unit ] ''i'', that has the property ''i''<sup>2</sup>&nbsp;=&nbsp;&minus;1, and is defined to multiplicatively commute with the quaternions '''i''', '''j''' and '''k'''.

This generates three further new linearly independent bases for the algebra, conventionally '''e'''<sub>1</sub> = − ''i'' '''i''', '''e'''<sub>2</sub> = &minus;''i'' '''j''' and '''e'''<sub>3</sub> = − ''i'' '''k''', which have the property that
:('''e'''<sub>1</sub>)<sup>2</sup> = ('''e'''<sub>2</sub>)<sup>2</sup> = ('''e'''<sub>3</sub>)<sup>2</sup> = +1,
and linear combinations of them '''a''' and '''b''' have the product
:'''ab''' = '''a''' · '''b''' (scalar) + ''i'' '''a''' &times; '''b''' (pseudovector).
The new bases therefore behave appropriately for the basis elements of a space of modern vectors, with the unit pseudoscalar ''i'' identifiable as the unit scalar triple product ('''e'''<sub>1</sub>&nbsp;^&nbsp;'''e'''<sub>2</sub>&nbsp;^&nbsp;'''e'''<sub>3</sub>).

This augmented system, which is in fact a ] of the ] ''C''ℓ<sub>3,0</sub>('''R''') with its defining Clifford property
:'''e'''<sub>&alpha;</sub>'''e'''<sub>&beta;</sub> = &minus;'''e'''<sub>&beta;</sub>'''e'''<sub>&alpha;</sub> for &alpha;&nbsp;1≠&nbsp;&beta;,
restores to the vectors the single product of Hamilton; and preserves the structure of the Hamiltonian quaternions with all their rotational magic; but it also cleanly distinguishes the different types of geometric objects – scalars, vectors, pseudovectors, and pseudoscalars; and satisfies the wish of the vector pioneers for vectors which square to +1, rather than &minus;1. Best of all, it generalises readily to any number of dimensions of the underlying ground space (taking quaternion-like rotation techniques with it), and so quaternions are no longer left perceived as isolated, a strange freak of the 3D world.

This re-integration into the mainstream has led to a renewed rediscovery and interest in the techniques for geometry pioneered by the ] methods in the 19th century.

== See also ==
* ]

== References ==
{{reflist}}


] ]
]

Latest revision as of 23:56, 3 January 2025

Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:
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 multiplicationi = j = k = ijk = −1& cut it on a stone of this bridge.

In mathematics, quaternions are a non-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.

Hamilton's discovery

In 1843, Hamilton knew that the complex numbers could be viewed as points in a 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 difficulty defining the appropriate multiplication.

According to a letter Hamilton wrote later to his son Archibald:

Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edwin 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."

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:

i = j = k = ijk = −1

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 Philosophical Magazine communicated Hamilton's exposition of quaternions. In 1853 he issued Lectures on Quaternions, a comprehensive treatise that also described biquaternions. The facility of the algebra in expressing geometric relationships led to broad acceptance of the method, several compositions by other authors, and stimulation of applied algebra generally. As mathematical terminology has grown since that time, and usage of some terms has changed, the traditional expressions are referred to as classical Hamiltonian quaternions.

Precursors

Hamilton's innovation consisted of expressing quaternions as an algebra over R. The formulae for the multiplication of quaternions are implicit in the four squares formula devised by Leonhard Euler in 1748. In 1840, Olinde Rodrigues used spherical trigonometry and developed a formula closely related to quaternion multiplication in order to describe the new axis and angle of two combined rotations.

Response

The special claims of quaternions as the algebra of four-dimensional space were challenged by James Cockle with his exhibits in 1848 and 1849 of tessarines and coquaternions as alternatives. Nevertheless, these new algebras from Cockle were, in fact, to be found inside Hamilton's biquaternions. From Italy, in 1858 Giusto Bellavitis responded to connect Hamilton's vector theory with his theory of equipollences of directed line segments.

Jules Hoüel led the response from France in 1874 with a textbook on the elements of quaternions. To ease the study of versors, he introduced "biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for spherical trigonometry introduced in chapter 9. Hoüel replaced Hamilton's basis vectors i, j, k with i1, i2, and i3.

The variety of fonts available led Hoüel to another notational innovation: A designates a point, a and a are algebraic quantities, and in the equation for a quaternion

A = cos ⁡ α + A sin ⁡ α , {\displaystyle {\mathcal {A}}=\cos \alpha +\mathbf {A} \sin \alpha ,}

A is a vector and α is an angle. This style of quaternion exposition was perpetuated by Charles-Ange Laisant and Alexander Macfarlane.

William K. Clifford expanded the types of biquaternions, and explored elliptic space, a geometry in which the points can be viewed as versors. Fascination with quaternions began before the language of set theory and mathematical structures was available. In fact, there was little mathematical notation before the Formulario mathematico. The quaternions stimulated these advances: For example, the idea of a vector space borrowed Hamilton's term but changed its meaning. Under the modern understanding, any quaternion is a vector in four-dimensional space. (Hamilton's vectors lie in the subspace with scalar part zero.)

Since quaternions demand their readers to imagine four dimensions, there is a metaphysical aspect to their invocation. Quaternions are a philosophical object. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of dot products and cross products in three-dimensional space, for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus Willard Gibbs and Oliver Heaviside made this accommodation, for pragmatism, to avoid the distracting superstructure.

For mathematicians the quaternion structure became familiar and lost its status as something mathematically interesting. Thus in England, when Arthur Buchheim prepared a paper on biquaternions, it was published in the American Journal of Mathematics since some novelty in the subject lingered there. Research turned to hypercomplex numbers more generally. For instance, Thomas Kirkman and Arthur Cayley considered the number of equations between basis vectors which would be necessary to determine a unique system. The wide interest that quaternions aroused around the world resulted in the Quaternion Society. In contemporary mathematics, the division ring of quaternions exemplifies an algebra over a field.

Principal publications

Octonions

Main article: Octonion

Octonions were developed independently by Arthur Cayley in 1845 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 , why should you stop there?"

Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843, presenting a kind of double quaternion, which he called octaves, and showed that they were what we now call a normed division algebra. Hamilton observed in reply that they were not associative, which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it. Hamilton needed a way to distinguish between two different types of double quaternions, the associative biquaternions 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.

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 eight squares theorem, which follows directly from the product rule from octonions. It had also been previously discovered as a purely algebraic identity by Carl Ferdinand Degen in 1818. This sum-of-squares identity is characteristic of composition algebra, a feature of complex numbers, quaternions, and octonions.

Mathematical uses

Main article: Quaternions

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 octonions, the sedenions, the trigintaduonions; they are also a 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 quaternions 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 rings, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.

Quaternions as rotations

Main article: Quaternions and spatial rotation

Quaternions are a concise method of representing the automorphisms 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.

For this reason, quaternions are used in computer graphics, control theory, robotics, 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. Quaternions have received another boost from number theory because of their relation to quadratic forms.

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.

References

Notes

  1. Simon L. Altmann (1989). "Hamilton, Rodrigues and the quaternion scandal". Mathematics Magazine. Vol. 62, no. 5. pp. 291–308. doi:10.2307/2689481. JSTOR 2689481.
  2. W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College Dublin
  3. Friedberg, Richard M. (2022). "Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un système solide...", translation and commentary". arXiv:2211.07787.
  4. ^ John H. Conway & Derek A. Smith (2003) On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, ISBN 1-56881-134-9
  5. Giusto Bellavitis ( 1858) Calcolo dei Quaternioni di W.R. Hamilton e sua Relazione col Metodo delle Equipollenze, link from HathiTrust
  6. Charles Laisant (1881) Introduction a la Méthode des Quaternions, link from Google Books
  7. A. Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org
  8. Michael J. Crowe (1967) A History of Vector Analysis, University of Notre Dame Press
  9. Lectures on Quaternions, Royal Irish Academy, weblink from Cornell University Historical Math Monographs
  10. Elements of Quaternions, University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author
  11. Elementary Treatise on Quaternions
  12. J. Hoüel (1874) Éléments de la Théorie des Quaternions, Gauthier-Villars publisher, link from Google Books
  13. Abbott Lawrence Lowell (1878) 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
  14. Introduction to Quaternions with Numerous Examples
  15. "A Memoir on biquaternions", American Journal of Mathematics 7(4):293 to 326 from Jstor early content
  16. Gustav Plarr (1887) Review of Valentin Balbin's Elementos de Calculo de los Cuaterniones in Nature
  17. Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co., now in Internet Archive
  18. C. G. Knott (editor) (1904) Introduction to Quaternions, 3rd edition via Hathi Trust
  19. Alexander Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, weblink from Cornell University Historical Math Monographs
  20. Charles Jasper Joly (1905) A Manual for Quaternions (1905), originally published by Macmillan Publishers, now from Cornell University Historical Math Monographs
  21. Penrose 2004 pg 202
  22. Baez 2002, p. 146.
  23. See Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...'
  24. Brown, Ezra; Rice, Adrian (2022), "An accessible proof of Hurwitz's sums of squares theorem", Mathematics Magazine, 95 (5): 422–436, doi:10.1080/0025570X.2022.2125254, MR 4522169
  25. 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
  26. See Hamilton's talk to the Royal Irish Academy on the subject
  27. Baez 2002, p. 146-7.
  28. Hardy and Wright, Introduction to Number Theory, §20.6-10n (pp. 315–316, 1968 ed.)
  29. Ken Shoemake (1985), Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), 245–254. Presented at SIGGRAPH '85.
  30. J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press
  31. Nick Bobick (February 1998) "Rotating Objects Using Quaternions", Game Developer (magazine)
  32. Hamilton walk at the National University of Ireland, Maynooth.
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