In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s are a non-
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
number system that extends the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. Quaternions and their applications to rotations were first described in print by
Olinde Rodrigues
Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors, ...
in all but name in 1840, but independently discovered by Irish mathematician Sir
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s could be viewed as
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
s in a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
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
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
. 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
The Royal Canal ( ga, An Chanáil RÃoga) is a canal originally built for freight and passenger transportation from Dublin to Longford in Ireland. It is one of two canals from Dublin to the River Shannon and was built in direct competition ...
in
Dublin
Dublin (; , or ) is the capital and largest city of Republic of Ireland, Ireland. On a bay at the mouth of the River Liffey, it is in the Provinces of Ireland, province of Leinster, bordered on the south by the Dublin Mountains, a part of th ...
. While they walked across Brougham Bridge (now
Broom Bridge
Broom Bridge ( Irish: ''Droichead Broome''), also called Broome Bridge, and sometimes Brougham Bridge, is a bridge along Broombridge Road which crosses the Royal Canal in Cabra, Dublin, Ireland. Broome Bridge is named after William Broome, on ...
), 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:
:
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
The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Oxford Dictionary of National Biography, Oxford Univer ...
communicated Hamilton's exposition of quaternions. In 1853 he issued ''Lectures on Quaternions'', a comprehensive treatise that also described
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
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
classical Hamiltonian quaternions.
Precursors
Hamilton's innovation consisted of expressing quaternions as an
algebra over . The formulae for the multiplication of quaternions are implicit in the
four squares formula
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four square (algebra), squares, is itself a sum of four squares.
Algebraic identity
For any pair of quadruples from a commutative ring, th ...
devised by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in 1748;
Olinde Rodrigues
Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors, ...
applied this formula to representing rotations in 1840.
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
& Derek A. Smith (2003) ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry'', A K Peters
A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the '' Journal ...
,
Response
The special claims of quaternions as the algebra of
four-dimensional space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
were challenged by
James Cockle
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterho ...
with his exhibits in 1848 and 1849 of
tessarine
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as
:(u,v)(w,z) = (u w - v z, u z ...
s and
coquaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in ...
s as alternatives. Nevertheless, these new algebras from Cockle were, in fact, to be found inside Hamilton’s
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s. From Italy, in 1858
Giusto Bellavitis responded to connect Hamilton’s vector theory with his theory of
equipollences of directed line segments.
Jules Hoüel
Guillaume-Jules Hoüel (7 April 1823 – 14 June 1886) was a French people, French mathematician. He entered the École Normale Supérieure in 1843 and received his doctoral degree in 1855 from the University of Paris, Sorbonne. He was sought by ...
led the response from France in 1874 with a textbook on the elements of quaternions. To ease the study of
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s, he introduced "biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
introduced in chapter 9. Hoüel replaced Hamilton’s basis vectors , , with , , and .
The variety of fonts available led Hoüel to another notational innovation: designates a point, and are algebraic quantities, and in the equation for a quaternion
:
is a vector and is an angle. This style of quaternion exposition was perpetuated by
Charles-Ange Laisant
Charles-Ange Laisant (1 November 1841 – 5 May 1920), French politician and mathematician, was born at Indre, near Nantes on 1 November 1841, and was educated at the École Polytechnique as a military engineer. He was a Freemason and a liberta ...
and
Alexander Macfarlane
Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.
Life
Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He s ...
.
William K. Clifford expanded the types of biquaternions, and explored
elliptic space
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, a geometry in which the points can be viewed as versors. Fascination with quaternions began before the language of
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
s was available. In fact, there was little
mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
before the
Formulario mathematico
''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book
There are many editions. Here are two:
* (French) Published 1901 by Gauthier-Villars, Paris. 230p.OpenLibrary OL15255022W
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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 product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
s and
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus
Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
and
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
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
Arthur Buchheim (1859-1888) was a British mathematician.
His father Carl Adolf Buchheim was professor of German language at King's College London. After attending the City of London School, Arthur Buchheim obtained an open scholarship at New Colle ...
prepared a paper on biquaternions, it was published in the
American Journal of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press.
History
The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...
since some novelty in the subject lingered there. Research turned to
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
s more generally. For instance,
Thomas Kirkman
Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...
and
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, C ...
considered the number of equations between basis vectors would be necessary to determine a unique system. The wide interest that quaternions aroused around the world resulted in the
Quaternion Society
The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the ac ...
. In contemporary mathematics, the
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
of quaternions exemplifies an
algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
.
Principal publications
* 1853 ''Lectures on Quaternions''
* 1866 ''Elements of Quaternions''
* 1873 ''Elementary Treatise'' by
Peter Guthrie Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...
* 1874
Jules Hoüel
Guillaume-Jules Hoüel (7 April 1823 – 14 June 1886) was a French people, French mathematician. He entered the École Normale Supérieure in 1843 and received his doctoral degree in 1855 from the University of Paris, Sorbonne. He was sought by ...
: ''Éléments de la Théorie des Quaternions''
* 1878
Abbott Lawrence Lowell
Abbott Lawrence Lowell (December 13, 1856 – January 6, 1943) was an American educator and legal scholar. He was President of Harvard University from 1909 to 1933.
With an "aristocratic sense of mission and self-certainty," Lowell cut a large f ...
: Quadrics: Harvard dissertation:
* 1882 Tait and
Philip Kelland
Philip Kelland PRSE FRS (17 October 1808 – 8 May 1879) was an English mathematician. He was known mainly for his great influence on the development of education in Scotland.
Life
Kelland was born in 1808 the son of Philip Kelland (d.1847), ...
: ''Introduction with Examples''
* 1885
Arthur Buchheim
Arthur Buchheim (1859-1888) was a British mathematician.
His father Carl Adolf Buchheim was professor of German language at King's College London. After attending the City of London School, Arthur Buchheim obtained an open scholarship at New Colle ...
: Biquaternions
* 1887 Valentin Balbin: (Spanish) ''Elementos de Calculo de los Cuaterniones'', Buenos Aires
* 1899
Charles Jasper Joly
Charles Jasper Joly (27 June 1864 – 4 January 1906) was an Irish mathematician and astronomer who became Royal Astronomer of Ireland.Obituary, New York Times, 5 January 1906
Life
He was born at St Catherine's Rectory, Hop Hill, Tullamore, ...
: ''Elements'' vol 1, vol 2 1901
* 1901
Vector Analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
by
Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
and
Edwin Bidwell Wilson
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...
(quaternion ideas without quaternions)
* 1904
Cargill Gilston Knott
Cargill Gilston Knott FRS, FRSE LLD (30 June 1856 – 26 October 1922) was a Scottish physicist and mathematician who was a pioneer in seismological research. He spent his early career in Japan. He later became a Fellow of the Royal Society, ...
: third edition of Kelland and Tait's textbook
* 1904 ''Bibliography'' prepared for the
Quaternion Society
The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the ac ...
by
Alexander Macfarlane
Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.
Life
Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He s ...
* 1905 C.J. Joly's ''Manual for Quaternions''
* 1940
Julian Coolidge
Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian and a professor and chairman of the Harvard University Mathematics Department.
Biography
Born in Brookline, Massachusetts, he graduated from Har ...
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
In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of ...
s to describe screw displacement in
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
.
Octonions
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s were developed independently by
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, C ...
in 1845 and
John T. Graves
John Thomas Graves (4 December 1806 – 29 March 1870) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions in October 1843 and then discover ...
, 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
he three imaginary units 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 that is called an ''octonion'', and showed that they were what we now call a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
ed
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
; Graves called them ''octaves''. Hamilton needed a way to distinguish between two different types of double quaternions, the associative
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
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. observed in reply that they were not
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
, 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
eight squares theorem, which follows directly from the product rule from octonions, 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
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution c ...
, a feature of complex numbers, quaternions, and octonions.
Mathematical uses
Quaternions continued to be a well-studied ''mathematical'' structure in the twentieth century, as the third term in the
Cayley–Dickson construction
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by ...
of
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
systems over the reals, followed by the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s and the
sedenion
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s; they are also a useful tool in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, 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
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
, is also the simplest non-commutative
Sylow group
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...
.
The study of
integral quaternions began with
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...
in 1886, whose system was later simplified by
Leonard Eugene Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ...
; but the modern system was published by
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
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
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include number ...
. Both systems are closed under subtraction and multiplication, and are therefore
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
s, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.
Quaternions as rotations
Quaternions are a concise method of representing the
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of three- and four-dimensional spaces. They have the technical advantage that
unit quaternion
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s form the
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
cover of the space of three-dimensional rotations.
[
For this reason, quaternions are used in ]computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
, signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, attitude control
Attitude control is the process of controlling the orientation of an aerospace vehicle with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc.
Controlling vehicle ...
, physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
, and orbital mechanics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. ''Tomb Raider
''Tomb Raider'', also known as ''Lara Croft: Tomb Raider'' from 2001 to 2008, is a media franchise that originated with an action-adventure video game series created by British gaming company Core Design. Formerly owned by Eidos Interactive, th ...
'' (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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
because of their relation to quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s.
Memorial
Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth
The National University of Ireland, Maynooth (NUIM; ga, Ollscoil na hÉireann Mhá Nuad), commonly known as Maynooth University (MU), is a constituent university of the National University of Ireland in Maynooth, County Kildare, Ireland. It w ...
has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
in 2002, Steven Weinberg
Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interactio ...
in 2005, Frank Wilczek
Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direct ...
in 2007, and mathematician Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
in 2003) take a walk from Dunsink Observatory
The Dunsink Observatory is an astronomical observatory established in 1785 in the townland of Dunsink in the outskirts of the city of Dublin, Ireland.Alexander Thom''Irish Almanac and Official Directory''7th ed., 1850 p. 258. Retrieved: 2011-02-2 ...
to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.Hamilton walk
at the National University of Ireland, Maynooth
The National University of Ireland, Maynooth (NUIM; ga, Ollscoil na hÉireann Mhá Nuad), commonly known as Maynooth University (MU), is a constituent university of the National University of Ireland in Maynooth, County Kildare, Ireland. It w ...
.
References
*
*
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and
E. M. Wright
Sir Edward Maitland Wright (13 February 1906, Farnley – 2 February 2005, Reading) was an English mathematician, best known for co-authoring ''An Introduction to the Theory of Numbers'' with G. H. Hardy.
Career
He was born in Farnl ...
, ''Introduction to Number Theory''. Many editions.
* Johannes C. Familton (2015
Quaternions: A History of Complex Non-commutative Rotation Groups in Theoretical Physics Ph.D. thesis in
Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
Department of Mathematics Education.
Notes
{{reflist, 2
Historical treatment of quaternions