William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics,^{[1]} geometry,^{[2]} and computing.^{[3]} Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mindstuff".
William Clifford  

William Kingdon Clifford (1845–1879)  
Born  4 May 1845 
Died  3 March 1879 (aged 33) 
Residence  England 
Nationality  English 
Alma mater  King's College London Trinity College, Cambridge 
Known for  Clifford algebra Clifford's theorem Clifford–Klein form Clifford parallel Bessel–Clifford function Dual quaternion Elements of Dynamic 
Spouse(s)  Lucy Clifford (18751879) 
Scientific career  
Fields  Mathematics Philosophy 
Institutions  University College London 
Doctoral students  Arthur Black 
Influences  Georg Friedrich Bernhard Riemann Nikolai Ivanovich Lobachevsky 
Born at Exeter, William Clifford showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fellow in 1868, after being second wrangler in 1867 and second Smith's prizeman.^{[4]} ^{[5]} Being second was a fate he shared with others who became famous mathematicians, including William Thomson (Lord Kelvin) and James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe the solar eclipse of December 22, 1870. During that voyage he survived a shipwreck along the Sicilian coast.^{[6]}
In 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society.^{[4]} He was also a member of the London Mathematical Society and the Metaphysical Society.
On 7 April 1875 Clifford married Lucy Lane.^{[7]} In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A halfyear holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children.
Clifford enjoyed entertaining children and wrote a collection of fairy stories, The Little People.^{[8]}
Clifford and his wife are buried in London's Highgate Cemetery just north of the grave of Karl Marx, and near the graves of George Eliot and Herbert Spencer.
"Clifford was above all and before all a geometer." (H. J. S. Smith).^{[4]} The discovery of nonEuclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry".^{[9]} In 1870 he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation^{[10]} of Riemann's paper was published in Nature in 1873. His report at Cambridge, On the SpaceTheory of Matter, was published in 1876, anticipating Albert Einstein’s general relativity by 40 years. Clifford elaborated elliptic space geometry as a nonEuclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels.
Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook Elements of Dynamic. His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe.^{[11]}
In 1878 Clifford published a seminal work, building on Grassmann's extensive algebra.^{[12]} He had succeeded in unifying the quaternions, developed by William Rowan Hamilton, with Grassmann's outer product (also known as the exterior product). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz. The inner product equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vectorlike properties, including a directional bias.
Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal^{[13]} of creating an algebra that mirrors the movements and projections of objects in 3dimensional space.^{[14]}
Moreover, Clifford's algebraic schema extends to higher dimensions. The algebraic operations have the same symbolic form as they do in 2 or 3dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes  as real algebras  have been identified in other mathematical systems beyond simply the quaternions.^{[15]}
The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a threedimensional sphere embedded in a fourdimensional space. Quaternion versors, which inhabit this 3sphere, provide a representation of the rotation group SO(3). Clifford noted that Hamilton’s biquaternions were a tensor product of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from splitcomplex numbers D or from the dual numbers N. In terms of tensor products, produces splitbiquaternions, while forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics.
As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, "mindstuff" and the "tribal self". The former symbolizes his metaphysical conception, suggested to him by his reading of Spinoza.^{[4]} Sir Frederick Pollock wrote about Clifford as follows:
Briefly put, the conception is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mindstuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon. Matter and the sensible universe are the relations between particular organisms, that is, mind organized into consciousness, and the rest of the world. This leads to results which would in a loose and popular sense be called materialist. But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism.^{[4]}
Clifford himself defined "mindstuff" as follows (1878, "On the Nature of ThingsinThemselves", Mind, Vol. 3, No. 9, pp. 57–67):
That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mindstuff. A moving molecule of inorganic matter does not possess mind or consciousness ; but it possesses a small piece of mindstuff. When molecules are so combined together as to form the film on the under side of a jellyfish, the elements of mindstuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mindstuff are so combined as to form some kind of consciousness; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mindstuff takes the form of a human consciousness, having intelligence and volition.
The other phrase, "tribal self", gives the key to Clifford's ethical view, which explains conscience and the moral law by the development in each individual of a "self", which prescribes the conduct conducive to the welfare of the "tribe." Much of Clifford's contemporary prominence was due to his attitude toward religion. Animated by an intense love of his conception of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism, and to put the claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism; and Clifford was regarded as a dangerous champion of the antispiritual tendencies then imputed to modern science.^{[4]} There has also been debate on the extent to which Clifford’s doctrine of "concomitance" or "psychophysical parallelism" influenced John Hughlings Jackson's model of the nervous system and through him the work of Janet, Freud, Ribot, and Ey.^{[16]}
In his essay, “The Ethics of Belief” published in 1877, Clifford argued that it was immoral to believe things for which one lacks evidence. He describes a shipowner who planned to send to sea an old and not well built ship full of passengers. The shipowner had doubts suggested to him that the ship might not be seaworthy. “These doubts preyed upon his mind, and made him unhappy.” He considered having the ship refitted even though it would be expensive. At last, “He succeeded in overcoming these melancholy reflections.” He watched the ship depart, “with a light heart… and he got his insurance money when she went down in midocean and told no tales.” ^{[17]}
Clifford argued that the shipowner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound. “[H]e had no right to believe on such evidence as was before him.” (The italics are in the original.) Clifford famously concludes, “it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." ^{[18]}
As such, he was arguing in direct opposition to religious thinkers for whom "blind faith" (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper was famously attacked by pragmatist philosopher William James in his "Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism, faith, and overbelief.
Though Clifford never constructed a full theory of spacetime and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book Elements of Dynamic (1878), he introduced "quasiharmonic motion in a hyperbola". He wrote an expression for a parametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states,
This passage makes reference to biquaternions, though Clifford made these into splitbiquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity. Clifford also discussed his views in On the SpaceTheory of Matter in 1876.
In 1910 William Barrett Frankland quoted the SpaceTheory of Matter in his book on parallelism.^{[20]} He wrote:
Years later, after general relativity had been advanced by Albert Einstein, various authors noted that Clifford had anticipated Einstein:
In 1923 Hermann Weyl mentioned Clifford^{[21]} as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity.
In 1940 Eric Temple Bell published his The Development of Mathematics. There on pages 359 and 360 he discusses the prescience of Clifford on relativity:
Also in 1960, at Stanford University for the International Congress for Logic, Methodology, and Philosophy of Science, John Archibald Wheeler introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator.^{[22]}
In his The Natural Philosophy of Time (1961, 1980) Gerald James Whitrow recalls Clifford's prescience by quoting him to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology (1st ed pp 246,7; 2nd ed p 291).
In 1970 Cornelius Lanczos summarizes Clifford's premonitions this way:
In 1973 Banesh Hoffmann wrote:
In 1990 Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight. They conclude "it was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity". To explain the lack of recognition of Clifford's prescience, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued." ^{[25]} In 1992 Farwell and Knee continued their study with "The Geometric Challenge of Riemann and Clifford"^{[26]} They "hold that once tensors had been used in the theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered."
The academic journal Advances in Applied Clifford Algebras publishes on Clifford’s legacy in kinematics and abstract algebra.
Alexander McAulay (9 December 1863 – 6 July 1931) was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biquaternions".
McAulay was born on 9 December 1863 and attended Kingswood School in Bath. He proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra. In 1883 he published an article "Some general theorems in quaternion integration". McAulay took his degree in 1886, and began to reflect on the instruction of students in quaternion theory. In an article "Establishment of the fundamental properties of quaternions" he suggested improvements to the texts then in use. He also wrote a technical article on integration.
Departing for Australia, he lectured at Ormond College, University of Melbourne from 1893 to 1895. As a distant correspondent, he participated in a vigorous debate about the place of quaternions in physics education. In 1893 his book Utility of Quaternions in Physics was published. A. S. Hathaway contributed a positive review and Peter Guthrie Tait praised
it in these terms:
Here, at last, we exclaim, is a man who has caught the full spirit of the quaternion system: the real aestus, the awen of the Welsh Bards, the divinus afflatus that transports the poet beyond the limits of sublunary things! Intuitively recognizing its power, he snatches up the magnificent weapon which Hamilton tenders us all, and at once dashes off to the jungle on the quest of big game.McAulay took up the position of Professor of Physics in Tasmania from 1896 until 1929, at which time his son Alexander Leicester McAulay took over the position for the next thirty years.
Following William Kingdon Clifford who had extended quaternions to dual quaternions, McAulay made a special study of this hypercomplex number system. In 1898 McAulay published, through Cambridge University Press, his Octonions: a Development of Clifford's Biquaternions.
McAulay died on 6 July 1931.
His brother Francis Macaulay, who stayed in England, also contributed to ring theory. The University of Tasmania has commemorated the McAulays' contributions in Winter Public Lectures.
Bessel–Clifford functionIn mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If
is the entire function defined by means of the reciprocal Gamma function, then the Bessel–Clifford function is defined by the series
The ratio of successive terms is z/k(n + k), which for all values of z and n tends to zero with increasing k. By the ratio test, this series converges absolutely for all z and n, and uniformly for all regions with bounded z, and hence the Bessel–Clifford function is an entire function of the two complex variables n and z.
Chris J. L. DoranChris J. L. Doran is a physicist, Director of Studies in Natural Sciences for Sidney Sussex College, Cambridge. He founded Geomerics, and is its Chief Operating Officer.
Doran obtained his Ph.D. in 1994 on the topic of Geometric Algebra and its Application to Mathematical Physics. He was an EPSRC Advanced Fellow from 1999 to 2004. In 2004, he became Enterprise Fellow of the Royal Society of Edinburgh.
Doran has been credited, together with Anthony N. Lasenby, Joan Lasenby and Steve Gull, for raising the interest of the physics community to the mathematical language and methods of geometric algebra and geometric calculus. These have been rediscovered and refined by David Hestenes, who built on the fundamental work of William Kingdon Clifford and Hermann Grassmann. In 1998, together with Lasenby and Gull, he proposed the gauge theory gravity.He took a break from academics in 2005, and he subsequently founded the software company Geomerics, making use of his knowledge of mathematics. His research interests relate to applied mathematics and theoretical physics, in particular quantum theory, gravitation, geometric algebra and computational geometry.
Doran has authored more than 50 scientific papers.
Clifford's circle theoremsIn geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.
Clifford analysisClifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on and their conformal equivalents on the sphere, the Laplacian in euclidean nspace and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on Spinc manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations.
Clifford parallelIn elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, but in fact the "lines" of elliptic geometry are curves, and they have finite length, unlike the lines of Euclidean geometry. The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
Clifford torusIn geometric topology, the Clifford torus is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles S^{1}_{a} and S^{1}_{b}. It is named after William Kingdon Clifford. It resides in R^{4}, as opposed to in R^{3}. To see why R^{4} is necessary, note that if S^{1}_{a} and S^{1}_{b} each exist in their own independent embedding spaces R^{2}_{a} and R^{2}_{b}, the resulting product space will be R^{4} rather than R^{3}. The historically popular view that the cartesian product of two circles is an R^{3} torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R^{3} is an asymmetric reduceddimension projection of the maximally symmetric Clifford torus embedded in R^{4}. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lowerdimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If S^{1}_{a} and S^{1}_{b} each has a radius of , their Clifford torus product will fit perfectly within the unit 3sphere S^{3}, which is a 3dimensional submanifold of R^{4}. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C^{2}, since C^{2} is topologically equivalent to R^{4}.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in threedimensional Euclidean space, the square torus can also be embedded into threedimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.
Elements of DynamicElements of Dynamic is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positively, has remained a standard reference since its appearance, and is now available online as a Historical Math Monograph from Cornell University.
On page 95 Clifford deconstructed the quaternion product of William Rowan Hamilton into two separate "products" of two vectors: vector product and scalar product, anticipating the complete severance seen in Vector Analysis (1901). Elements of Dynamic was the debut of the term crossratio for a fourargument function frequently used in geometry.
Clifford uses the term twist to discuss (pages 126 to 131) the screw theory that had recently been introduced by Robert Stawell Ball.
KingdonKingdon may refer to:
Billy Kingdon (1907–1977), English footballer who played, as a winghalf, over 240 games for Aston Villa
Edith Kingdon (1864–1921), the actress wife of George Jay Gould I (1864–1923)
Edith Kingdon Gould (1920–2004), socialite, linguist, and poet
Francesca Kingdon, British actress best known for modelling on K8TIE
Frank KingdonWard (1885–1958), English botanist, explorer, plant collector and author
Guy Kingdon Natusch (born 1921), noted New Zealand architect
John W. Kingdon (born 1940), political scientist
Jonathan Kingdon (born 1935), science author focusing on taxonomic illustration and evolution of the mammals of Africa
Kingdon Gould disambiguation page
Mark D. Kingdon, the CEO of Linden Lab, the American company responsible for the Internetbased 3D virtual world Second Life
Mark E. Kingdon, hedge fund manager and president of the 4 billion USD Kingdon Capital Management
William Kingdon Clifford FRS (1845–1879), English mathematician and philosopher
Lucy CliffordLucy Clifford (2 August 1846 – 21 April 1929), better known as Mrs. W. K. Clifford, was an English novelist and journalist, and the wife of philosopher William Kingdon Clifford.
Nikolai LobachevskyNikolai Ivanovich Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj] (listen); 1 December [O.S. 20 November] 1792 – 24 February [O.S. 12 February] 1856) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry and also his fundamental study on Dirichlet integrals known as Lobachevsky integral formula.
William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.
Olaus HenriciOlaus Magnus Friedrich Erdmann Henrici, FRS (9 March 1840, Meldorf, Duchy of Holstein – 10 August 1918, Chandler's Ford, Hampshire, England) was a German mathematician who became a professor in London.
After three years as an apprentice in engineering, Henrici entered Karlsruhe Polytechnium where he came under the influence of Alfred Clebsch who encouraged him in mathematics. He then went to Heidelberg where he studied with Otto Hesse. Henrici attained his Dr. phil. degree on 6 June 1863 at University of Heidelberg. He continued his studies in Berlin with Karl Weierstrass and Leopold Kronecker. He was briefly docent of mathematics and physics at the University of Kiel, but ran into financial difficulties.Henrici moved to London in 1865 where he worked as a private tutor. In 1869 Hesse introduced him to J. J. Sylvester who in turn brought him into contact with Arthur Cayley, William Kingdon Clifford, and Thomas Archer Hirst. It was Hirst that gave him some work at University College London. Henrici also became a professor at Bedford College. When Hirst fell ill, Henrici filled his position at University College. He held the position until 1884, turning to applied mathematics after 1880.
From 1882 to 1884 Henrici was President of the London Mathematical Society.
In 1884 he moved to Central Technical College where he directed a Laboratory of Mechanics which included calculating machines, planimeters, moment integrators, and a harmonic analyzer.Henrici was impressed by the work of Robert Stawell Ball in screw theory as presented in a German textbook by Gravelius. In 1890 Henrici wrote a book review for Nature outlining the program of the theory.In 1911 he retired and took up gardening at Chandler's Ford in Hampshire.
Rotor (mathematics)A rotor is an object in geometric algebra (or more generally Clifford algebra) that rotates any blade or general multivector about the origin. They are normally motivated by considering an even number of reflections, which generate rotations (see also the Cartan–Dieudonné theorem).
The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). Hestenes defined a rotor to be any element of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies , where is the "reverse" of —that is, the product of the same vectors, but in reverse order.
Ruth FarwellRuth Sarah Farwell CBE DL retired as ViceChancellor and Chief Executive of Buckinghamshire New University in February 2015.
Farwell held a research fellowship in theoretical physics at Imperial College, London, in the early eighties. Her research is at the boundary between applied mathematics and theoretical physics. Her use of Clifford algebras in her mathematics generated her interest in the Victorian mathematician William Kingdon Clifford. She continues to research the mathematical contribution of Clifford, and mathematical models of particle physics, as well as undertaking research on higher education policy. Farwell actively promotes closer collaborative efforts between universities.The University of Kent, from which she received a master's degree, awarded her an honorary degree in 2010.Farwell is the chair of higher education representative body GuildHE, chair of the Board of Trustees of the Open College Network, South East Region, and a board member of the Higher Education Funding Council for England (HEFCE), and the Universities and Colleges Employers Association. She serves on a number of committees including HEFCE’s Teaching, Quality and the Student Experience Committee, the Quality in Higher Education Group, and Universities UK’s Student Experience Policy Committee and Health and Social Care Policy Committee.She is a member of the London Mathematical Society and the Higher Education Academy.
She was appointed Commander of the Order of the British Empire (CBE) in the 2015 New Year Honours for services to higher education, appointed a Deputy Lieutenant of Buckinghamshire in 2015 and High Sheriff of Buckinghamshire for 2018–19.
SplitbiquaternionIn mathematics, a splitbiquaternion is a hypercomplex number of the form
where w, x, y, and z are splitcomplex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the splitbiquaternion is an element of an eightdimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the splitcomplex numbers form the ring. This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of algebras. The splitbiquaternions have been identified in various ways by algebraists; see § Synonyms below.
Stephen GullStephen Gull is a British physicist based at St John's College, Cambridge credited, together with Anthony N. Lasenby, Joan Lasenby and Chris J. L. Doran, with raising the interest of the physics community to the mathematical language and methods of geometric algebra and geometric calculus. These have been rediscovered and refined by David Hestenes, who built on the fundamental work of William Kingdon Clifford and Hermann Grassmann. In 1998, together with Lasenby and Doran, he proposed gauge theory gravity.
Timothy MadiganTimothy J. Madigan (born 1962) is an American philosopher, author and editor, and a noted humanist. He is particularly notable for having been the Editor of Free Inquiry, a leading journal of secular humanist discussion and commentary.
Madigan graduated in philosophy from the State University of New York at Buffalo in 1984, later gaining an MA and a PhD from the same institution. His PhD supervisor was Peter Hare. Madigan's PhD was on the 19th century mathematician and philosopher William Kingdon Clifford, and he wrote a 2009 book about Clifford.
From the mid1980s Madigan was employed by the journal Free Inquiry. He became its Executive Editor (1987–1996) and then Editor (19961998). He left to become the Editorial Director of the University of Rochester Press, in Rochester, New York. He is currently Professor and Chair of the Philosophy Department at St John Fisher College, also in Rochester, NY. Madigan is also one of the US Editors of Philosophy Now magazine.As Secular Humanist Mentor of the Council for Secular Humanism, Madigan was active in helping establish local secular humanist societies throughout the United States. Since 1993 he has been a member of the board of directors of the Bertrand Russell Society. In 2015 he was elected President of the Bertrand Russell Society.
Madigan is a frequent speaker and panel chair at academic conferences on a wide range of humanities subjects. His own advice on chairing conference sessions has been published in Academe, the journal of the American Association of University Professors.
William CliffordWilliam Clifford may refer to:
William Clifford (priest) (died 1670), English Roman Catholic theologian
William Clifford (cricketer) (1811–1841), English cricketer
William Clifford (actor) (1877–1941), American actor of the silent era
William Clifford (bishop) (1823–1893), English prelate of the Roman Catholic Church
William Kingdon Clifford (1845–1879), mathematician and philosopher
William Clifford Heilman (1877–1946), American composer
Billy Clifford (soccer), American professional soccer player
Billy Clifford (footballer) (born 1992), English professional footballer
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