William Kingdon Clifford

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 "mind-stuff".

William Clifford
Clifford William Kingdon
William Kingdon Clifford (1845–1879)
Born4 May 1845
Died3 March 1879 (aged 33)
ResidenceEngland
NationalityEnglish
Alma materKing's College London
Trinity College, Cambridge
Known forClifford algebra
Clifford's theorem
Clifford–Klein form
Clifford parallel
Bessel–Clifford function
Dual quaternion
Elements of Dynamic
Spouse(s)Lucy Clifford (1875-1879)
Scientific career
FieldsMathematics
Philosophy
InstitutionsUniversity College London
Doctoral studentsArthur Black
InfluencesGeorg Friedrich Bernhard Riemann
Nikolai Ivanovich Lobachevsky

Biography

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 half-year 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.

Mathematician

"Clifford was above all and before all a geometer." (H. J. S. Smith).[4] The discovery of non-Euclidean 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 Space-Theory of Matter, was published in 1876, anticipating Albert Einstein’s general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels.

William Kingdon Clifford by John Collier
Clifford by John Collier

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]

Algebras

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 vector-like 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 3-dimensional 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 3-dimensions. 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 three-dimensional sphere embedded in a four-dimensional space. Quaternion versors, which inhabit this 3-sphere, 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 split-complex numbers D or from the dual numbers N. In terms of tensor products, produces split-biquaternions, while forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics.

Philosopher

Clifford William Kingdon desk
William Kingdon Clifford

As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, "mind-stuff" 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 mind-stuff, 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 "mind-stuff" as follows (1878, "On the Nature of Things-in-Themselves", 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 Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness ; but it possesses a small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff 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 mind-stuff 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 mind-stuff 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]

Ethics

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 ship-owner who planned to send to sea an old and not well built ship full of passengers. The ship-owner 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 mid-ocean and told no tales.” [17]

Clifford argued that the ship-owner 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.

Premonition of relativity

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 "quasi-harmonic 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,

The geometry of rotors and motors ... forms the basis of the whole modern theory of the relative rest (Static) and the relative motion (Kinematic and Kinetic) of invariable systems.[19]

This passage makes reference to biquaternions, though Clifford made these into split-biquaternions 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 Space-Theory of Matter in 1876.

In 1910 William Barrett Frankland quoted the Space-Theory of Matter in his book on parallelism.[20] He wrote:

The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight.

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:

Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein’s (1915–16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford’s rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot.

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:

[He] with great ingenuity foresaw in a qualitative fashion that physical matter might be conceived as a curved ripple on a generally flat plane. Many of his ingenious hunches were later realized in Einstein's gravitational theory. Such speculations were automatically premature and could not lead to anything constructive without an intermediate link which demanded the extension of 3-dimensional geometry to the inclusion of time. The theory of curved spaces had to be preceded by the realization that space and time form a single four-dimensional entity.[23]

In 1973 Banesh Hoffmann wrote:

Riemann, and more specifically Clifford, conjectured that forces and matter might be local irregularities in the curvature of space, and in this they were strikingly prophetic, though for their pains they were dismissed at the time as visionaries.[24]

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."

Selected writings

Legacy

The academic journal Advances in Applied Clifford Algebras publishes on Clifford’s legacy in kinematics and abstract algebra.

Quotations

Highgate Cemetery - East - William Kingdon Clifford 01
Marker for W. K. Clifford and his wife in Highgate Cemetery (c. 1986)
  • "I ... hold that in the physical world nothing else takes place but this variation [of the curvature of space]." — Mathematical Papers (1882).
  • "There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture — that it came to him from outside, and that he did not consciously create it from within." (From an 1868 lecture to the Royal Institution titled "Some of the conditions of mental development")
  • "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." — The Ethics of Belief (1879 [1877])
  • "I was not, and was conceived. I loved and did a little work. I am not and grieve not." — Epitaph.
  • "If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it — the life of that man is one long sin against mankind." — Contemporary Review (1877)

See also

Notes

  1. ^ Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge, England: Cambridge University Press. p. 592. ISBN 9780521715959.
  2. ^ Hestenes, David (2011). "Grassmann's legacy". Grassmann's Legacy in From Past to Future: Graßmann's Work in Context, Petsche, Hans-Joachim, Lewis, Albert C., Liesen, Jörg, Russ, Steve (ed). Basel, Germany: Springer. pp. 243–260. doi:10.1007/978-3-0346-0405-5_22. ISBN 978-3-0346-0404-8.
  3. ^ Dorst, Leo (2009). Geometric Algebra for Computer Scientists. Amsterdam: Morgan Kaufmann. p. 664. ISBN 9780123749420.
  4. ^ a b c d e f g h i Chisholm 1911, p. 506.
  5. ^ "Clifford, William Kingdon (CLFT863WK)". A Cambridge Alumni Database. University of Cambridge.
  6. ^ Chisholm, M. (2002). Such Silver Currents. Cambridge: The Lutterworth Press. p. 26. ISBN 978-0-7188-3017-5.
  7. ^ Stephen, Leslie; Pollock, Frederick (1901). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. 1. New York: Macmillan and Company. p. 20.
  8. ^ Eves, Howard W. (1969). In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes. 3–4. Prindle, Weber and Schmidt. pp. 91–92.
  9. ^ Bernhard Riemann (1854, 1867) On the hypotheses which lie at the bases of geometry, Habilitationsschrift and posthumous publication, translated by Clifford, link from School of Mathematics, Trinity College Dublin
  10. ^ W. K. Clifford (1873) "On the hypotheses which lie at the bases of geometry", Nature 8:14 to 17, 36, 37; also Paper #9 in Mathematical Papers (1882), page 55, synopsis pp 70,1
  11. ^ Norman L. Biggs; Edward Keith Lloyd; Robin James Wilson (1976). Graph Theory: 1736-1936. Oxford University Press. p. 67. ISBN 978-0-19-853916-2. Retrieved 30 July 2013.
  12. ^ Clifford, William (1878). "Applications of Grassmann's extensive algebra". American Journal of Mathematics. 1 (4): 350–358. doi:10.2307/2369379. JSTOR 2369379.
  13. ^ Gottfried Leibniz, letter to Christian Huygens (8 September 1679) "I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometrical or linear and which will express situation directly as algebra expresses magnitude directly.", in Gottfried Leibniz (2nd edition 1976) Philosophical Papers and Letters, Springer
  14. ^ Hestenes, David. "On the Evolution of Geometric Algebra and Geometric Calculus".
  15. ^ Dechant, Pierre-Philippe (March 2014). "A Clifford algebraic framework for Coxeter group theoretic computations". Advances in Applied Clifford Algebras. 14 (1): 89–108. arXiv:1207.5005. Bibcode:2012arXiv1207.5005D. doi:10.1007/s00006-013-0422-4.
  16. ^ Berrios, G E (2000). "Body and Mind: C K Clifford". History of Psychiatry. 11 (43): 311–338. doi:10.1177/0957154x0001104305. PMID 11640231.
  17. ^ Clifford, William, Kingdon, essay “The Ethics of Belief.” published in 1877.
  18. ^ Clifford, William, Kingdon, essay “The Ethics of Belief.” published in 1877.
  19. ^ Common Sense of the Exact Sciences (1885), page 214 (page 193 of the Dover reprint), immediately followed by a section on "The bending of space". However, according to the preface (p.vii) this section was written by Karl Pearson
  20. ^ William Barrett Frankland (1910) Theories of Parallelism, pp 48,9, Cambridge University Press
  21. ^ Raum Zeit Materie, page 101, Springer-Verlag, Berlin
  22. ^ J. Wheeler (1960) "Curved empty space as the building material of the physical world: an assessment", in Ernest Nagel (1962) Logic, Methodology, and Philosophy of Science, Stanford University Press
  23. ^ Cornelius Lanczos (1970) Space through the Ages: The evolution of geometrical ideas from Pythagoras to Hilbert and Einstein, page 222, Academic Press
  24. ^ Banesh Hoffmann (1973) "Relativity" in Dictionary of the History of Ideas 4:80, Charles Scribner's Sons
  25. ^ Farwell & Knee (1990)Studies in History and Philosophy of Science 21:91–121
  26. ^ Farwell & Knee (1992) in 1830–1930: A Century of Geometry, pages 98 to 106, Lecture Notes in Physics #402, Springer-Verlag ISBN 3-540-55408-4

References

Further reading

External links

Alexander McAulay

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 function

In 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. Doran

Chris 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 theorems

In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.

Clifford analysis

Clifford 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 n-space 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 parallel

In 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 torus

In geometric topology, the Clifford torus is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles S1a and S1b. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1a and S1b each exist in their own independent embedding spaces R2a and R2b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 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 R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional 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 S1a and S1b each has a radius of , their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.

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 2-torus (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 three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional 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 Dynamic

Elements 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 cross-ratio for a four-argument 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.

Kingdon

Kingdon may refer to:

Billy Kingdon (1907–1977), English footballer who played, as a wing-half, 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 Kingdon-Ward (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 Internet-based 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 Clifford

Lucy 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 Lobachevsky

Nikolai 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 Henrici

Olaus 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 Farwell

Ruth Sarah Farwell CBE DL retired as Vice-Chancellor 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.

Split-biquaternion

In mathematics, a split-biquaternion is a hypercomplex number of the form

where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional 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 split-complex 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 split-biquaternions have been identified in various ways by algebraists; see § Synonyms below.

Stephen Gull

Stephen 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 Madigan

Timothy 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 mid-1980s Madigan was employed by the journal Free Inquiry. He became its Executive Editor (1987–1996) and then Editor (1996-1998). 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 Clifford

William 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

Concepts in religion
Conceptions of God
Existence of God
Theology
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Problem of evil
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