A hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science. The idea was criticized for its failure to conform to associativity of multiplication, but it has a legacy in Minkowski space and as an extension of split-complex numbers. Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ... 1891 (MDCCCXCI) was a common year starting on Thursday (see link for calendar). ... The American Association for the Advancement of Science (AAAS) is an organization that promotes cooperation between scientists, defends scientific freedom, encourages scientific responsibility and supports scientific education for the betterment of all humanity. ... In mathematics, associativity is a property that a binary operation can have. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Please refer to Real vs. ... 2-dimensional renderings (ie. ...

A linear combination In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

q = a + bi + cj + dk

is a hyperbolic quaternion when a, b, c, and d are real numbers and the basis set {1,i,j,k} has these products:

ij = k = ( − j)i

jk = i = ( − k)j

ki = j = ( − i)k

i² = 1 = j² = k²

Though these basis products do not obey associativity, the set

{1,i,j,k, − 1, − i, − j, − k}

forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...

q ^* = a − bi − cj − dk

is the conjugate of q, then the product

q(q ^* ) = a² − b² − c² − d²

is the quadratic form used in Minkowski space. In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Later, Macfarlane published an article in the Proceedings of the Royal Society at Edinburgh in 1900. In it he establishes a model for hyperbolic space on the hyperboloid Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...

q(q ^* ) = 1

This isotropic model serves as a means to relativize velocity calculations within the limits of the speed of light. Writing in 1967, M.J. Crowe summarized the status of hyperbolic quaternions as follows: Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... Cherenkov effect in a swimming pool nuclear reactor. ...

The introduction of another system of vector analysis, even a sort of compromise system such as MacFarlane's, could scarcely be well received by the advocates of the already existing systems and moreover probably acted to broaden the question beyond the comprehension of the as-yet uninitiated reader.

History of Vector Analysis, p. 191 .

Basis quasigroup

The basis {1,i,j,k} of the vector space of hyperbolic quaternions is not closed under multiplication: for example, ji = − k. Nevertheless, the set {1,i,j,k, − 1, − i, − j, − k} is closed under multiplication. In the 1890s there was no structural theory of abstract algebras so this mathematical object could not be labeled, except as a latin square. Loss of the associativity property of multiplication in quasigroup theory is not tenable in linear algebra since all linear transformations compose in an associative manner. Yet physical scientists were calling in the 1890s for mutation of the squares of i,j, and k to be + 1 instead of − 1 : American physicists Willard Gibbs and Alexander MacFarlane made their cases in pamphlets, and Oliver Heaviside in England wrote columns in the Electrician, a trade paper. The Americans had chairs at Yale University and Texas, while Heaviside expounded in print with vector algebra and differential equations. Cargill Gilston Knott was moved to offer the following: In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English engineer, mathematician and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and magnetic... Yale University is a private university in New Haven, Connecticut. ... The University of Texas at Austin, often called UT or Texas, is the flagship institution of the University of Texas System. ... Cargill Gilston Knott (June 30, 1856—October 26, 1922) was born on June 30, 1856 at Penicuik, Scotland. ...

Theorem (Knott, 1893) If a 4-algebra on basis {1,i,j,k} is associative and off-diagonal products are given by Hamilton's rules, then i² = − 1 = j² = k².

Proof i²j( − i) = ij(i + j) = i(k + j²) = − j + ij² Therefore i² = − 1 and j² = − 1.

Use jk(j + k) for k² = − 1.

This theorem needed statement to justify resistance to the call of the physicists and the electrician. The quasigroup stimulated a considerable stir in the 1890s: the journal Nature was especially conducive to an exhibit of what was known by giving two digests of Knott's work as well as those of several other vector theorists. Michael J. Crowe devotes chapter six of his book History of Vector Analysis to the various published views. Crowe has the benefit of hindsight on vector analysis and the nabla operator, but he does not recognize the quasigroup, being content with the comment: The deepest visible-light image of the universe, the Hubble Ultra Deep Field. ...

MacFarlane constructed a new system of vector analysis more in harmony with Gibbs-Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system.

In retrospect, this quasigroup, with its unusual non-associativity, evoked an attitude of interest in axiomatic basics, an attitude that evolved into abstract algebra with its great variety of axiomatic structures.The contributions of Alfred Tarski, B. L. van der Waerden, and Bourbaki preceded the the category and functor theory now used to locate mathematical objects. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Alfred Tarski (January 14, 1901 in Warsaw – October 26, 1983 in Berkeley, USA) was a Polish mathematician, and widely considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel. ... Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ...

MacFarlane's hyperbolic quaternion paper of 1900

The Proceedings of the Royal Society at Edinburgh published "Hyperbolic Quaternions" in 1900, a paper in which MacFarlane regains associativity for multiplication by reverting to complexified quaternions. While there he used some expressions later made famous by Wolfgang Pauli: where MacFarlane wrote In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century. ... Wolfgang Pauli Wolfgang Ernst Pauli (April 25, 1900 – December 15, 1958) was an Austrian physicist noted for his work on the theory of spin, and in particular the discovery of the Exclusion principle, which underpins the whole of chemistry. ...

The Pauli matrices satisfy The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Wolfgang Pauli. ...

while referring to the same complexified quaternions.

The opening sentence of the paper is "It is well known that quaternions are intimately connected with spherical trigonometry and in fact they reduce the subject to a branch of algebra." This statement may be verified by reference to the contemporary work Vector Analysis by J.W. Gibbs and E.B. Wilson.In MacFarlane's paper there is an effort to produce "trigonometry on the surface of the equilateral hyperboloids" through the algebra of hyperbolic quaternions, now re-identified in an associative ring of eight real dimensions.The effort is re-enforced by a plate of nine figures on page 181.They illustrate the descriptive power of his "space analysis" method.For example, figure 7 is the common Minkowski diagram used today in special relativity to discuss change of velocity of a frame of reference and simultaneous events. Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... The Minkowski diagram is a graphical tool used in special relativity to visualize spacetime with regard to an inertial reference frame. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ...

On page 173 MacFarlane expands on his greater theory of quaternion variables.By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation. Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ... The Wikipedia article on quaternions describes the history and purely mathematical properties of the algebra of quaternions. ...

References

• MacFarlane (1891) "Principles of the Algebra of Physics" Proceedings of the American Association for the Advancement of Science 40:65-117.
• C.G. Knott (1892) "Recent Innovations in Vector Theory" Proceedings of the Royal Society in Edinburgh and Nature 47:590-3.
• MacFarlane (1900) "Hyperbolic Quaternions" Proceedings of the Royal Society at Edinburgh, 1899-1900 session, pp. 169-181.
• J.W. Gibbs and E.B. Wilson (1901) Vector Analysis, Yale.
• M.J. Crowe (1967) History of Vector Analysis, University of Notre Dame

External link

• Two Propositions in hyperbolic quaternion theory