- Hyperbolic quaternion
In

mathematics , a**hyperbolic quaternion**is a mathematical concept first suggested by Alexander MacFarlane in1891 in a speech to theAmerican Association for the Advancement of Science . The idea was criticized for its failure to conform toassociativity of multiplication, so the collection**M**of all hyperbolic quaternions forms anon-associative ring . It has a legacy inMinkowski space and as an extension ofsplit-complex number s.This article describes their algebra and traces a development that preceded prevailing modern insight on what is desirable in a structure. The non-associativity of hyperbolic quaternions poses a complication for constructing transformation geometry from its multiplication, but for a while it was considered very promising. At that time, the split-complex arithmetic was suppressed due to concerns about

zero-division , andlinear representation was not at all common. Thus, hyperbolic quaternions were succeeded bybiquaternion s as the popular ring of choice for a while, before generalizing concepts like e.g.tensor algebra ,Lie algebra , andClifford algebra took over. Nevertheless, they remain the only quaternionic number system where the square of all bases is +1 .Writing in 1967, M.J. Crowe summarized the status of hyperbolic quaternions as follows::"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.

**Algebraic structure**Like the

quaternions , the set of hyperbolic quaternions form avector space over thereal numbers ofdimension 4. Alinear combination :$q\; =\; a+bi+cj+dk$

is a

**hyperbolic quaternion**when $a,\; b,\; c,$ and $d$ are real numbers and the basis set $\{1,i,j,k\}$ has these products::$ij=k=-ji$:$jk=i=-kj$:$ki=j=-ik$:$i^2=1=j^2=k^2$

Unlike Hamilton's quaternions, of which these are a mutant form, the hyperbolic quaternions are not

associative . For example, $(ij)j\; =\; kj\; =\; -i$, while $i(jj)\; =\; i$. The first three relations show that products of the (non-real) basis elements are "anti-commutative". Although this basis set does not form an group, 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 ofsplit-complex number s. If:$q^*=a-bi-cj-dk$

is the conjugate of $q$, then the product

:$q(q^*)=a^2-b^2-c^2-d^2$

is the

quadratic form used in spacetime theory.In fact, thebilinear form called the Minkowski inner product arises as the negative of the real part of the hyperbolic quaternion product pq* ::$-p\_0q\_0\; +\; p\_1q\_1\; +\; p\_2q\_2\; +\; p\_3q\_3$.Note that the set of units U = {q : qq* ≠ 0 } is not closed under multiplication. See the external reference below for details of this singularity of multiplication within U.**Geometry**Later, Macfarlane published an article in the "Proceedings of the Royal Society at Edinburgh" in 1900. In it he treats a model for

hyperbolic space H^{3}on thehyperboloid :$H^3\; =\; \{\; q\; in\; M:\; q(q^*)=1\; \}\; !$.

This

isotropic model is called thehyperboloid model and consists of all the hyperbolic versors in the ring of hyperbolic quaternions.**Historical review**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 alatin square . Loss of the associativity property of multiplication as found inquasigroup 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 physicistsWillard Gibbs and Alexander MacFarlane made their cases in pamphlets, andOliver Heaviside in England wrote columns in the "Electrician", a trade paper. The Americans had chairs atYale University and Texas, while Heaviside expounded in print with vector algebra and differential equations.Cargill Gilston Knott was moved to offer the following:**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 $i^2=-!1=j^2=k^2$.**Proof:**:$j\; =\; ki\; =\; (-ji)i\; =\; -j(ii)$, so $i^2\; =\; -1$. Cycle the letters $i$, $j$, $k$ to obtain $i^2=-1=j^2=k^2$. "QED".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 bookA 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::"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."

The hyperbolic quaternions had an appearance on page 163 of Charles Jasper Joly’s second edition, 1899, of Hamilton’s "Elements of Quaternions". Joly speaks in square brackets, and ascribes the system to

Oliver Heaviside , another proponent of the positive dot product. Joly makes use of theanti-commutative property to show that associativity is broken.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 ofAlfred Tarski , B. L. van der Waerden, andBourbaki preceded the category and functor theory now used to locate mathematical objects. Furthermore, the unwieldy nature of hyperbolic quaternions is not encountered when the formal method inring theory of quotient rings is applied.**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:$ij=ksqrt\{-1\}$ :$jk=isqrt\{-1\}$:$ki=jsqrt\{-1\}$,thePauli matrices satisfy:$sigma\_1sigma\_2=sigma\_3sqrt\{-1\}$:$sigma\_2sigma\_3=sigma\_1sqrt\{-1\}$:$sigma\_3sigma\_1=sigma\_2sqrt\{-1\}$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 workVector Analysis (Gibbs/Wilson) which works with a reduced quaternion system based ondot product andcross product . 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 reinforced 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 commonMinkowski diagram used today inspecial relativity to discuss change of velocity of a frame of reference and simultaneous events.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 ofQuaternions and spatial rotation .**Minkowski space**Recall that the "scalar part" of quaternion $q\; =\; a+bi+cj+dk$ is the variable $a$. Using quaternion conjugation $q^*\; =\; a\; -\; bi\; -\; cj\; -\; dk$ one can express the "Minkowski inner product" with $eta(p,q)$ being the scalar part of $(pq*)$ , where there is a hyperbolic quaternion product of $p$ with $q^*$. The inner product generates two structures in Minkowski space: simultaneity of events relative to a given velocity and the "Minkowski squared interval":$eta(q,q)\; =\; q\; q*\; =\; a^2\; -\; b^2\; -\; c^2\; -d^2.$In particular, the hyperboloid $\{q:\; a\; >\; 0,\; qq^*\; =\; 1\; \}$ presents a kinematic model since (with appropriate units for $a,b,c,$ and $d$) it represents the locus of temporal potential for a particle passing through the origin after a moment of

local time .**imultaneity**Select an arbitrary point from the hyperboloid: $u\; =\; cosh(a)\; +\; r\; sinh(a)$. Then relative to $u$, arbitrary hyperbolic quaternions $p$ and $q$ represent "simultaneous" events in Minkowski space if the scalar part of the product $(p\; -\; q)u^*$ is zero. Clearly simultaneity is a function of

rapidity a and direction $r$. Geometrically, the hyperbolic quaternions $p\; -\; q$ and $u$ arehyperbolic-orthogonal .**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)A History of Vector Analysis , University of Notre Dame

* [*http://ca.geocities.com/macfarlanebio/hypquat Alexander MacFarlane and Hyperbolic Quaternions*]

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