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contents hyperbolic quaternion 7 languages español français 日本語 português slovenščina українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia mutation of quaternions where unit vectors square to 1 hyperbolic quaternion multiplication 1 i j k 1 1 i j k i i 1 k j j j k 1 i k k j i 1 in abstract algebra the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form q a b i c j d k a b c d r displaystyle q a bi cj dk quad a b c d in mathbb r where the squares of i j and k are 1 and distinct elements of i j k multiply with the anti commutative property the four dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of biquaternions they both contain subalgebras isomorphic to the split complex number plane furthermore just as the quaternion algebra h can be viewed as a union of complex planes so the hyperbolic quaternion algebra is a pencil of planes of split complex numbers sharing the same real line it was alexander macfarlane who promoted this concept in the 1890s as his algebra of physics first through the american association for the advancement of science in 1891 then through his 1894 book of five papers in space analysis and in a series of lectures at lehigh university in 1900 algebraic structure edit like the quaternions the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4 a linear combination q a b i c j d k displaystyle q a bi cj dk is a hyperbolic quaternion when a b c displaystyle a b c and d displaystyle d are real numbers and the basis set 1 i j k displaystyle 1 i j k has these products i j k j i displaystyle ij k ji j k i k j displaystyle jk i kj k i j i k displaystyle ki j ik i 2 j 2 k 2 1 displaystyle i 2 j 2 k 2 1 using the distributive property these relations can be used to multiply any two hyperbolic quaternions unlike the ordinary quaternions the hyperbolic quaternions are not associative for example i j j k j i displaystyle ij j kj i while i j j i displaystyle i jj i in fact this example shows that the hyperbolic quaternions are not even an alternative algebra the first three relations show that products of the non real basis elements are anti commutative although this basis set does not form a group the set 1 i j k 1 i j k displaystyle 1 i j k 1 i j k forms a loop that is a quasigroup with an identity element 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 q a b i c j d k displaystyle q a bi cj dk is the conjugate of q displaystyle q then the product q q a 2 b 2 c 2 d 2 displaystyle q q a 2 b 2 c 2 d 2 is the quadratic form used in spacetime theory in fact for events p and q the bilinear form η p q p 0 q 0 p 1 q 1 p 2 q 2 p 3 q 3 displaystyle eta p q p_ 0 q_ 0 p_ 1 q_ 1 p_ 2 q_ 2 p_ 3 q_ 3 arises as the negative of the real part of the hyperbolic quaternion product pq and is used in minkowski space note that the set of units u q qq 0 is not closed under multiplication see the references external link for details discussion edit the hyperbolic quaternions form a nonassociative ring the failure of associativity in this algebra curtails the facility of this algebra in transformation theory nevertheless this algebra put a focus on analytical kinematics by suggesting a mathematical model when one selects a unit vector r in the hyperbolic quaternions then r 2 1 the plane d r t x r t x r displaystyle d_ r lbrace t xr t x in r rbrace with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split complex number plane the hyperbolic versor exp a r cosh a r sinh a displaystyle exp ar cosh a r sinh a transforms d r by t x r exp a r t x r cosh a t x sinh a sinh a t x cosh a r displaystyle begin aligned t xr mapsto quad exp ar t xr quad cosh a t x sinh a sinh a t x cosh a r end aligned since the direction r in space is arbitrary this hyperbolic quaternion multiplication can express any lorentz boost using the parameter a called rapidity however the hyperbolic quaternion algebra is deficient for representing the full lorentz group see biquaternion instead writing in 1967 about the dialogue on vector methods in the 1890s historian michael j crowe commented 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 1 geometry edit later macfarlane published an article in the proceedings of the royal society of edinburgh in 1900 in it he treats a model for hyperbolic space h 3 on the hyperboloid h 3 q m q q 1 displaystyle h 3 q in m q q 1 this isotropic model is called the hyperboloid model and consists of all the hyperbolic versors in the ring of hyperbolic quaternions historical review edit the 1890s felt the influence of the continuous groups of sophus lie an example of a one parameter group is the hyperbolic versor with the hyperbolic angle parameter this parameter is part of the polar decomposition of a split complex number but it is a startling aspect of finite mathematics that makes the hyperbolic quaternion ring different the basis 1 i j k displaystyle 1 i j k of the vector space of hyperbolic quaternions is not closed under multiplication for example j i k displaystyle ji k nevertheless the set 1 i j k 1 i j k displaystyle 1 i j k 1 i j k is closed under multiplication it satisfies all the properties of an abstract group except the associativity property being finite it is a latin square or quasigroup a peripheral mathematical structure loss of the associativity property of multiplication as found in quasigroup theory is not consistent with 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 displaystyle i j displaystyle j and k displaystyle k to be 1 displaystyle 1 instead of 1 displaystyle 1 the yale university physicist willard gibbs had pamphlets with the plus one square in his three dimensional vector system 2 oliver heaviside in england wrote columns in the electrician a trade paper advocating the positive square in 1892 he brought his work together in transactions of the royal society a 3 where he says his vector system is simply the elements of quaternions without quaternions with the notation simplified to the uttermost and with the very inconvenient minus sign before scalar product done away with so the appearance of macfarlane s hyperbolic quaternions had some motivation but the disagreeable non associativity precipitated a reaction cargill gilston knott was moved to offer the following theorem knott 4 1892 if a 4 algebra on basis 1 i j k displaystyle 1 i j k is associative and off diagonal products are given by hamilton s rules then i 2 1 j 2 k 2 displaystyle i 2 1 j 2 k 2 proof j k i j i i j i i displaystyle j ki ji i j ii so i 2 1 displaystyle i 2 1 cycle the letters i displaystyle i j displaystyle j k displaystyle k to obtain i 2 1 j 2 k 2 displaystyle i 2 1 j 2 k 2 qed this theorem revealed a flaw in elements of dynamic 1978 by w k clifford who described products in 2x2 real matrices 5 in that system some basis squares are positive identity yet clifford had distinct basis elements multiplying as in the quaternion group rather that the symmetry group of a square the theorem justified 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 a history of vector analysis to the various published views and notes the hyperbolic quaternion 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 1 in 1899 charles jasper joly noted the hyperbolic quaternion and the non associativity property 6 while ascribing its origin to oliver heaviside the hyperbolic quaternions as the algebra of physics undercut the claim that ordinary quaternions made on physics as for mathematics the hyperbolic quaternion is another hypercomplex number as such structures were called at the time by the 1890s richard dedekind had introduced the ring concept into commutative algebra and the vector space concept was being abstracted by giuseppe peano in 1899 alfred north whitehead promoted universal algebra advocating for inclusivity the concepts of quasigroup and algebra over a field are examples of mathematical structures describing hyperbolic quaternions macfarlane s hyperbolic quaternion paper of 1900 edit the proceedings of the royal society of 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 i j k 1 displaystyle ij k sqrt 1 j k i 1 displaystyle jk i sqrt 1 k i j 1 displaystyle ki j sqrt 1 the pauli matrices satisfy σ 1 σ 2 σ 3 1 displaystyle sigma _ 1 sigma _ 2 sigma _ 3 sqrt 1 σ 2 σ 3 σ 1 1 displaystyle sigma _ 2 sigma _ 3 sigma _ 1 sqrt 1 σ 3 σ 1 σ 2 1 displaystyle sigma _ 3 sigma _ 1 sigma _ 2 sqrt 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 work vector analysis which works with a reduced quaternion system based on dot product and cross 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 common minkowski diagram used today in special relativity to discuss change of velocity of a frame of reference and relativity of simultaneity 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 references edit 1 2 crowe m j 1967 a history of vector analysis university of notre dame p 191 gibbs 1881 4 elements of vector analysis arranged for students in physics via internet archive heaviside 1892 pp 427 430 knott c g 1893 recent innovations in vector theory nature 47 1225 590 3 bibcode 1893natur 47r 590 doi 10 1038 047590b0 read before the royal society of edinburgh 19 december 1892 and published in proceedings w k clifford 1878 elements of dynamic page 170 hamilton 1899 joly c j ed elements of quaternions 2nd ed london longmans green and co p 163 heaviside oliver 1892 on the forces stresses and fluxes of energy in the electromagnetic field philosophical transactions of the royal society of london a 183 423 480 bibcode 1892rspta 183 423h doi 10 1098 rsta 1892 0011 jstor 90590 macfarlane a 1891 principles of the algebra of physics proceedings of the american association for the advancement of science 40 65 117 macfarlane a 1894 paper 2 the imaginary of the algebra papers on space analysis new york b westerman macfarlane a 1900 space analysis a brief of twelve lectures lehigh university macfarlane a january 1902 hyperbolic quaternions proceedings of the royal society of edinburgh 23 169 180 doi 10 1017 s0370164600010385 internet archive free or google books free note p 177 and figures plate incompletely scanned in free versions mathews g b m 1913 an algebra for physicists nature 91 2284 595 6 bibcode 1913natur 91 595g doi 10 1038 091595b0 alexander macfarlane and the ring of hyperbolic quaternions v t e number systems sets of definable numbers natural numbers n displaystyle mathbb n integers z displaystyle mathbb z rational numbers q displaystyle mathbb q constructible numbers algebraic numbers a displaystyle mathbb a closed form numbers periods p displaystyle mathcal p computable numbers arithmetical numbers set theoretically definable numbers gaussian integers gaussian rationals eisenstein integers composition algebras division algebras real numbers r displaystyle mathbb r complex numbers c displaystyle mathbb c quaternions h displaystyle mathbb h octonions o displaystyle mathbb o split types over r displaystyle mathbb r split complex numbers split quaternions split octonions over c displaystyle mathbb c bicomplex numbers biquaternions bioctonions other hypercomplex dual numbers dual quaternions dual complex numbers hyperbolic quaternions sedenions s displaystyle mathbb s trigintaduonions t displaystyle mathbb t split biquaternions multicomplex numbers geometric algebra clifford algebra algebra of physical space spacetime algebra plane based geometric algebra infinities and infinitesimals cardinal numbers extended natural numbers extended real numbers projective extended complex numbers hyperreal numbers levi civita field ordinal numbers supernatural numbers surreal numbers superreal numbers other types irrational numbers fuzzy numbers transcendental numbers p adic numbers p adic solenoids profinite integers normal numbers classification list retrieved from https en wikipedia org w index php title hyperbolic_quaternion oldid 1355686901 categories non associative algebra history of quaternions minkowski space hidden categories articles with short description short description is different from wikidata pages that use a deprecated format of the math tags this page was last edited on 23 may 2026 at 08 45 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal 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