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Complex numbers actually describe the algebra of rotations It thus appeared at first that quaternions may indeed be for the plane, which is two dimensional. Hence this has the the ideal algebra for three-dimensional physical space that same dimension as a two-dimensional vector space. This is had been sought. Unfortunately there was one cloud on the why complex numbers can doubly serve as rotation operators horizon, the fact that a vector quaternion squares to the nega- as well as vectors for the plane. That is, a rotation of a vector tive Pythagorean length. Indeed, Maxwell commented on this given by a complex number z can be written unusual fact, noting that the kinetic energy, which involves the ′ square of the velocity vector, would therefore be negative [9]. z =eiθz. (1) Maxwell, despite these reservations, formulated the equations In this case both the rotation operator eiθ and the vector z are of electromagnetism in quaternionic form. Maxwell, however, represented by complex numbers. backed away from a complete endorsement of the quaternions, However, following the generally successful use of complex recommending in his treatise on electricity and magnetism ‘the numbers in describing vectors in the plane, researchers of the introduction of the ideas, as distinguished from the operations nineteenth century then turned their attention to the generaliza- and methods of Quaternions’ [10]. tion of complex numbers to three-dimensional space in order The difficulties with quaternions led to a breakaway for- to naturally describe these vectors. malism of the Gibbs vector system. Gibbs considered that the Hamilton led this program, and in 1843 he succeeded in most useful function of the quaternions was in their forming generalizing the complex number algebra to the quaternion of the dot product and the cross product operations. That algebra [5]. A quaternion can be written is, for two vector quaternions v = v1i + v2j + v3k and w = w1i+w2j+w3k we find through expanding the brackets, q = a + v1i + v2j + v3k, (2) that where the three basis vectors have a negative square i2 = 2 2 vw i j k i j k j = k = 1 and are anticommuting with each other. = (v1 + v2 + v3 )(w1 + w2 + w3 ) (4) − Hamilton’s quaternions form a four-dimensional associative = v1w1 v2w2 v3w3 + (v2w3 v3w2)i H − − − − division algebra over the real numbers represented by . Once +(v3w1 v1w3)j + (v1w2 v2w1)k − − again being a division algebra, like complex numbers, they = v w + v w, are amenable to all the common arithmetic operations. The − · × sequence of algebras , C and H are constructed to be division using the negative square and the anticommuting properties of algebras, that is, theyℜ are closed with an inverse operation. the basis vectors. We can also see how the vector quaternion Indeed, the required algebraic rules for quaternions follow squared v2 = v v is thus the negative of the Pythagorean from this closure property [6]. These properties also, in fact, length as noted− previously.· We can see though how indeed make them naturally suited to describe rotations in space as the dot and cross products naturally arise from the product of they also have the closure property. two vector quaternions. Gibbs then considered that adopting Quaternions are also isotropic, as required, with a three the separate operations of the dot and cross products acting dimensional ‘vector’ represented as v = v1i + v2j + v3k. on three-vectors could thus form the basis for a more efficient Hamilton then claimed that being a generalization of complex and straightforward vector system. numbers to three dimensions it would therefore logically be This led to an intense and lengthy debate over several years the appropriate algebra to describe three-dimensional space. between the followers of Gibbs and the followers of Hamilton, Indeed he proposed, many years before Einstein or Minkowski, beginning in 1890, over the most efficient vectorial system to that if the scalar a, in Eq. (2), was identified with time then be adopted in mathematical physics [1]. The supporters of the four-dimensional quaternion can be a representation for Hamilton were able to claim that quaternions being general- a unified spacetime framework [7]. Indeed, when Minkowski ized complex numbers were clearly preferable as they had a came to develop the idea of a unified spacetime continuum, proper mathematical foundation. The Gibbs’ side of the debate after considering the merits of quaternions, he chose rather to though argued that the non-commutativity of the quaternions extend the Gibbs vector system with the addition of a time added many difficulties to the algebra compared with the much coordinate to create a four-component vector [8]. simpler three-vector formalism in which the dot and cross Using quaternions, we have rotations in three dimensions products were each transparently displayed separately. Ulti- given by the bilinear form mately, with the success of the Gibbs formalism in describing ′ a −a electromagnetic theory, exemplified by the developments of q =e /2qe /2, (3) Heaviside [11], and with an apparently more straightforward where a = a1i + a2j + a3k is a vector quaternion describing formalism, the Gibbs system was adopted as the standard the axis of rotation. One of the objections raised against vector formalism to be used in engineering and physics. This the quaternions at the time was their lack of commutativity, outcome to the debate is perhaps surprising in hindsight as in however in order to describe rotations in three dimensions this comparison with quaternions, the Gibbs vectors do not have is actually a requirement for the algebra as three dimensional a division operation and two new multiplication operations rotations themselves are non-commuting in general. Indeed are required beyond elementary algebra and so therefore did this form of three-dimensional rotation is extremely useful as it not satisfy the basic principles of Descartes or Hankel. The avoids problems such as gimbal lock and has other advantages trend of adopting the Gibbs vector system, however, continued such as being very efficient in interpolating rotations. in 1908 when Minkowski rejected the quaternions as his 3

2 2 2 description of spacetime and chose to rather extend the Gibbs e1 = e2 = e3 =1. We thus have a vector three-vector system, on the grounds that the quaternions were v = v1e1 + v2e2 + v3e3 (5) too restrictive. In fact the Lorentz boosts are indeed difficult 2 2 2 2 to describe with quaternions and interestingly this difficulty that now has a positive square v = v1 + v2 + v3, giving the arises from the square of vector quaternions being negative— Pythagorean length. This thus avoids Hamilton’s defect with the same problem that was identified earlier by Maxwell. the vector quaternions having a negative square. Also in 1927 with the development of quantum mechanics Using the elementary algebraic rules just defined, multiply- and the Schr¨odinger and Pauli equations, a vector in the form ing two vectors produces of the complex spinor was adopted to describe the wave vw = v w + jv w, (6) function, despite the fact that the spinor is in fact isomorphic to · × the quaternion! However with the arrival of the Dirac equation where j = e1e2e3. Thus Clifford’s vector product forms an in 1928, which required an eight-dimensional wave function, invertible product, as well as producing the dot and cross the four-dimensional quaternions were then clearly deficient. products in the form of a complex-like number. Indeed, this The quaternions, though, can be complexified that results in an allows us to write the dot and cross products as eight-dimensional algebra. However, in 1945, Dirac concluded 1 that the true value of quaternions lies in their unique algebraic v w = (vw + wv) , (7) · 2 properties, and so generalizing the algebra to the complex field 1 is therefore not the best approach [12]. jv w = (vw wv) × 2 − In summary, it therefore appears that Hamilton had indeed so that they become simply the symmetric and antisymmetric been wrong with his assertion that his quaternions were the components of a more general Clifford product. This also natural algebra for three-dimensional space, as they were becomes a convenient way to the define the wedge product ultimately deficient in describing the Dirac equation and had as v w v w. For the inverse of a vector we therefore difficulties with Lorentz transformations. As we have already = j have ∧ × noted, complex numbers define the algebra of rotations for the − v 1 = v/v2. (8) plane and indeed Hamilton had only generalized this algebra to produce the algebra of rotations in three dimensions. This fact That is, vv−1 = vv/v2 = v2/v2 =1, as required. Rotations demonstrates that Gibbs, in fact, was correct in asserting that use a similarly efficient form to the quaternions with quaternions were not suitable to describe Cartesian vectors in ′ −ja/2 ja/2 three-dimensions due to their natural role as rotation operators. v =e ve , (9) Indeed, the more natural identification of the quaternions as where a is the axis of rotation. Also similarly to quaternions 3 the even subalgebra of Cℓ( ), and hence rotation operators, we can combine scalars and the various algebraic components ℜ is illustrated in Table I. into a single number called a multivector The solution to this dilemma would therefore appear to be to add Hamiltons quaternion algebra to the Gibbs Cartesian a+v1e1 +v2e2 +v3e3 +w1e2e3 +w2e3e1 +w3e1e2 +be1e2e3. vectors in some way in order to provide a complete algebraic (10) description of space—a goal that was indeed achieved by Now using j = e1e2e3 we can form the dual relations je1 = Clifford. The Clifford geometric algebra Cℓ( 3), being an e2e3, je2 = e3e1 and je3 = e1e2. Therefore we can write the eight-dimensional linear space, is indeed ableℜ to subsume multivector as the quaternion algebra and the Gibbs vectors into a single M = a + v + jw + jb, (11) formalism, as required. We now therefore describe the Clifford geometric algebra in three dimensions. with the vectors v = v1e1 + v2e2 + v3e3 and w = w1e1 + w2e2 + w3e3. We also define Clifford conjugation on a multivector M = a + v + jw + jb as III. CLIFFORD’S VECTOR SYSTEM ¯ A Clifford geometric algebra Cℓ ( n) defines an associative M = a v jw + jb (12) ℜ − − real algebra over n dimensions [13]. In three dimensions that gives a general multivector inverse M −1 = M/M¯ M¯ . 3 the algebra Cℓ
forms an eight-dimensional linear space In order to restore isotropy to complex numbers and the and is isomorphicℜ to the 2 2 complex matrices. For three × Argand plane, we can introduce the Clifford geometric algebra dimensions, we adopt the three quantities e1,e2,e3 for basis Cℓ( 2) where the complex numbers are isomorphic to the 2 vectors that are defined to anticommute, so that e1e2 = evenℜ subalgebra. The multivector in Cℓ( 2) describes sep- e2e1, e1e3 = e3e1, and e2e3 = e3e2, but unlike the arately the rotation algebra and vectors. Thatℜ is, we have a − − − 3 quaternions these quantities square to positive one , that is multivector a + v1e1 + v2e2 + ib, (13) 2While it is possible to define Clifford geometric algebra in a coordinate free manner, it is more illustrative to follow Hamilton’s approach and firstly where e1,e2 are now isotropic anticommuting basis vectors, define a basis. 2 2 3The basis elements of a Clifford algebra can be defined more generally where e1 = e2 = 1 and i = e1e2, the bivector of the plane. to have a positive or a negative square and over any number of dimensions. We can then write for the rotation of a planar vector For example, a Clifford algebra is typically defined over several hundred ′ − dimensions in order to analyze signals in terahertz spectroscopy. [14] v =e iθv, (14) 4

where now we have the Cartesian vector v = v1e1+v2e2 acted that culminate in Clifford geometric algebra is shown in Fig 1 −iθ on by the even subalgebra re , where i = e1e2. We can see the equivalence with the complex number formula given earlier A. Comparison of physical theories because we can write a Cartesian vector v = e1(v1+e1e2v2)= e1z that then reverts to the complex number rotation formula Within Clifford’s multivectors we write a spacetime event in Eq. (1). Hence Cℓ( 2) is a more precise way to describe as ℜ the Cartesian plane. The identification of the complex numbers S = t + x1e1 + x2e2 + x3e3. (15) as the even subalgebra of Cℓ( 2) is illustrated in Table I. We then find Clifford’s system is also mathematicallyℜ quite similar to Hamilton’s system, with the main distinction being that the SS¯ = (t + x1e1 + x2e2 + x3e3)(t x1e1 x2e2 x3e3) basis vectors have a positive square. Unlike Hamilton’s system, 2 2 2 2 − − − = t x1 x2 x3, (16) we can also form the compound quantities with the basis − − − vectors such as bivectors e1e2,e3e1,e2e3 and trivectors j = thus producing the required metric. This is equivalent to e1e2e3. This greater dimensionality allows the inclusion of the Minkowski’s four-vectors [8] s = [t, x1, x2, x3] and where 2 2 2 2 Gibbs Cartesian vectors and the quaternions as a subalgebra. s s¯ = t x1 x2 x3. Indeed it can be shown that quaternions are isomorphic to ·Lorentz− boosts− also− follow immediately from the formalism the even subalgebra of the multivector, with the mapping simply by the exponentiation of vectors similar to the expo- i e2e3, j e1e3, k e1e2 and the Gibbs vector can nentiation of bivectors for rotations, as shown in Eq. (9), that be↔ replaced by↔ the vector↔ component of the multivector as is ′ v v shown in Eq. (10). The fact that the quaternions form the even S =e /2Se /2, (17) subalgebra and so are represented by the bivectors explains why the vector quaternions have the property of squaring to where v describes the boost direction. The form of the the negative of the Pythagorean length. The bivectors are in expression in Eq. (17) is not available with quaternions as we fact pseudovectors or axial vectors and so clearly not suitable require vectors with a positive square and is also not available for their use as polar vectors as Hamilton claimed. with Gibbs’ vectors because they do not allow us to form an While Hamilton’s non-commutivity was one of the things exponential series. In order to achieve this with four-vectors we need to generate a 4 4 boost matrix to act on the four- that counted against his vector system at the time it actually × is exactly what is needed in a vector system in three di- vector. With Clifford’s system we can also neatly summarize v+jw mensions, as three-dimensional rotations are intrinsically non- the special Lorentz group with the operator e that in commuting. A further oversight of the Gibbs vector system a single expression describes rotations and boosts. Alternate is that while attempting to describe linear vectorial quanti- descriptions of spacetime have been developed, using Clifford ties his system completely ignored the presence of directed algebra, that are isomorphic to this formulation such as the areal and volume objects within three-dimensional space. This algebra of physical space (APS) [15], [16], or the space-time situation is also remedied with the Clifford multivector that algebra (STA) [17]. algebraically describes the points, lines, areas and volumes Using the Gibbs vector system we can reduce Maxwell’s of three-dimensional space. The four geometric elements also field equations down to the following four equations describe the four types of physical variables of scalar, polar ρ E = , (Gauss’ law); (18) vector, axial vectors and pseudoscalars. So in fact both Gibbs ∇ · ǫ and Hamilton fell well short of the ultimate objective of 1 ∂E B = µ0J, (Amp`ere’s law); algebraically describing three-dimensional space and was only ∇× − c2 ∂t ∂B completed by Clifford. E + = 0, (Faraday’s law); Thus, the generalization of quaternions to Cℓ( 3), which ∇× ∂t includes the quaternions as the even subalgebra,ℜ provides a B = 0, (Gauss’ law of magnetism), ∇ · more complete generalization of the two-dimensional plane where = e1∂ + e2∂ + e3∂ . This form of the electromag- than quaternions and a more efficient algebraic description of x y z netic equations∇ is found today in most modern textbooks [18]. three-dimensional space. Using Clifford geometric algebra, Maxwell’s four equations Note that, while we have typified the three key vector can be written with the single equation [19], [20] systems with the name of their main originator, many other scientists were involved in their development. For the Gibb’s ρ (∂t + ) F = µcJ, (19) vector system one of the key developers and proponents ∇ ǫ − was Oliver Heaviside who applied it very effectively to where the field is F = E +jcB and = e1∂x +e2∂y +e3∂z. problems in electrical engineering. For Hamilton’s system of Thus the Clifford vector system allows∇ a single equation over quaternions, they were further developed and championed by the reals as opposed to four equations required using the Gibbs Peter Tait. Clifford [13] acknowledged the key theoretical vector system. This simplified form for Maxwell’s equations work of the algebra of extension developed previously by follows from the unification of the dot and cross products into Hermann Grassmann, with Clifford’s system further developed a single algebraic product [21]. However beyond this lack of and popularized more recently by David Hestenes [17], [24], economy of representation other issues arise within the Gibbs [28]. The path of development of the various vectorial systems formalism. 5

The Gibbs vector system defines both the electric field B. Bound vectors—Plucker¨ coordinates E = [E1, E2, E3] and the magnetic field B = [B1,B2,B3] Vectors are often typified as quantities with a magnitude as three component vectors. However, the magnetic field has and a direction and commonly represented as arrows acting different transformational properties to the electric field, which from the origin of a coordinate system. However, we would are normally taken into account through referring to the elec- like to generalize this idea to include vectors that are not tric field as a polar vector and the magnetic field as an axial acting through the origin. This will then allow us to naturally vector. As stated by Jackson: We see here....a dangerous aspect represent a quantity such as torque as a force offset from the a of our usual notation. The writing of a vector as ‘ ’ does origin, for example. This generalization can be achieved as an not tell us whether it is a polar or an axial vector [22]. The extension of Gibbs’ vector system through defining a Pl¨ucker Clifford vector system correctly distinguishes the electric and coordinate, which extends a normal three-vector of Gibbs to magnetic field as a vector and bivector respectively within the a six-dimensional vector, where the three extra components single field variable F = E+jcB, thus being an improvement represent the vector offset from the origin [25]. over the Gibbs notation. Within Clifford’s system this idea can be represented more Using a mixture of notation, the spinor, four-vector, complex easily through the use of the vector and bivector components numbers and matrix notation we can write the Dirac equation of the multivector µ γ ∂µψ = imψ, (20) V = v + jw = v + v r, (25) − ∧ where the Dirac gamma matrices satisfy γµγν + γνγµ = where v is the direction of the normal free vector and r is the 2ηµν I, where ηµν describes the Minkowski metric signature offset of this vector from the origin. Thus if the wedge product 0 1 2 (+, , , ). The four gradient being γ ∂0 + γ ∂1 + γ ∂2 + v r = jv r =0 then this implies that the vectors are parallel 3 − − − ∂ ∧ × γ ∂3, where ∂0 ∂ . The wave function is a four and so implies that the vector passes through the origin and so ≡ t ≡ ∂t component complex vector ψ = [z1,z2,z3,z4], where zµ are we are reverting to a pure position vector. Hence this naturally complex numbers. As we can see a whole suite of new notation generalizes the normal concept of a position vector. needs to be introduced in order to describe the Dirac equation. For example if we have two force vectors F1 = f1 + jt1 = Now, the Dirac wave-function is in fact eight-dimensional f1 + f1 r1 and F2 = f2 + jt2 = f2 + f2 r2, where each ∧ ∧ and so naturally corresponds with the eight-dimensional Clif- Clifford six-vector consists of two forces f1 and f2 which are ford multivector. Indeed, we can write the Dirac equation in each offset from the origin by r1 and r2 respectively. We then Clifford’s formalism [23] as find the trivector part of the Clifford product F1F2 is ∗ ∂M = jmM e3, (21) j(f1 t2 + f2 t1). (26) − · · where the trivector j = e1e2e3 replaces the unit imaginary Now, we would expect, the trivector part will give the torsion allowing us to stay within a real field. The four-gradient is of these two forces, which is indeed the case, with the sign of ∗ defined as ∂ = ∂t+ and the involution M = a v+jw jb. the product indicating a left or right hand screw direction. If Thus the Clifford multivector,∇ without any additional− form−al- this product is zero then the two forces are coplanar and there ism such as matrices or complex numbers, naturally describes is no net twist force. Hence the Clifford multivector allows the central equations of electromagnetism and quantum me- a natural extension to bound vectors not available with either chanics over a real field. quaternions or Gibbs vectors without significant additions to As already noted, Gibbs developed his system by splitting the notation. the single quaternion product into the separate dot and cross products. This is reflected in the standard calculus results for IV. DISCUSSION differentiation The first main defect of the Gibbs vector system is the lack d da db of an inverse operation due to the splitting of the Clifford prod- (a b) = b + a , (22) dt · dt · · dt uct into the separate dot and cross products. Also describing d da db a plane using the orthogonal vector is only workable in three (a b) = b + a . dt × dt × × dt dimensions. That is, an orthogonal vector does not exist in Clifford (and Hamilton) naturally unifies these results into the two dimensions and in four dimensions and higher there is single expression an infinitude of orthogonal vectors for a given plane. It is actually more natural to define an areal quantity as lying in d da db the plane of the two vectors under consideration, as is the case (ab)= b + a . (23) dt dt dt with Clifford’s system. This then allows planar quantities to be Also the div and curl relations are unified into a single represented uniformly in an arbitrary number of dimensions. expression For example, the Gibbs system can find the area of a a = a + j a. (24) parallelogram from the magnitude of the cross product a b ∇ ∇ · ∇× where the direction of the vector formed by the cross product× Clifford geometric algebra and the geometric product allow is orthogonal to the plane formed by the vectors a and b. The more generalizations in the field of calculus, such as unifying volume of a parallelepiped is typically found from the scalar the Gauss and Stokes theorem [24]. triple product a (b b). · × 6

TABLE I 2 3 THEALGEBRAOFSPACEFORTWOANDTHREEDIMENSIONSGIVEN Cℓ(ℜ ) AND Cℓ(ℜ ) RESPECTIVELY.THESE CONTAIN CARTESIAN VECTORS AND THEIR ROTATIONAL ALGEBRA THAT IS GIVEN BY THE EVEN SUBALGEBRA.

Dimension Clifford Even subalgebra (Rotations) Vector 2 2 Cℓ(ℜ ) Complex numbers v = v1e1 + v2e2 a + ib ∈ Z i = e1e2 3 3 Cℓ(ℜ ) Quaternions v = v1e1 + v2e2 + v3e3 q0 + q1i + q2j + q3k ∈ H j = e1e2e3

The Clifford product of two vectors a and b, on the other hand is ab where the magnitude of the bivector component gives the area formed by the two vectors. In this case, bivectors are a native descriptor of area that give the orientation of the plane it represents. The trivector components of the Clifford product abc will give the volume of the parallelepiped and is an obvious extension of the area formula as volume intuitively relates to the trivector components. Thus while the Gibbs formulas are essentially unmotivated, the Clifford product intuitively produces areal and volume quantities. In Clifford geometric algebra, in order to ascertain the geometric relationship between two vectors a and b we can simply form the product ab = a b + a b. (27) · ∧ If the scalar component a b = 0 then the vectors are orthogonal, or if the bivector· or area is zero, then they are parallel. The Clifford vector product thus provides a simple but general way to compare the orientation of two vectors in a single expression. If seeking to determine whether or not three vectors in space lie in a plane we can simply check the trivector component of abc and if it is zero then they are coplanar. More generally, because Clifford vectors, such as v are now elementary algebraic quantities all common functions are available such as logarithms, trigonometric, exponential functions as well as the general calculations of roots. For example, the sin v or log v or even the expressions such Fig. 1. The descent of the various vector systems. The main path of devel- as 2v of raising a number to a vector power, can now be opment beginning from Euclid geometry down through Grassman and then to Clifford. Other parallel developments using complex numbers, quaternions, calculated [26]. Gibbs’ vectors, tensors, matrices and spinor algebra subsumed into the general It is interesting that electrical engineers, such as Heaviside, formalism of Clifford geometric algebra with the inclusion of calculus. led the adoption of the original Gibbs’ vector system. Is it possible that forward thinking electrical engineers will also lead the adoption of Clifford’s vector system? algebra to the four-dimensional quaternion algebra. We noted though that the quaternions only provide the algebra describing V. CONCLUSION rotations in three dimensions and so do not provide suitable We conclude that the mathematical formalism developed Cartesian vectors and so is not a complete description. The by Clifford of Cℓ( 3) is the natural algebra to describe confusion between the rotational algebra and Cartesian vectors three-dimensional space.ℜ Clifford geometric algebra Cℓ( 3) can actually only arise in three dimensions in which we thus fulfills this important goal of nineteenth century science.ℜ have both three translational degrees of freedom and three Clifford’s geometric algebra also extends seamlessly to spaces rotational degrees of freedom. This simple fact was thus at the with an arbitrary number of dimensions Cℓ( n), where n is basis of the misunderstanding between Hamilton and Gibbs. an integer, and so is scalable. That is, theℜ algebra in two The resolution of the confusion is supplied by the Clifford 3 dimensions Cℓ( 2) naturally extends to Cℓ( 3) and up to geometric algebra Cℓ( ) that absorbs both the quaternions ℜ Cℓ( n) in anℜ orderly manner. Each additionalℜ dimension and complex numbers as subalgebras as well as including doublingℜ the size of the space. a Cartesian vector component, as shown in Eq. (11) and in In attempting to find the natural algebraic description of Table I. physical space Hamilton generalized the complex number We also showed how Clifford’s system provides a very 7 natural formalism for Maxwell’s electromagnetism, special [20] W. E. Baylis, Electrodynamics: A Modern Geometric Approach. relativity and the Dirac equation. The competing formalism of Birkh¨auser, Boston, 2001. [21] Y. Friedman and Y. Gofman, “Why does the geometric product simplify quaternions requires the addition of complex numbers and the the equations of physics?” International Journal of Theoretical Physics, Gibb’s vector system requires the addition of matrices, spinors vol. 41, no. 10, pp. 1841–1855, 2002. [22] J. Jackson, Classical Electrodynamics. John Wiley and Sons, 1998. and complex numbers. As the Clifford trivector j = e1e2e2 is [23] P. Lounesto, Clifford Algebras and Spinors. New York: Cambridge a commuting quantity having a negative square, it can replace University Press, 2001. the abstract unit imaginary and allows all relationships to be [24] D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A described over a strictly real space [27], [28], [6]. We also Unified Language for Mathematics and Physics. Dordrecht: D. Reidel publishing, 1984. showed how Clifford multivectors can describe both free and [25] L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer bound vectors efficiently thus providing a generalization of Science: An Object-Oriented Approach to Geometry. USA: Morgan vectors not found in either competing system. Kaufmann, 2009. [26] J. M. Chappell, A. Iqbal, L. J. Gunn, and D. Abbott, “Functions of Thus the goal of the nineteenth century to find the natural multivector variables,” PLOSONE, vol. 10, no. 3, art. no. e0116943, algebraic description of three-dimensional space was achieved 2015. by Clifford with eight-dimensional multivectors, as shown in [27] W. E. Baylis, J. Huschilt, and J. Wei, “Why i?” American Journal of Physics, vol. 60, pp. 788–797, 1992. Eq. (11) and as we have shown can supersede to a large degree [28] D. Hestenes, “Oersted medal lecture 2002: Reforming the mathematical the various vector systems used today. New applications for language of physics,” American Journal of Physics, vol. 71, pp. 104– Clifford algebra are steadily appearing in many fields of 121, 2003. [29] W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light- engineering and physics such as electromagnetism [29], op- polarization: A geometric-algebra approach,” Am. J. 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Dirac, “Application of quaternions to Lorentz transformations,” Dordrecht, 1993, pp. 387–396. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 50, no. 16, pp. 261–270, November 1945. [13] W.K. Clifford, “Applications of Grassmann’s Extensive Algebra,” Amer- ican Journal of Mathematics, vol. 1, no. 4, pp. 350–358, 1878. [14] W. Xie, J. Li, and J. Pei, “THz-TDS signal analysis and substance James M. Chappell BE (civil, 1984), identification,” Journal of Physics: Conference Series, vol. 276, no. 1, Grad.Dip.Ed. (1993), B.Sc. (Hons, 2006) and pp. 49–63, 2011. PhD (quantum computing, 2011) from the [15] W. E. Baylis, “Geometry of paravector space with applications to University of Adelaide, began his career in civil relativistic physics,” in Computational Noncommutative Algebra and PLACE engineering followed by employment as a computer Applications. Netherlands: Springer, 2004, pp. 363–387. PHOTO programmer before retraining as a school teacher. [16] W. E. Baylis, “Relativity in introductory physics,” Canadian Journal of HERE He then returned to academia, completing a Physics, vol. 82, no. 11, pp. 853–873, 2004. science degree, followed by a PhD in quantum [17] D. Hestenes, Spacetime Algebra. New York: Gordon and Breach, 1966. computing. During his PhD he specialized in [18] D. J. Griffiths, Introduction to Electrodynamics. Prentice Hall, 1999. quantum computing and Clifford’s geometric [19] J. M. Chappell, S. P. Drake, C. L. Seidel, L. J. Gunn, A. Iqbal, A. Al- algebra. Dr Chappell is presently a Visiting Scholar lison, and D. Abbott, “Geometric algebra for electrical and electronic at the School of Electrical & Electronic Engineering, at the University of engineers,” Proceedings of the IEEE, vol. 102, no. 9, pp. 1340–1363, Adelaide, working on applications of geometric algebra. Sept 2014. 8

Azhar Iqbal received the degree in physics from the University of Sheffield, U.K., in 1995 and re- ceived the Ph.D. degree in applied mathematics from the University of Hull, U.K., in 2006. In 2006, PLACE he won the Postdoctoral Research Fellowship for PHOTO Foreign Researchers from the Japan Society for the HERE Promotion of Science (JSPS) to work under Prof. T. Cheon at the Kochi University of Technology, Japan. In 2007, he won the prestigious Australian Postdoctoral (APD) Fellowship from the Australian Research Council, under Derek Abbott, at the School of Electrical and Electronic Engineering, the University of Adelaide. For two years took up a post as an assistant professor at the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. Dr Iqbal is currently an adjunct senior lecturer with the School of Electrical and Electronic Engineering, The University of Adelaide.

John G. Hartnett received a B.Sc. (Hons) and Ph.D. (with distinction) in Physics from The Uni- versity of Western Australia (UWA), Crawley, W.A., Australia. He served as a Research Professor with PLACE the Frequency Standards and Metrology Research PHOTO Group, at the The University of Western Australia. HERE In 2013, he took up an Australian Research Council (ARC) Discovery Outstanding Researcher Award (DORA) fellowship based within the Institute for Photonics & Advanced Sensing, the School of Phys- ical Sciences, the University of Adelaide, where he is currently an Associate Professor. His research interests include planar metamaterials, development of ultra-stable microwave oscillators based on sapphire resonators, and tests of fundamental theories of physics such as special and general relativity using precision oscillators. Prof. Hartnett was the recipient of the 2010 IEEE UFFC Society W. G. Cady Award. He was a corecipient of the 1999 Best Paper Award presented by the Institute of Physics Measurement Science and Technology.

Derek Abbott was born in South Kensington, Lon- don, UK, in 1960. He received a B.Sc. (Hons) in physics from Loughborough University, U.K. in 1982 and the Ph.D. degree in electrical and elec- PLACE tronic engineering from the University of Adelaide, PHOTO Adelaide, Australia, in 1995, under K. Eshraghian HERE and B. R. Davis. From 1978 to 1986, he was a research engineer at the GEC Hirst Research Centre, London, U.K. From 1986–1987, he was a VLSI design engineer at Austek Microsystems, Australia. Since 1987, he has been with the University of Adelaide, where he is presently a full Professor with the School of Electrical and Electronic Engineering. He holds over 800 publications/patents and has been an invited speaker at over 100 institutions. Prof. Abbott is a Fellow of the Institute of Physics (IOP) and a Fellow of the IEEE from 2005. He has won a number of awards including the South Australian Tall Poppy Award for Science (2004), the Premier’s SA Great Award in Science and Technology for outstanding contributions to South Australia (2004), an Australian Research Council (ARC) Future Fellowship (2012), and the David Dewhurst Medal (2015). He has served as an Editor and/or Guest Editor for a number of journals including IEEE JOURNALOF SOLID-STATE CIRCUITS, Journal of Optics B (IOP), Microelectronics Journal (Elsevier), PLOSONE, PROCEEDINGSOF THE IEEE, and the IEEE Photonics Journal. He is currently on the editorial boards of IEEE ACCESS, Scientific Reports (Nature), and Royal Society Open Science (RSOS). Prof Abbott co-edited Quantum As- pects of Life, Imperial College Press (ICP), co-authored Stochastic Resonance, Cambridge University Press (CUP), and co-authored Terahertz Imaging for Biomedical Applications, Springer-Verlag. Prof. Abbott’s interests are in the areas of multidisciplinary physics and electronic engineering applied to complex systems. His research programs span a number of areas of stochastics, game theory, photonics, biomedical engineering, and computational neuroscience.