Conformal Mappings in Geometric Algebra Garret Sobczyk n 1878 William Kingdon Clifford wrote down the rules for his geometric algebra, also known as Clifford algebra. We argue in this paper that in doing so he laid down the groundwork that is profoundly altering the Ilanguage used by the mathematical community to express geometrical ideas. In the real estate busi- ness everyone knows that what is most important is location. We demonstrate here that in the busi- ness of mathematics what is most important to the clear and concise expression of geometrical ideas is notation. In the words of Bertrand Russell, …A good notation has a subtlety Herman Gunther Grassmann (1809–1877) was and suggestiveness which at times a high school teacher. His far-reaching make it seem almost like a live Ausdehnungslehre, “Theory of extension”, laid teacher. the groundwork for the development of the Heinrich Hertz expressed much the same thought exterior or outer product of vectors. William when he said, Rowan Hamilton (1805–1865) was an Irish One cannot escape the feeling physicist, astronomer, and mathematician. His that these mathematical formulae invention of the quaternions as the natural have an independent existence and generalization of the complex numbers of the an intelligence of their own, that plane to three-dimensional space, together they are wiser than we are, wiser with the ideas of Grassmann, set the stage for even than their discoverers, that William Kingdon Clifford’s definition of we get more out of them than we geometric algebra. William Kingdon Clifford originally put into them. (1845–1879) was a professor of mathematics and mechanics at the University College of The development of the real and complex num- London. Tragically, he died at the early age of ber systems represents a hard-won milestone in 34 before he could explore his profound ideas. the robust history of mathematics over many cen- turies and many different civilizations [5], [29]. Without it mathematics could progress only halt- ingly, asis evidentfrom the history of mathematics Alexandria, the illustrious inventor of the wind- and even the terminology that we use today. Neg- mill and steam engine during the first century AD ative numbers were referred to by Rene Descartes [18]. What most mathematicians fail to see even (1596–1650) as “fictitious”, and “imaginary” num- today is that geometric algebra represents the bers were held up to even greater ridicule, though grand culmination of that process with the com- they were first conceived as early as Heron of pletion of the real number system to include the Garret Sobczyk is professor of mathematics at Universi- concept of direction. Geometric algebra combines dad de Las Américas-Puebla. His email address is garret_ the two silver currents of mathematics, geometry [email protected]. and algebra, into a single coherent language. As DOI: http://dx.doi.org/10.1090/noti793 David Hestenes has eloquently stated, 264 Notices of the AMS Volume 59, Number 2 Algebra without geometry is blind, to complete a new manuscript New Foundations geometry without algebra is dumb. in Mathematics: The Geometric Concept of Num- David Hestenes, a newly hired assistant profes- bers which has been accepted for publication by sor of physics, introduced me to geometric algebra Birkhäuser [24]. at Arizona State University in 1966. I immediately Distilling geometric algebra down to its core, became captivated by the idea that geometric al- it is the extension of the real number system to include new anticommuting square roots of 1, gebra is the natural completion of the real number ± system to include the concept of direction. If this which represent mutually orthogonal unit vectors idea is correct, it should be possible to formulate in successively higher dimensions. Whereas every- all the ideas of multilinear linear algebra, tensor body is familiar with the idea of the imaginary analysis, and differential geometry. I considered number i √ 1, the idea of a new square root u √ 1 ∉= R has− taken longer to be assimilated it to be a major breakthrough when I succeeded = + in proving the fundamental Cayley-Hamilton the- into the main body of mathematics. However, in orem within the framework of geometric algebra. [22] it is shown that the hyperbolic numbers, which The next task was mastering differential geome- have much in common with their more famous try. Five years later, with Hestenes as my thesis brothers the complex numbers, also lead to an director, I completed my doctoral dissertation, efficacious solution of the cubic equation. “Mappings of Surfaces in Euclidean Space Using The time has come for geometric algebra to Geometric Algebra” [26]. become universally recognized for what it is: the Much work of Hestenes and other theoretical completion of the real number system to include physicists has centered around understanding the the concept of direction. Let twenty-first century geometric meaning of the appearance of imaginary mathematics be a century of consolidation and numbers in quantum mechanics. Ed Witten, in his streamlining so that it can bloom with all the Josiah Willard Gibbs Lecture “Magic, Mystery, and renewed vigor that it will need to prosper in the Matrix” says, “Physics without strings is roughly future [10]. analogous to mathematics without complex num- bers” [31]. In a personal communication with the Fundamental Concepts author, Witten points out that “Pierre Ramond, in Let Rp,q be the pseudoeuclidean space of the 1970, generalized the Clifford algebra from field indefinite nondegenerate real quadratic form q(x) theory (where Dirac had used it) to string theory.” with signature p, q , where x Rp,q . With this { } ∈ Dale Husemöllerin his book Fibre Bundles includes quadratic form, we define the inner product x y · a chapter on Clifford algebras, recognizing their for x, y Rp,q to be ∈ importance in “topological applications and for 1 x y [q(x y) q(x) q(y)]. giving a concrete description of the group Spin(n)” · = 2 + − − [13]. Other mathematicians have developed an ex- Let (e)(n) (e1, e2,..., ep q) denote the row of tensive new field which has come to be known as + standard orthonormal= anticommuting basis vec- “Clifford analysis” [3], [7] and “spin geometry” [2]. tors of Rp,q , where e2 q(e ) for j 1, . , n, During the academic year 1972–73 I continued j j i.e., = = work at Arizona State University with David on dif- 2 2 2 2 ferential geometry as a visiting assistant professor. e1 ep 1 and ep 1 ep q 1 = · · · = = + = · · · = + = − The following year, at the height of the academic and n p q. Taking the transpose of the row crunch, I was on my own—a young unemployed = + T of basis vectors (e)(n) gives the column (e)(n) of mathematician, but with a mission: To convince these same basis vectors. The metric tensor [g] of the world of the importance of geometric algebra. I Rp,q is defined to be the Gramian matrix travelled to Europe and European universities giv- T ing talks when and where I was graciously invited. (1) [g] : (e)(n) (e)(n), = · During the year 1973–74, Professor Roman Duda which is a diagonal matrix with the first p entries of the Polish Academy of Sciences offered me a 1 and the remaining entries 1. The standard postgraduate research position, which I gratefully + − reciprocal basis to (e)(n) is then defined, with accepted. raised indices, by My collaboration with David culminated a num- (e)(n) [g](e)T . ber of years later with the publication of our book = (n) Clifford Algebra to Geometric Calculus: A Unified In terms of the standard basis (e)(n), and its re- Language for Mathematics and Physics [11]. Over ciprocal basis (e)(n), a vector x Rp,q is efficiently the years I have been searching for the best way expressed by ∈ to handle ideas in linear algebra and differential (n) (n) x (e)(n)(x) (x)(n)(e) . geometry, particularly with regard to introducing = = students to geometric algebra at the undergrad- Both the column vector of components (x)(n) and uate level. To this end, I have been working the row vector of components of x are, of course, February 2012 Notices of the AMS 265 Figure 2. The vector a is reflected in the xy -plane to give the vector b e12ae12 , where = e1 and e2 are unit vectors along the x and y axis, respectively, and e12 is the unit bivector of the xy -plane. for each λk i1 . ik where 1 i1 < < ik n. = ≤ ··· ≤ When k 0, λ0 0 and e0 1. For example, the 3 = = = 3 2 -dimensional geometric algebra G3 of R has the standard basis G 3 3 span eλk k 0 = { } = span 1, e1, e2, e3, e12, e13, e23, e123 . = { } The geometric numbers of space are pictured in Figure 1. Figure 1. The great advantage of geometric algebra is that deep geometrical relationships can be ex- pressed directly in terms of the multivectors of (n) real numbers. If y (e)(n)(y) is a second such the algebra without having to constantly refer to vector, then the inner= product between them is a basis; see Figures 2 and 3. On the other hand, (n) (n) the language gives geometric legs to the power- x y (x)(n)(e) (e)(n)(y) · = · ful matrix formalism that has developed over the p n n last one hundred fifty years. As a real associative x y i x y x y , i i i j j algebra, each geometric algebra under geometric = i 1 = i 1 − j p 1 = = =+ multiplication is isomorphic to a corresponding where n p q. algebra or subalgebra of real matrices, and, in- Let (a)= be+ a second basis of Rp,q .