User:Cronholm144/Mathematical Physics

First excerpt citation
The Elementary Foundation of Mathematical Physics Ronold King The American Mathematical Monthly, Vol. 44, No. 1. (Jan., 1937), pp. 14-22.

Stable URL: http://links.jstor.org/sici?sici=0002-9890%28193701%2944%3A1%3C14%3ATEFOMP%3E2.0.CO%3B2-6 The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Fri May 18 03:03:52 2007

14 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS
By RONOLD KING, Lafayette

For many centuries man has been curious about the structure of nature. In attempting to understand the mysteries of its changing form and the rhythm of its stupendous continuity, he has progressed slowly, yet amazingly. Thus, he has flown blindly on the powerful wings of imagination; he has trudged boldly along the winding road of consistent logic; he has used his senses to look with penetration and to listen with patience. But it was not until he learned to send a logical mind soaring high into the abstract spaces of mathematics, while a skillful hand guided its flight with the compass of observed reality, that he began to find in nature not the whims of capricious gods, but unity and consistency. In the early nineteenth century the great mathematician Gauss* called mathematics the "Queen of the sciences." In the year 1935 a distinguished American scholar wrote: "Had Gauss known-that in respect of aim, method, and content, the enterprise of mathematics and the enterprise of natural science are separated by a chasm as deep and unbridgeable as that which sunders logical deduction from experimental observation, it is certain-that the great 'prince of mathematicians' would not deliberately have so spoken as to imply that one of two essentially disparate enterprises could be the 'Queen' of the other."

THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS 1.5 If there is a bottomless gulf between mathematics and the natural sciences, how is it possible that it is precisely in the application of mathematics to these sciences that the human mind has been so successful? If it is true that mathematics rises above the clouds into the boundless sunshine of an infinite space, while physics is tied by its very nature to earth itself with its tides and its tremors, with winter and the magic of spring, then by what agency are they connected? This is a modern example of an age-old problem, that of explaining mind and matter. Mathematics,is mind-stuff; natural science is earth-stuff. But it is not the purpose of this analysis to become entangled in the confusing "isms" of philosophy, however interesting these may be. For, whatever may be the relationship between mind and matter, it is certainly true that a correspondence has been discovered between the logic of mathematics and the laws of nature as revealed by experimental science. This correspondence is the cornerstone of applied mathematics, and in particular of mathematical physics. It is to a clearer insight into those mental and physical processes which are the foundation of that phenomenal structure, theoretical physics, that this study is dedicated. 1. Mathematics. The position of mathematics in the scheme of knowledge has occupied the keenest minds. The Pythagorean Greeks characterized number as "great and perfect and omnipotent and the principle and guide of divine and human life." More recently it has been described as the principle by which the world "instead of being a chaos becomes a cosmos." These are eloquent words, but are they not questions rather than explanations? The modern concept of mathematics is that of a pure, formal science completely identified with logic. It emphasizes that pure mathematics does not deal with the material world, and, hence, is not concerned with natural science. This view is in striking contrast with historically earlier ones which, by failing to distinguish clearly between mathematics and its applications, encountered insurmountable logical obstacles. The true field cf mathematics is that of relations. Specifically, it is the field of relations characterized by precision, completeness, and sharpness. Ordinary ideas, most often interrelated in vague and indefinite ways, are outside its realm. Since mathematics thus limits itself to those precise relations which are independent of the uncertainties of a physical universe, its laws are fundamentally and invariably true. But, clearly, the correctness of mathematical relationship is due not to any especially charmed mode of thought, but rather to the nature of the ideas and concepts with which it operates. The method of mathematics is that of logical deduction. It depends upon one of the great achievements of the human mind, namely, the discovery by Pythagoras and Thales of the art of proving theorems. This art is based on the fundamental principle that no proposition is considered established until it has been proved, i.e., logically deduced from other propositions previously estab16 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS [January, lished. Evidently the application of this principle presupposes the specification of certain initial assumptions or premises. These may be selected arbitrarily to suit the convenience or the interest of the investigator. This method is illustrated in the development of Euclidean geometry which, although originally a practical art of land measurement, has become a model of abstract logical coherency. In this system all geometrical quantities can be defined in terms of a few indefinables such as "point" and "straight line," and all propositions can be deduced from a relatively small number of axioms about these indefinables. The axioms (for example, the well-known statement that only one straight line can be drawn through two points) used to be regarded as self-evident truths. They are now recognized to be merely assumptions, since it has been found possible to construct other no less consistent systems of geometry on the basis of quite different premises. And it is entirely outside the sphere of the mathematician to attempt to decide whether Euclidean or Riemannian geometry is true in any particular universe. The mathematician merely states that if anything whatsoever has the properties specified in the axioms, then of necessity it will also have the properties described in the theorems. Mathematics is an excursion into the uncharted spaces of the logically thinkable. Give a mathematician a group of entities satisfying certain conditions specified in a set of hypotheses, and he will proceed to investigate what logical deductions can be made. To accomplish this end he will invent convenient symbols and a code of rules to govern their use. This he will do on the basis of the principle of logical continuity and to suit his own convenience. His only guide is his mathematical curiosity; his compass is his experience in dealing with highly abstract relationships. Thus, urged on by an adventurous mind, he blazes new trails in ever more intricate regions of abstract space. His range is unlimited, it is bound by no dimensions, tied to no facts. It is the hunting ground of pure mind stalking the logically thinkable. It is the domain of correspondence, of definite precision, of positive relationships. In it value answers to value, state corresponds to state, condition relates to condition, change responds to change. Can anything new be created in a process of such complete logical sequence? Is it possible for more to be derived from conclusions than is first included in the premises? It is beside the point to state that no human process can ever create in the sense of divine creation. But if age-old stone is cut from a quarry and shaped into man-made building blocks; and if these are piled upon each other to build a cathedral, then man has created something new. What is new is not the structure of the stone, which may persist into eternity, but the form in which it appears. In much the same way mathematics creates new forms in the shape of deductions. But it does more than this. It reveals in such deductions many an implication never supposed to be contained in the hypotheses. In other words, the quest for mathematical truth is in a very large degree an attempt to discover what the assumed hypotheses really mean. And in the measure that the human mind reveals to itself what it never suspected to exist in its postulates, it has created something new-new, not in an absolute or uni19371 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS 17 versa1 sense, but in the restricted meaning of enlightened human understanding. Thus a complex number, a vector, or a determinant is a new form created by mathematics. But let it not be supposed that mathematics is unlimited and all-powerful. Such is not the case. Mathematics is a very human endeavor, and as such it is restricted by the limitations of the human mind to think what is logical and to comprehend what is complex. 2. Experimental Physics. The aim of experimental physics in particular, and of all empirical science in general, is unlike that of mathematics. Experimental science is not at all interested in what is logically thinkable; its entire concern is with a material universe the structure of which it seeks to reveal in the form of observed fact. As Thales' and Pythagoras' art of proving theorems is the basis of mathematical logic, so Galileo's example of experimental demonstration is the foundation of scientific exactitude. This demands that no statement about the structure of nature be accepted as true, until it has been exhaustively verified by direct observation. All observation begins with one of the five senses, and it depends (as was enunciated by Kant) upon an elementary intuition or awareness of natural phenomena. But even a well-trained eye can provide only a kind of qualitative knowledge which is piecemeal and inexact. The scientist has, therefore, been compelled to reinforce the simple senses with delicate instruments which he has been able to invent as his knowledge about natural phenomena increased. With their aid he has succeeded in recording quantitative knowledge about selected events which he isolates more or less completely in his laboratory. In the last analysis the science of measurement is an ingenious technique for comparing particular phenomena with previously established standards. It depends, therefore, upon the establishment, the preservation and the reproduction of a few fundamental standards, and upon experimental observations which may be duplicated at will. If it is found convenient, new quantities may be derived from the established standards and defined in terms of these and of the actual operations performed in measuring them. All experimental measurements are essentially in the form of pointer readings on an arbitrary scale. They are definitely and inevitably limited in accuracy by the size of the smallest scale division, and by the fact that they are necessarily made in terms of small areas and not in terms of abstract points and lines. Thus, for example, if velocity is to be defined in terms of the ratio of a measured distance traversed in a measured time interval, it must always be an average and an approximate velocity. The mathematical concept of instantaneous velocity, or of velocity at a point, is excluded as an unmeasurable abstraction. Every measurement involves a calibrated instrument, a phenomenon to be measured, and a contact in the form of an energy transfer between them. The correctness of every experimental observation must, therefore, depend upon the effect of this contact on the measurement. In ordinary large scale determinations this may be made entirely insignificant. In the atomic field, on the other hand, 18 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS [January, even the smallest possible energy transfer, namely that due to a single quantum, may represent so large a contact as to make the measurement totally meaningless. This is found to be the case when an attempt is made to measure the position and the velocity of an electron in a microscopic sense. Furthermore, since the nature of the measurement depends upon an interaction between an observer and a given phenomenon, the results observed may vary with interactions due to different types of measuring contrivances. For example, clouds of electrons interacting with one measuring device are observed to give an effect interpreted as due to waves, whereas an interaction with a different kind of instrument produces a so-called particle effect. It is evident, then, that the experimental method is limited not only by technical problems of precision and sensitivity, but by the structure of nature itself. The observer, regardless of his mechanical or electrical disguise, is always a part of this structure and unavoidably plays a role in every measurement. Thus a so-called experimental fact must always be merely a more or less exact estimate of nature itself. The method of mathematics was described as logical deduction from selfimposed hypotheses; the method of experimental science is careful comparison of specific observations with self-selected standards. But whereas the blade of abstract logic is infinitely sharp and thin, that of experimental comparison is only finitely and relatively so; at times it proves to be a hammer. 3. Mathematical Physics. Knowledge which is purely factual is sterile. Nothing new can be learned by repeatedly measuring the same fact no matter how accurately it is ascertained. But, if when a large number of similar facts has been carefully observed, the investigator is led to suspect the existence of an invariant relation between the phenomena and the circumstances under which they were measured, the seed for something new has been planted. Such is the case, for example, when a smooth curve is drawn through the plotted points corresponding to the empirically observed times and distances of a falling body. This is a simple illustration of the process of induction as practiced in the physical sciences, a process of generalization from a few discrete facts to infinitely many. The essence of physical induction is that in the presence of experimental observations showing that all examined x's have certain properties, one concludes that all existing x's have the same properties. The process might be described as the discovery of a frame into which the observed facts will fit. The intuitive awareness of a correspondence between a series of empirical observations and a definite, though perhaps not readily visualized relation, is the transition from matter to mind, from experimental to theoretical physics..But the field of positive relations is the field of mathematics, and the expression of such a relationship is the expression of a mathematical function. Thus, the fundamental basis of mathematical physics is the observed fact that the logically necessary relations which hold between mathematical expressions hold for natural phenomena themselves. The representation of natural phenomena in any form which is to persist 19371 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS 19 in time depends on the existence cf something in nature which is permanent. The old metaphysics conceived this ultimate invariant substance to be matter. Modern physics finds the element of permanence in the functional relationships of pure mathematics. Thus, the reason why logic and mathematics apply to nature is because they describe the invariant relations which are actually found in it. This discovery at once makes available the entire mathematical superstructure for the study not of experimental facts of observation, but of the relations which govern them, of those abstract properties commonly called the laws of nature. "When we consider natural objects purely as the embodiment of the invariant relations found in nature, we are said to idealize those objects, or to consider them as ideal limits. But such idealization gives us the essential conditions for what truly exists." In this light one may represent rigid bodies by points and consider their motion along mathematical lines. One may take advantage of the concepts of the infinitesimal calculus such as instantaneous velocity and acceleration at a point. In other words, relations between idealized physical objects may be expressed in the foi-m of differeritial or integral equations. For example, the motion of the center of gravity of a rigid body is described by the differential equation due to Newton, F(s, v, t) =d(mv)/dt. But let it be clearly noted that this entire mathematical representation is an abstraction which in itself has no real measureable physical significance. One must not forget that one is dealing with relations, not with measured facts. To demand a physical significance for all mathematical symbols occurring in relations which are derived from observed facts is to presume the experimentally verifiable existence of a framework in nature corresponding to every conceivable logically performed operation in formal mathematics. Such experimental verification must be forever lacking in any general sense, since experimental measurement can be performed and expressed in terms of only a limited number of physically real quantities. Thus, mathematical physics does not bridge the chasm between pure mathematics and experimental science. It operates entirely on the mathematical side. In form and in method it resembles mathematics, but its aim is different. The purpose of theoretical physics is not to think the logically thinkable, but by logical thinking to calculate relations which can be verified by direct experiment. Birkhoff writes that "the chief function of mathematical symbolism is to enable the mind to carry through certain processes of logical thought." In mathematical physics the process is immaterial, the logical character is presumed, and the interest centers on specific results which it is desired to obtain in as simple a form as possible. Thus the only condition imposed by the mathematical physicist upon his symbolism is that it ultimately must be useful to calculate definite relations corresponding to a set of empirical facts. In pure mathematics one begins with arbitrary hypotheses and concerns one's self with possible logical deductions. In mathematical physics the procedure is essentially the reverse. Given is a mathematical relation obtained inductively from a set of experimental facts; to find is a general hypothesis from which it may be deduced. 20 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS [January, The principal and most successful method of mathematical physics is to guess at a solution, i.e., an hypothesis, and then to verify its correctness deductively. This may, perhaps, be done most elegantly and most significantly by assuming a variety of possible hypotheses of a very general nature, and by then proceeding mathematically to obtain a number of possible deductions. These may then be compared with the measured data of a series of related experiments to determine whether the observed facts fit into any of thein exactly or approximately. But since the functional relations found in nature are usually highly complex, it is frequently impossible to establish mathematical forms which are at the same time sufficiently general to be exact and sufficiently simple to be soluble with available mathematics in a form useful for purposes of calculation. It is then convenient to derive relations involving only a few variables and parameters which have been selected with due regard for the nature of the problem and reasonable mathematical simplicity. From such relations it is often possible to compute approximate but nevertheless useful results which possess what the German calls Uebersichtlichkeit. Indeed such idealized solutions are the ones from which one may obtain the clearest insight into the individual processes of nature. With increasingly powerful mathematical tools and a broader understanding for relations in the physical world, the mathematical physicist has become bolder and more successful. Instead of merely seeking relations to fit known facts, he proposes broad generalizations from which he predicts what experimental facts must be found if the superstructure he has invented is to be a correct one. There are innumerable examples of successful prediction from theory, indicating that the mathematical physicist is gaining a real understanding of the structure of nature. It is by this method of bold mathematical generalization that simple relations suggested by specialized experiments are transformed according to the logical rules of mathematics into symbolically more imposing but theoretically simpler and more universal forms. It is in such forms that unity in nature becomes apparent. For, from extremely general relations a great variety of special ones may be logically deduced and made to serve as frameworks for the observations of new experiments. 4. Consistency in Nature. The method of mathematical physics of calculating from differential equations the observable facts of the physical world depends upon four fundamental conditions. First, it presupposes the experimental availability of accurate quantitative facts about natural phenomena. Second, it depends upon the discovery in sets of such facts of relations between conveniently selected variables. Third, it assumes the invariance in space and time of the discovered relations. Fourth, it depends upon the invention of mathematical functions and equations which represent variables and relations discoverable in nature. The statement that physical facts are connected according to invariant laws condenses these four basic assumptions into a single principle of consistency in 19371 THE ELEMENTARY FOUNDATION OF MATHEMATICAL PHYSICS 21 nature or of causality. This principle is nothing more than the assertion that the mathematical-logical sequence from hypothesis to deduction is valid in mathematical physics as, indeed, it must be, since it is the very foundation of the mathematical method. So long as physical laws are expressed in terms of mathematical relations, they must be presumed to obey this sequence. The law of cause and effect in physics is, therefore, in the nature of a comprehensive postulate or maxim. Obviously it cannot be proved by a method which presupposes its validity and which alone explains its meaning. It merely asserts and thus assumes that mathematical functional relationships can, in principle, be invented to fit permanently the observed facts in nature. In mathematics one states that every deduction is the logical consequence of definite, pregstablished hypotheses because deduction was so defined. In mathematical physics one asserts that every hypothesis (cause) has for its logical deductions (effects) the observed facts of nature, because the hypothesis was constructed so that this logical sequence would be found true. Its validity, moreover, must be invariant to transformations in space and time. It is assumed, for example, that the differential equation which is used to describe the motion of a vibrating string at a given instant and place will be useful in the same way at a different time and a different place. If nature were found to be inconsistent in terms of a certain differential equation, then one would be compelled to assume that the mathematical representation, since it does not fit this physical world, has no general significance in it. A different mathematical formulation using other variables to represent the physical state in question would then have to be sought. In quantum mechanics, for example, it has been found impossible to calculate certain observed facts from relations using position and momentum coordinates to represent the physical state. On the other hand, the experimental data are correctly calculated from relations in which the variable is a complicated probability function. The aim of mathematical physics, to calculate observed facts, is thus achieved in the latter but not in the former case. To be sure, it may be of metaphysical interest to speculate on the kind of universe in which the former representation would lead to correct results, but this is outside the realm of physics. For the problem in physics is not to consider the properties of conceivable universes, but to find mathematical relations from which the experimentally observed facts of this universe may be calculated. While mathematical relations are successfully used to represent nature, the rules of logic as expressed in the law of cause and effect must be found verified. Let it be emphasized that the aim of physics is to fit mathematics to relations found experimentally in nature, and not to explain the physical world in terms, of a mathematical tradition. The elementary foundation of mathematical physics has been analysed in three major parts. These are, first, the experimental technique which makes it possible to intercompare natural phenomena to a'high degree of precision; second, the framework of permanent relations which is discovered in experi22 A GEOMETRIC REPRESENTATION OF A LINE INTEGRAL [January, mental facts and which is expressible in terms of mathematical functions; third, the vast and highly articulate system of mathematical logic. Upon this foundation the human mind has constructed its most powerful and most dependable tool-the mathematical-scientific method. It is used wherever mathematics serves a practical purpose. It is the leaven of modern knowledge.


 * Read before the American Mathematical Society, December 1, 1928.

Second excerpt citation
A Unified Mathematical Language for Physics and Engineering in the 21st Century Joan Lasenby; Anthony N. Lasenby; Chris J. L. Doran Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol. 358, No. 1765, Science into the Next Millenium: Young Scientists Give Their Visions of the Future: II. Mathematics, Physics and Engineering. (Jan., 2000), pp. 21-39.

Stable URL: http://links.jstor.org/sici?sici=1364-503X%28200001%29358%3A1765%3C21%3AAUMLFP%3E2.0.CO%3B2-H Philosophical Transactions: Mathematical, Physical and Engineering Sciences is currently published by The Royal Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/rsl.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Fri May 18 03:04:34 2007

THE ROYAL Bza SOCIETY

A unified mathematical language for physics and engineering in the 21st century
Department of Engineering, University of Cambridge, Trz~mpington Street, Cambridge CB2 IPZ, UK (jl@eng.cam.ac.uk) Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK (a.n.lasenby@mrao.cam.ac.ukc;.d oran@mrao.cam.ac.uk; http://www.mrao.cam.ac.uk/-clifford) The late 18th and 19th centuries were times of great mathematical progress. Many new mathematical systems and languages were introduced by some of the millennium's greatest mathematicians. Amongst tliese were tlie algebras of Clifford and Grassmann. While tliese algebras caused considerable interest at the time, they were largely abandoned with the introduction of what people saw as a more straightforward and more generally applicable algebra: tlie vector algebra of Gibbs. This was effectively the end of the search for a unifying mathematical language and the beginning of a proliferation of novel algebraic systems, created as and when they were needed; for example, spinor algebra, matrix and tensor algebra, differential forms, etc. In this paper we will cliart the resurgence of the algebras of Clifford and Grassmann in tlie form of a framework known as geometric algebra (GA). Geometric algebra was pioneered in the mid-1960s by the American physicist and mathematician, David Hestenes. It has taken tlie best part of 40 years but tliere are signs that his claim that GA is the universal language for physics and mathematics is now beginning to take a very real form. Througliout tlie world there are an increasing number of groups wlio apply GA to a range of problems from many scientific fields. While providing an immensely powerful mathematical framework in which the most advanced concepts of quantum mechanics, relativity, electromagnetism, etc., can be expressed, it is claimed that GA is also simple enough to be taught to schoolchildren! In this paper we will review the development and recent progress of GA and discuss whether it is indeed tlie unifying language for tlie physics and matliematics of tlie 21st century. The examples we will use for illustration will be taken from a number of areas of physics and engineering. Keywords: geometric/Clifford algebra; geometry; quantum mechanics; relativity; gravity; computer vision; buckling 1. Introduction Today, high scliool students studying for A levels, or their equivalent, in tlie sciences will be introduced to tlie concept of vectors-directed line segments-and taught how to manipulate vectors using classical vector algebra. Tliis is effectively the algebra introduced by Gibbs towards the end of the 19th century; it lias changed little Phil. Trans. R. Soc. Lond. A (2000) 358, 21-39 @ 2000 The Royal Society 21 J. Lasenby, A. N. Lasenby and C. J. L. Doran Figure 1. William Rowan Hamilton 1805-1865. Inventor of quaternions, and one of the key scientific figures of the 19th century. since then. Those students become practised in the art of vector algebra and see how successful it is in expressing much of two- and three-dimensional geometry. Manipulation of the system becomes almost second nature. One can see how hard it then is to abandon this familiar, and apparently successful, system in favour of a new algebra (geometric algebra (GA)) that has additional rules and unconventional concepts. However, for a moderate investment of time and effort put into learning GA, the reward is to have at one's disposal a tool that allows the user to penetrate into even the most high-powered areas of current scientific research. As we move into the 21st century, we have reached the stage where to do research in the physical sciences is often to specialize in one, usually very limited, area. However, it has always been the case that great advantages are to be gained from interactions between fields, something that is becoming increasingly difficult but increasingly desirable. We envisage that the new millennium will see the push for interdisciplinary activity increase manyfold. In the following sections, we attempt to give the reader some evidence that GA may be the best hope we currently have of attaining the goal of a unifying mathematical language for modern science. 2. Some history A problem that occupied many eminent mathematicians of the early 19th century was how best to represent rotations mathematically in three dimensions, i.e. ordinary space. Hamilton (see figure 1) spent much of his later life working on this problem Phil. Trans. R.Soc. Lond. A (2000) A unzfied mathematical language for the 2lst century 23 Figure 2. Hermann Gunther Grassmann (1809-1877). German mathematician and schoolteacher, famous for the algebra that now bears his name. and eventually produced the quaternions, which were a generalization of the complex numbers (see later) to three dimensions (Hamilton 1844). The algebra contains four elements, {I, i ,j?kl-? which satisfy i2= j2= k2 = i j k = -1. While the elements i,j,k are often referred to as vectors, we shall see later that they do not have the properties of vectors. Despite the clear utility of the quaternions, there was always a slight mystery and confsioonv er their nature and use. Today, quaternions are still used to represent three-dimensional rotations in many fields since it is recognized that they are a very efficient way of carrying out such operations. However, the confusion still persists, aiid a deep and detailed uiiderstanding of the quaternions has been lost to a generation. While Hamilton was developing his quaternionic algebra, Grassmann (see figure 2) was formulating his own algebra (Grassman 1844, 1877), the key to which was the introduction of the exterior or outer product; we denote this outer product by A, so that the outer product of two vectors a and b is written as a A b. This product has certain features. One such feature is its associativity, i.e. This tells us that the way in which we group the terms together in the outer product does not matter. The other feature is anticommutativity, that is, if we reverse the Phil. Trans. R.Soc. Lond. A (2000) 24 J. Lasenby, A. N. Lasenby and C. J. L. Doran Figure 3. A portrait of William Kingdon Clifford, FRS (1845-1879), mathematician and philosopher, by the Hon. John Collier. (Royal Society Library and Archives.) order of vectors in the outer product we change its sign: We are more used to dealing with a product that is commutative, i.e. nlultiplicatioil between two numbers, 2 x 5 = 5 x 2 = 10, but it turns out to be extremely useful in many areas of physics, nlaths and engineering to have a product that does not necessarily commute. By contrast, the inner product between two vectors, a and b, written as a. b (this produces a scalar whose magnitude is abcos 8,where 6' is the angle between the vectors), is commutative, i.e. Grassmann, a German schoolteacher, was largely ignored during his lifetime, but since his death his work has stimulated the fashionable areas of diflerential forms and Grassmann (anticommuting) variables. The latter are fundamental to the foundation of much of modern supersyminetry and superstring theory. The next crucial stage of the story occurs in 1878 with the work of the English mathematician, William Kingdon Clifford (Clifford 1878; see figure 3). Clifford was one of the few mathematicians who had read and understood Grassmann's work, and in an attempt to unite the algebras of Hamilton and Grassmanil into a single structure, he introduced his own geometric algebra. In this algebra we have a single geometric product formed by uniting the inner and outer products; this is associative Phil. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st century 2 5 like Grassmann's product but also invertible, like products in Hamilton's algebra. In Clifford's geometric algebra, an equation of the type ab = C has the solution b = a-'C, where a-' exists and is known as the inverse of a. Neither the inner nor the outer product possess this invertibility on their own. Much of the power of geometric algebra lies in this property of invertibility. Clifford's algebra combined all the advantages of quaternions with those of vector geometry, so geometric algebra should then have gone forward as the main system for mathematical physics. However, two events conspired against this. The first was Clifford's untimely death at the age of just 34, and the second was Gibbs's introduction of his vector calculus. Vector calculus was well suited to the theory of electromagnetism as it stood at the end of the 19th century; this, and Gibbs's considerable reputation, meant that his system eclipsed the work of Clifford and Grassmann. It is ironic that Gibbs himself seems to have been convinced that Grassnlann's approach to multiple algebras was the correct one.? With the arrival of special relativity, physicists realized that they were in need of a system capable of handling four-dimensional space, but, by this time, the crucial insights of Grassmann and Clifford had long been lost in the papers of the late 19th century. In the 1920s, Clifford algebra resurfaced as the algebra underlying quantum spin. In particular, the algebra of Pauli and Dirac spin matrices became indispensable in quantunl theory. However, they were treated just as algebras: the geometrical meaning had been lost. Accordingly, we will employ the term 'Clifford algebras' when the use is solely in formal algebra. When applied in its proper, geometric setting, we use Clifford's own name of geometric algebra. This is also a concession to Grassmann, who was actually the first to write down a geometric (Clifford) product! The situation remained largely unchanged until the 1960s, when David Hestenes began to recover the geometric meaning behind the Pauli and Dirac algebras (Hestenes 1966). Although his original motivation was to gain some insight into the nature of quantum mechanics, he very soon realized that, properly applied, Clifford's system was nothing less than a universal language for mathematics, physics and engineering. Again, this remarkable work was largely ignored for around 20 years, but today interest in Hestenes's system (Hestenes & Sobczyk 1984; Hestenes 1986) is gathering nlomentum. There are now many groups around the world working on applying geometric algebra to topics as diverse as black holes and cosn~ology, quantum tunnelling and quantum field theory, beam dynamics and buckling, computer vision and robotics, protein folding, neural networks, and computer-aided design (Sommer 2000; Doran et al. 1996; Baylis 1996; Lasenby et al. 1998). Exactly the same algebraic system is used throughout, making it possible for people to make contributions across a nunlber of these fields simultaneously. 3. Geometric algebra, a brief outline In our geometric algebra we start out with scalars, i.e. ordinary numbers that have a nlagnitude but no associated orientation, and vectors, i.e. directed line segments with both magnitude and orientation/direction. Let us now take these vectors and look a little more closely at the geometry behind Grassmann's outer product. The outer product between two vectors a and b is written as a A b and is a new quantity t In the chapter on multzple algebras in Gibbs (1906), Cibbs goes to great length in his discussioll of the merits of the Grassmannian system. Phil. Trans. R. Soc. Lond. A (2000) J. Lasenby, A. N. Lasenby and C. J. L. Doran scalar /vector -directed line segment a aAb=-bAa bivectors -oriented areas (aAb)A c=aA ( ~ A c ) trivectors -oriented volumes Figure 4. Vectors, bivectors and trivectors shown as oriented geometric objects. called a bivector. The bivector a A b is the directed area swept out by the two vectors a and b; thus the outer product of two vectors is a new mathematical entity that encodes the notion of an oriented plane. If we sweep b out along a,we obtain the same bivector but with the opposite sign (orientation). Now, by extending this idea, we see that the outer product between three vectors, a A b A c, is obtained by sweeping the bivector aA b out along c, thus giving an oriented volume or trivector. If we sweep a across the area represented by the bivector bA c, we get the same trivector (it can be shown that it has the same 'orientation'); this fact expresses the associativity of the outer product. Figure 4 summarizes these ideas of the basic elements of the algebra as geometric objects. In an n-dimensional space, we can have n-vectors, which are simply oriented n-volumes; thus we see that the outer product is easily generalizable to higher dimensions, unlike the Gibbs's vector product, which is restricted to three dimensions. The crucial step in developing geometric algebra now comes with the introduction of the geometric product. We already know what a. b and a A b are: the geometric product unites these in the single product ab, Phil. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st century -el el Figure 5. Multiplication on the right by the bivector elez rotates 90' anticlockwise This step of summing two diflerent objects is not a totally foreign act; in fact, we have long been doing a similar thing when carrying out operations with complex numbers. It turns out that many quantities in physics can be expressed very concisely and efficiently in terms of multivectors (linear combinations of n-vectors, e.g. a scalar plus a bivector, etc.); indeed, this combining of objects of different types appears to occur at a deep level in physical theory. (a) Geometric algebra in two dimensions In two dimensions (a plane), any point can be reached by taking different linear combinations of two vectors with different directions; we say the space is then spanned by these two basis vectors. Now let these two vectors be orthonormal, i.e. of unit length and perpendicular to each other, and call them el and e2. They satisfy which are the equations that encode these properties. The only other element in our two-dimensional geometric algebra is the bivector el A e 2 ; this is the highest grade element in the algebra (often called the pseudoscalar). Let us now look at the properties of this bivector. The first thing to note is that i.e. the geometric product is a pure bivector because the perpendicularity of the vectors guarantees that el. e2 vanishes. Now let us square this bivector: Note that we have a real geometric quantity that squares to -I! It is therefore tempting to relate this quantity with the unit imaginary of the complex number system (a complex number takes the form x + iy where the i is known as the unit imaginary and has the property that i2 = -1). Thus, in two dimensions, geometric algebra reproduces the properties of the complex numbers but uses only geometric objects. In fact, going to geometric algebras of higher dimensions, we begin to see that there are many objects that square to -1, and that we can use them all in their correct geometric setting. Phil. Trans. R. Soc. Lond. A (2000) 28 J. Lasenby, A. N. Lasenby and C. J. L. Doran Let us now see what happens when the bivector ele2 multiplies vectors from the left and right. I\/lultiplying el and e 2 from the left gives We therefore see that left multiplication by the bivector rotates vectors 90' clockwise. Similarly, if we multiply on the right we rotate 90" anticlockwise (see figure 5): 4. Rotations From the properties of the bivector ele2, it is then very easy to show that a rotation of a vector a through an angle 0 to a vector a' is achieved by the equation where R is a quantity we shall call a rotor and is made up of a scalar plus a bivector, R = cos $0 - ele2 sin $0, and R is the same expression but with a '+'. This may at first seem like a rather cumbersome expression to deal with in order to carry out a simple two-dimensional rotation; however, it turns out that it is generalizable to higher dimensions and therefore has enormous power. The above equation, a' = R ~ Ris,, in fact, the formula that is used to rotate a vector in any dimension; if we go to three dimensions, the rotor R will rotate by an angle 6' in the plane described by a bivector. Therefore, all we need do is replace the bivector ele2 with the bivector that defines the plane of rotation (see figure 6). And that is all there is to it; using this very simple expression we find that we can not only rotate vectors, but also bivectors and higher-grade quantities. To carry out rotations in three dimensions in a manner that extended the concepts we understood in two dimensions was a problem Hamilton struggled with for many years, before finally producing, as his solution, the quaternions. In fact, the elements of Hamilton's quaternion algebra are nothing other than elementary bivectors (planes). Having this very simple idea of a rotor that performs rotations, we can give amazingly simple geometric interpretations of many otherwise complicated fields; some examples are given below. 5. Special relativity Special relativity was introduced in 1905 and heralded the beginning of a new era in physics; the departure from the purely classical regime of Newtonian physics. In special relativity (SR), we deal with a four-dimensional space; the three dimensions of ordinary Euclidean space, and time. Suppose we have a stationary observer with whom we can associate coordinates of space and time, this observer will observe events from his space-time position. Now suppose that we have another observer travelling at a velocity v;he too will observe events from his continuously changing space-time position. The problem of how the two observers perceive different events Phzl. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st century Figure 6. The rotor R taking the vector a to the vector a'. Note that the concept of the perpendicular vector is 110 longer needed; it is the bivector or plane of rotation that is important. Figure 7. Illustration of the four-dimensional space-time axes. One of the time-space bivectors is shown; as before it defines a plane in our space and can therefore be used in rotating the axes. is relatively easy when the speed, (v(,is small. But, wl~enl v( approacl~est he speed of light, c, and we add in the fact that c must be constant in any frame, the mathematics is no longer so straightforward. Conventionally, one can derive a coordinate transformation between the frames of the two observers, and to move between these Phil. Trans. R. Soc. Lond.A (2000) J. Lasenby, A. N. Lasenby and C. J. L. Doran incoming particle packet Figure 8. Particle packet incident on a barrier of higher energy than itself. two frames we apply a matrix transformation known as a Lorentz boost. Geometric algebra provides us with a beautifully simple way of dealing with special relativistic transformations using nothing other than the formula for rotations discussed above, namely a' = R ~ R(Hestenes 1966; Gull et al. 1993). Our space now has four dimensions and our basis vectors are the three space directions and one time direction; let us call these basis vectors yo, yl, 72 and 73. Because we have four dimensions we have six bivectors (the three spatial bivectors plus the bivectors made up of spacetime 'planes'). The Lorentz boost turns out to be simply a rotor R, which takes the time axis to a different position in four dimensions: R ~ (~see Rfigure 7). So, in an elegant coordinate-free way we are able to give the transformations of SR an intuitive geometric meaning. All the usual results of SR follow very quickly from this starting point. For example, the complicated formulae for the transformation of the electric (E)and magnetic (B)fields under a Lorentz boost are replaced by the (much simpler!) result E' +IB' = R(E + IB)R, where I = 70717273is the pseudoscalar of four-dimensional space (a 4-volume) and primes denote transformed quantities. 6. Quantum mechanics In non-relativistic quantum mechanics, there are important quantities known as Pauli spinors; using these spinors we are able to write down an equation (the Pauli equation) that governs the behaviour of a quantum mechanical state in some external field. The equation involves quantities called spin operators, which are conventionally seen as completely different entities to the states. Using the three-dimensional geometric algebra we are able to write down the equivalent to the Pauli equation in which the operators and states are all real-space multivectors; indeed, the spinors become rotors of the type we have discussed earlier. Now, the extension to relativistic quantum mechanics is easy. Conventionally, this is described by the Dirac algebra, where the Dirac equation again tells us about the state of the particle in an external field. This time we use the four-dimensional spacetime geometric algebra, and once again the wavefunction in conventional quantum Phil. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st century 31 mechanics becomes an instruction to rotate a basis set of axes and align them in certain directions: analogous to the theory of rigid-body mechanics! The simplicity of this approach has some interesting consequences. The Dirac equation for some external potential A can be solved, and, by seeing where the time axis, yo,has been rotated to, we can plot streamlines (lines that give the direction of the particle's velocity at each point) of the particle's motion. We can illustrate the comparison with conventional theory by a simple example. Consider the case of an incident particle packet, of energy 5 eV, say, encountering a rectangular barrier potential of height 10 eV and finite width, 5 A say (see figure 8). The theory of quantum mechanics enables us to predict that despite the seemingly impenetrable barrier, some of the packet indeed emerges the other side-an effect called tunnelling, which is of fundamental importance in many of today's semiconductor devices. However, when we ask the apparently obvious question of how long does a tunnelling particle spend inside the barrier, quantum theory provides us with a variety of answers: (a) this cannot be discussed as time is not a Hermitian observable; (b) the time is identically zero; (c) the time taken is imaginary. Why should quantum mechanics make such strange predictions? The main reason for the inability to deal with the path of the particlelpacket within the barrier lies in the use of i, the uninterpreted scalar imaginary (i2 = -1); conventionally, the momentum of the particle within the barrier is taken as a multiple of i and this leads to these rather unhelpful ideas of imaginary time. However, the geometric algebra approach tells us that we should plot the streamlines representing the path of the particle within the barrier, and, hence, find how much time they really spend inside the barrier. Not too far into the next millennium it may be possible to compare the times given by this theory with times measured in actual experiments. Figure 9 shows the predicted streamlines of particles starting at different positions within the wave packet of energy 5 eV incident on a barrier of height 10 eV and width 5 A,as depicted above. It can be seen that the particle streamlines slow up while in the barrier. This is in contrast with some recent discussions of superluminal velocities within such barriers, which have been inferred from the experimentally observed fact that particles tunnelling through a barrier reach a target before those travelling an equivalent distance in free space. This apparent contradiction is explained here by the fact that it is particles near the front of the wave packet, which already have a head start, that are transmitted and able to reach the target.? 7. Gravity Electromagnetism is a gauge theory. A gauge theory occurs if we stipulate that global symmetries must also become local symmetries (in electromagnetism these symmetries are called phase rotations); the price one has to pay to achieve this is the introduction of forces. In geometric algebra, gravity can also be regarded as a gauge t It is interesting to note that much of the currently fashionable area of quantum cosmology is based on the concepts of imaginary time. Phil. Dans. R. Soc. Lond. A (2000) J. Lasenby, A. N. Lasenby and C. J. L. Doran time (10-14 s) Figure 9. Streamlines of particles (energy 5 eV) incident on a barrier (energy 10 eV). z is the distance (in angstroms) in the direction of travel and the barrier is between z = 0 and z = 5. Particles start out at different positions within the wave packet, illustrated by the spread of lines along the z = -10 axis. Particles near the front of the wave packet are transmitted whereas those near the back are reflected. theory, and here the symmetries are much easier to understand. Suppose we require that physics at all points of space-time is invariant under arbitrary local displacements and rotations (recall that by rotations in four dimensions we are referring to Lorentz boosts); the gauge field that results from such a requirement is the gravitational field. A consequence of this theory is the huge simplification of being able to discuss gravity entirely in a flat space-time background (Lasenby et al. 1998). There is no need for the complex notions of curved space-time that we are all used to associating with Einstein's theory of general relativity (GR). This is where the GA approach differs from past gauge-theoretic approaches to gravity-these past theories have still retained the ideas of a curved space-time background. Locally, the GA gauge theory of gravity reproduces all the results of general relativity, but globally, the two theories will differ when issues of topology are at hand. For example, whenever there is discussion of singularities or horizons (as with black holes), the GA theory can give different predictions to conventional GR. Some improved methods for solution, working entirely with physical quantities, have also been found in GA. The GA gauge theory of gravity deals with extreme fields (i.e. fields in which singularities occur) in a different manner to GR. These singularities are treated simply in a manner analogous to that employed in electromagnetism (using integral theorems). The interaction with quantum fields is also different and suggests an alternative route to a quantum theory of gravity. In this context, it is also interesting to note that many of the other fashionable attempts at uniting gravity and quantum theory (twistors, supergravity, superstrings) also sit naturally within the GA framework. Phil. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st centurg Figure 10. Model of a beam split into very small segments; the deformation is described by the position and orientation of each segment. 8. Rods, shells and buckling beams It is not only in the areas of fundamental physics that geometric algebra is a useful tool. The concept of a frame of reference that varies in either space or time (or both) is at the heart of much work that tries to understand deforming bodies. Let us take, as a simple example, a beam of uniform cross-section that is subject to some loading along its length; the properties of the beam and the loading will determine how the beam deforms. Mathematically, we can describe the deformation by splitting up the beam into very small segments and attaching a frame (three mutually perpendicular axes) to the centre of mass of each segment. Initially, under no loading and no torsion, we expect the origin, 0, of each frame to be along the centreline of the beam and that each frame is aligned so that the x-axis points along the length of the beam and the z-axis vertically upwards. Now, as the beam deforms, we can describe its position at a given time by specifying the position of the origin and the orientation of the frame for each segment. Suppose we have a fixed frame at one end of the beam, the frame at segment i will then be related to this fixed frame by some rotor, Ri.Thus, as we move along the beam, the orientations are described by a rotor that varies with distance x (see figure 10). For a given loading and specified boundary conditions one might want to solve for the rotors to give information on the buckling properties of the beam. Conventionally, this task has been carried out using a variety of means to encode rotations; Euler angles, rotational parameters, direction cosines, rotation matrices, etc. The advantage of using rotors is twofold; firstly, they automatically have the correct number of degrees of freedom (three), unlike, for example, direction cosines Phil. Trans. R. Soc. Lond. A (2000) 34 J. Lasenby, A. N. Lasenby and C. J. L. Doran (where we have nine parameters, only three of which are independent), and secondly, we can solve the full equations (without approximations) in an efficient manner. One can take this idea of varying frames one stage further. Today, much of the research in modern structural mechanics has become the province of the mathematician. In order to deal with thin structures such as rods and shells, where, under deformation, the surface structure can be fairly complicated, people saw that areas of mathematics such as dzflerentzal geometry and dzflerential topology might provide useful tools. Indeed, much of the finite-element code used today in standard structural engineering packages is written from algorithms based on this mathematics. The outcome is, however, that many of the engineers can no longer understand the working of such packages, and must take for granted that what they are using is correct. On the other hand, using geometric algebra, the problem again reduces to having rotors that may vary in time and/or space across any given surface; the mathematics is no harder than one would use to solve simple mechanics problems (McRobie & Lasenby 1999). The internal finite-element code thus becomes accessible to engineers and modifications are possible. 9. Computer vision and motion analysis Computer vision is essentially the art of reconstructing or inferring things about the real three-dimensional world from views of the scene taken by one or several cameras. The positions and orientations of the cameras may or may not be known and the internal parameters of the cameras (which determine how the images we see differ from those that would result from a perfect projection onto an image plane) may also be unknown. From this rather simplified description, one can see that a significant amount of three-dimensional geometry will be involved. In fact, since the mid-1980s much of computer vision has been written in the language of projective geometry (Faugeras 1993). In classical projective geometry, we define a three-dimensional space whose points correspond to lines through some origin (specified point) in a fourdimensional space. Using such a system, the algebra of incidence (intersections of lines, planes, etc.) is extremely elegant, and, moreover, transformations that appear complicated in three dimensions (e.g. projection of points, lines, etc., down onto a given plane) now become simple. In recent years, people have started to use an algebra called the Grassmann-Cayley algebra for projective geometry calculations and manipulations; this is effectively Grassmann's exterior algebra as it restricts itself to using only the outer product. Geometric algebra contains the exterior algebra as a subset and is, therefore, an ideal language for expressing all the ideas of projective geometry (Hestenes & Ziegler 1991; Lasenby & Bayro-Corrochano 1997). However, GA also has the notion of an inner product, which often allows us to do things that would be very difficult with only an outer product. To illustrate another way in which geometric algebra can be used in computer vision, let us look at a problem that occurs in motion analysis (the reconstruction of the three-dimensional motion of an object from the image coordinates of matched points in several camera views), in scene reconstruction and image registration (mosaicking a number of different, overlapping images when limited information is available). Suppose we have a number of cameras observing an object, we suppose also, for convenience, that markers are placed on the object so that these points Phil. Trans. R. Soc. Lond. A (2000) A unzfied mathematical language for the 21st century Figure 11. Schematic showing a marked object observed by a system of cameras feeding data back to the processor. can be easily extracted from the images. Figure 11shows a sketch of a three-camera system. Now, if we observe a scene with, say, M cameras, we will find that in each pair of cameras there is a subset of the total number of markers that are visible. The first problem is to find, using these M images, the best estimates of the relative positions and orientations of each camera. Once we know the positions of the cameras we would like to triangulate in order to find the three-dimensional coordinates of other world points visible in a number of images; these problems are not too difficult for exactly known image points but become much harder if these points are noisy. There do of course exist conventional techniques for solving these problems, indeed, photogrammetrists have been doing precisely this for many years. However, the solutions generally involve large optimizations, which are often unstable. This is where geometric algebra can help. Using GA, it is possible to solve both the calibration and triangulation problems in a way that takes into account all the data from each Phil. Trans. R. Soc. Lond. A (2000) 36 J. Lasenby, A. N. Lasenby and C. J. L. Doran camera simultaneously. The optimizations involved in the solutions are able to use both first and second analytic (as opposed to numerical) derivativest of all quantities to be estimated in a consistent way. Conventionally, it is much harder to take derivatives of quantities representing rotations. Using GA in this way, it is possible to produce accurate solutions while reducing the computational load, thus making it useful in applications that require many such optimizations. 10. Conclusions We have attempted to give a brief introduction to the mathematical system we refer to as geometric algebra and to illustrate its usefulness in a variety of fields. While we have discussed a range of topics from quantum mechanics to buckling beams, there are many persuasive examples of the use of GA in physics and engineering that we have not discussed. These include electromagnetics, polarization, geometric modelling and linear algebra. The modern tools of mathematics, of which most of us are familiar with but a few, are varied and complex. In one lifetime of research we can hope only to master a very few areas. However, if most of physics and mathematics were to use the same language, the situation would perhaps be different. We hope that we have shown in this paper that geometric algebra is a candidate for such a unified language. References Baylis, W. (ed.) 1996 Clifford (geometric) algebras: with applications in physics, mathematics and engineering. Boston, MA: Birkhauser. Clifford, W. K. 1878 Applications of Grassmann's extensive algebra. Am. J. Math. 1,350-358. Doran, C. J. L., Lasenby, A. N., Gull, S. F., Somaroo, S. & Challinor, A. 1996 Spacetime algebra and electron physics. Adv. Electronics Electron Phys. 95, 272-383. Faugeras, 0. 1993 Three-dimensional computer vision: a geometric viewpoint. Artificial intelligence. Cambridge, MA: MIT Press. Gibbs, W. 1906 The scientific papers of Willard Gibbs, vol. 3. London: Longmans Green. Grassmann, H. 1844 Die Wissenschaft der extensiven Grosse oder die Ausdehnungslehre, eine neue mathematischen Disciplin. Leipzig. Grassmann, H. 1877 Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehre. Math. Ann. 12, 375. Gull, S. F., Lasenby, A. N. & Doran, C. J. L. 1993 Imaginary numbers are not real-the geometric algebra of spacetime. Found. Phys. 23, 1175. Hamilton, W. R. 1844 On quaternions: or a new system of imaginaries in algebra. Phil. Mag. 3rd Series 25, 489-495. Hestenes, D. 1966 Space-time algebra. London: Gordon and Breach. Hestenes, D. 1986 New foundations for classical mechanics. Dordrecht: Reidel. Hestenes, D. & Sobczyk, G. 1984 Clifford algebra to geometric calculus: a unified language for mathematics and physics. Dordrecht: Reidel. Hestenes, D. & Ziegler, R. 1991 Projective geometry with Clifford algebra. Acta Applicandae Mathematicae 23, 25-63. t A derivative is simply the rate of change of a quantity; for example, speed and acceleration are the first and second derivatives of distance with respect to time. Phil. Trans. R. Soc. Lond. A (2000) A unified mathematical language for the 21st century 37 Lasenby, J. & Bayro-Corrochano, E. 1997 Computing 3D projective invariants from points and lines. In Proc. 7th Int. Conf. Computer Analysis of Images and Patterns (CAIP-97), Kiel, Germany, September 10-12, pp. 334-338. Lasenby, A. N., Doran, C. J. L. & Gull, S. F. 1998 Gravity, gauge theories and geometric algebra. Phil. Trans. R. Soc. Lond. A 356, 487-582. McRobie, F. A. & Lasenby, J. 1999 Simo-Vu Quoc rods using Clifford algebra. Int. J. Num. Methods. Engng 45, 377-398. Sommer, G. (ed.) 2000 Applications of geometric algebra in engineering. Springer. (In the press.) Phil. Trans. R.Soc. Lond. A (2000) AUTHOR PROFILES Born in Aldershot, Hants, Chris Doran studied at Cambridge University obtaining a Distinction in Part I11 Mathematics and a PhD in 1994. He was elected a Junior Research Fellow of Churchill College in 1993, and was made a Lloyd's of London Fellow in 1996. He currently holds an EPSRC Advanced Fellowship, and is the Schlumberger Interdisciplinary Research Fellow of Darwin College, Cambridge. Chris has published widely on aspects of mathematical physics and is currently researching the applications of geometric methods in engineering. His interests include geometric algebra, computer vision, robotics, general relativity, and quantum field theory. Recreations include hockey, mountain biking and running. He is also a keen squash player and a poor golfer. J. Lasenby Joan Lasenby was born in Liverpool in 1960 and studied mathematics at Cambridge University, graduating with first-class honours in 1981. She gained a Distinction in Part I11 Mathematics in 1983 and a PhD in radio astronomy in 1987. She then held a Junior Research Fellowship at Trinity Hall College from 1986 to 1989 and worked for Marconi Research Laboratory from 1989 to 1990. She is currently a Royal Society University Research Fellow in the Signal Processing Group of the Cambridge University Engineering Department and a Fellow of Newnham College. Her research interests include applications of geometric algebra in computer vision and robotics, motion analysis, constrained optimization and structural mechanics. She has two children and time for just one hobby, running. A. Lasenby Born in Malvern in 1954, Anthony Lasenby read mathematics at King's College, Cambridge, and then worked for Decca in London while obtaining a part-time MSc in astrophysics from Queen Mary College. He then returned full time to academic studies, obtaining a PhD from Jodrell Bank in 1981 and working for the National Radio Astronomy Observatory in America from 1983 to 1984. He was appointed Lecturer in Radio Astronomy at the University of Cambridge in 1984 and was made a Reader in Physics in 1996. He is also a Fellow and Director of Studies in Physics at Queens' College. His research interests include cosmology, microwave background astronomy and applications of geometric algebra to physics, particularly astrophysics.