Download Reforming the Mathematical Language of Physics and more Lecture notes Quantum Mechanics in PDF only on Docsity!
Oersted Medal Lecture 2002: Reforming
the Mathematical Language of Physics
David Hestenes Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287- The connection between physics teaching and research at its deepest level can be illuminated by Physics Education Research (PER). For students and scientists alike, what they know and learn about physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmenta- tion of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathemati- cal language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geo- metric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics. Introducing research-based re- form into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun!
I. Introduction
The relation between teaching and research has been a perennial theme in academia as well as the Oersted Lectures, with no apparent progress on re- solving the issues. Physics Education Research (PER) puts the whole matter into new light, for PER makes teaching itself a subject of research. This shifts attention to the relation of education research to scientific research as the central issue. To many, the research domain of PER is exclusively pedagogical. Course content is taken as given, so the research problem is how to teach it most effec- tively. This approach to PER has produced valuable insights and useful results. However, it ignores the possibility of improving pedagogy by reconstructing course content. Obviously, a deep knowledge of physics is needed to pull off anything more than cosmetic reconstruction. It is here, I contend, in addressing the nature and structure of scientific subject matter, that PER and scientific research overlap and enrich one another with complementary perspectives. The main concern of my own PER has been to develop and validate a sci- entific Theory of Instruction to serve as a reliable guide to improving physics teaching. To say the least, many physicists are dubious about the possibility. Even the late Arnold Arons, patron saint of PER, addressed a recent AAPT session with a stern warning against any claims of educational theory. Against this backdrop of skepticism, I will outline for you a system of general principles that have guided my efforts in PER. With sufficient elaboration (much of which
already exists in the published literature), I believe that these principles provide quite an adequate Theory of Instruction. Like any other scientific theory, a Theory of Instruction must be validated by testing its consequences. This has embroiled me more than I like in developing suitable instruments to assess student learning. With the help of these instru- ments, the Modeling Instruction Program has amassed a large body of empirical evidence that I believe supports my instructional theory. We cannot review that evidence here, but I hope to convince you with theoretical arguments. A brief account of my Theory of Instruction sets the stage for the main subject of my lecture: a constructive critique of the mathematical language used in physics with an introduction to a unified language that has been developed over the last forty years to replace it. The generic name for that language is Geometric Algebra (GA). My purpose here is to explain how GA simplifies and clarifies the structure of physics, and thereby convince you of its immense implications for physics instruction at all grade levels. I expound it here in sufficient detail to be useful in instruction and research and to provide an entre´e to the published literature. After explaining the utter simplicity of the GA grammar in Section V, I explicate the following unique features of the mathematical language: (1) GA seamlessly integrates the properties of vectors and complex numbers to enable a completely coordinate-free treatment of 2D physics. (2) GA articulates seamlessly with standard vector algebra to enable easy contact with standard literature and mathematical methods. (3) GA Reduces “grad, div, curl and all that” to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it. (4) The GA formulation of spinors facilitates the treatment of rotations and rotational dynamics in both classical and quantum mechanics without coordi- nates or matrices. (5) GA provides fresh insights into the geometric structure of quantum me- chanics with implications for its physical interpretation. All of this generalizes smoothly to a completely coordinate-free language for spacetime physics and general relativity to be introduced in subsequent papers. The development of GA has been a central theme of my own research in theoretical physics and mathematics. I confess that it has profoundly influenced my thinking about PER all along, though this is the first time that I have made it public. I have refrained from mentioning it before, because I feared that my ideas were too radical to be assimilated by most physicists. Today I am coming out of the closet, so to speak, because I feel that the PER community has reached a new level of maturity. My suggestions for reform are offered as a challenge to the physics community at large and to the PER community in particular. The challenge is to seriously consider the design and use of mathematics as an important subject for PER. No doubt many of you are wondering why, if GA is so wonderful – why have you not heard of it before? I address that question in the penultimate Section by discussing the reception of GA and similar reforms by the physics community and their bearing on prospects for incorporating GA
- Expert learning requires deliberate practice with critical feed- back. There is substantial evidence that practice does not significantly improve intellectual performance unless it is guided by critical feedback and deliberate attempts to improve.^4 Students waste an enormous amount of time in rote study that does not satisfy this principle. I believe that all five principles are essential to effective learning and instruc- tional design, though they are seldom invoked explicitly, and many efforts at educational reform founder because of insufficient attention to one or more of them. The terms “concept” and “conceptual learning” are often tossed about quite cavalierly in courses with names like “Conceptual Physics.” In my expe- rience, such courses fall far short of satisfying the above learning principles, so I am skeptical of claims that they are successful in teaching physics concepts without mathematics. The degree to which physics concepts are essentially mathematical is a deep problem for PER. I should attach a warning to the First (Constructivist) Learning Principle. There are many brands of constructivism, differing in the theoretical context afforded to the constructivist principle. An extreme brand called “radical con- structivism” asserts that constructed knowledge is peculiar to an individual’s experience, so it denies the possibility of objective knowledge. This has radi- calized the constructivist revolution in many circles and drawn severe criticism from scientists.^5 I see the crux of the issue in the fact that the constructivist principle does not specify how knowledge is constructed. When this gap is closed with the other learning principles and scientific standards for evidence and inference, we have a brand that I call scientific constructivism. I see the five Learning Principles as equally applicable to the conduct of research and to the design of instruction. They support the popular goal of “teaching the student to think like a scientist.” However, they are still too vague for detailed instructional design. For that we need to know what counts as a scientific concept, a subject addressed in the next Section.
III. Modeling Theory
Modeling Theory is about the structure and acquisition of scientific knowledge. Its central tenet is that scientific knowledge is created, first, by constructing and validating models to represent structure in real objects and processes, and second, by organizing models into theories structured by scientific laws. In other words, Modeling Theory is a particular brand of scientific epistemology that posits models as basic units of scientific knowledge and modeling (the process of creating and validating models) as the basic means of knowledge acquisition. I am not alone in my belief that models and modeling constitute the core of scientific knowledge and practice. The same theme is prominent in recent History and Philosophy of Science, especially in the work of Nancy Nercessian and Ronald Giere,^6 and in some math education research.^7 It is also proposed as a unifying theme for K-12 science education in the National Science Education
Standards and the AAAS Project 2061. The Modeling Instruction Project has led the way in incorporating it into K-12 curriculum and instruction.^3 Though I first introduced the term “Modeling Theory” in connection with my Theory of Instruction,^8 I have always conceived of it as equally applicable to scientific research. In fact, Modeling Theory has been the main mechanism for transferring what I know about research into designs for instruction. In so far as Modeling Theory constitutes an adequate epistemology of sci- ence, it provides a reliable framework for critique of the physics curriculum and a guide for revising it. From this perspective, I see the standard curriculum as seriously deficient at all levels from grade school to graduate school. In partic- ular, the models inherent in the subject matter are seldom clearly delineated. Textbooks (and students) regularly fail to distinguish between models and their implications.^2 This results in a cascade of student learning difficulties. However, we cannot dwell on that important problem here. I have discussed Modeling Theory and its instructional implications at some length elsewhere,2, 8–10^ although there is still more to say. The brief account above suffices to set the stage for application to the main subject of this lecture. Modeling Theory tells us that the primary conceptual tools mentioned in the 4th Learning Principle are modeling tools. Accordingly, I have devoted considerable PER effort to classification, design, and use of modeling tools for instruction. Heretofore, emphasis has been on the various kinds of graphs and diagrams used in physics, including analysis of the information they encode and comparison with mathematical representations.9, 10^ All of this was motivated and informed by my research experience with mathematical modeling. In the balance of this lecture, I draw my mathematics research into the PER domain as an example of how PER can and should be concerned with basic physics research.
IV. Mathematics for Modeling Physical Reality
Mathematics is taken for granted in the physics curriculum—a body of im- mutable truths to be assimilated and applied. The profound influence of math- ematics on our conceptions of the physical world is never analyzed. The pos- sibility that mathematical tools used today were invented to solve problems in the past and might not be well suited for current problems is never considered. I aim to convince you that these issues have immense implications for physics education and deserve to be the subject of concerted PER. One does not have to go very deeply into the history of physics to discover the profound influence of mathematical invention. Two famous examples will suffice to make the point: The invention of analytic geometry and calculus was essential to Newton’s creation of classical mechanics.^2 The invention of tensor analysis was essential to Einstein’s creation of the General Theory of Relativity. Note my use of the terms “invention” and “creation” where others might have used the term “discovery.” This conforms to the epistemological stance of Modeling Theory and Einstein himself, who asserted that scientific theories
sented in several different systems, but one of them is invariably better suited than the others for a given application. For example, Goldstein’s textbook on mechanics^12 gives three different ways to represent rotations: coordinate matri- ces, vectors and Pauli spin matrices. The costs in time and effort for translation between these representations are considerable.
- Deficient integration. The collection of systems in Fig. 1 is not an integrated mathematical structure. This is especially awkward in problems that call for the special features of two or more systems. For, example, vector algebra and matrices are often awkwardly combined in rigid body mechanics, while Pauli matrices are used to express equivalent relations in quantum mechanics.
- Hidden structure. Relations among physical concepts represented in different symbolic systems are difficult to recognize and exploit.
- Reduced information density. The density of information about physics is reduced by distributing it over several different symbolic systems. Evidently elimination of these defects will make physics easier to learn and apply. A clue as to how that might be done lies in recognizing that the various symbolic systems derive geometric interpretations from a common coherent core of geometric concepts. This suggests that one can create a unified mathematical language for physics by designing it to provide an optimal representation of geometric concepts. In fact, Hermann Grassmann recognized this possibility and took it a long way more than 150 years ago.^13 However, his program to unify mathematics was forgotten and his mathematical ideas were dispersed, though many of them reappeared in the several systems of Fig. 1. A century later the program was reborn, with the harvest of a century of mathematics and physics to enrich it. This has been the central focus of my own scientific research. Creating a unified mathematical language for physics is a problem in the design of mathematical systems. Here are some general criteria that I have applied to the design of Geometric Algebra as a solution to that problem:
- Optimal algebraic encoding of the basic geometric concepts: magni- tude, direction, sense (or orientation) and dimension.
- Coordinate-free methods to formulate and solve basic equations of physics.
- Optimal uniformity of method across classical, quantum and relativis- tic theories to make their common structures as explicit as possible.
- Smooth articulation with widely used alternative systems (Fig. 1) to facilitate access and transfer of information.
- Optimal computational efficiency. The unified system must be at least as efficient as any alternative system in every application. Obviously, these design criteria ensure built-in benefits of the unified lan- guage. In implementing the criteria I deliberately sought out the best available mathematical ideas and conventions. I found that it was frequently necessary to modify the mathematics to simplify and clarify the physics. This led me to coin the dictum: Mathematics is too important to be left to the mathematicians! I use it to flag the following guiding principle for Modeling Theory: In the development of any scientific theory, a
major task for theorists is to construct a mathematical language that optimizes expression of the key ideas and consequences of the theory. Although existing mathematics should be consulted in this endeavor, it should not be incorporated without critically evaluating its suitability. I might add that the process also works in reverse. Modification of mathematics for the purposes of science serves as a stimulus for further development of mathematics. There are many examples of this effect in the history of physics. Perhaps the most convincing evidence for validity of a new scientific theory is successful prediction of a surprising new phenomenon. Similarly, the most impressive benefits of Geometric Algebra arise from surprising new insights into the structure of physics. The following Sections survey the elements of Geometric Algebra and its application to core components of the physics curriculum. Many details and derivations are omitted, as they are available elsewhere. The emphasis is on highlighting the unique advantages of Geometric Algebra as a unified mathe- matical language for physics.
V. Understanding Vectors
A recent study on the use of vectors by introductory physics students summa- rized the conclusions in two words: “vector avoidance!”^14 This state of mind tends to propagate through the physics curriculum. In some 25 years of graduate physics teaching, I have noted that perhaps a third of the students seem inca- pable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus! 15 Once the coordinate virus has been identified, it becomes evident that the entire physics curriculum, including most of the textbooks, is infected with the virus. From my direct experience, I estimate that two thirds of the graduate students have serious infections, and half of those are so damaged by the virus that they will never recover. What can be done to control this scourge? I suggest that universal inoculation with Geometric Algebra could eventually eliminate the coordinate virus altogether. I maintain that the origin of the problem lies not so much in pedagogy as in the mathematics. The fundamental geometric concept of a vector as a directed magnitude is not adequately represented in standard mathematics. The basic definitions of vector addition and scalar multiplication are essential to the vector concept but not sufficient. To complete the vector concept we need multiplication rules that enable us to compare directions and magnitudes of different vectors.
a
b
a
b
a b = = b a
Fig. 2. Bivectors a ∧ b and b ∧ a represent plane segments of op- posite orientation as specified by a “parallelogram rule” for drawing the segments.
From the geometric interpretations of the inner and outer products, we can infer an interpretation of the geometric product from extreme cases. For or- thogonal vectors, we have from (5)
a · b = 0 ⇐⇒ ab = −ba. (8)
On the other hand, collinear vectors determine a parallelogram with vanishing area (Fig. 2), so from (6) we have
a ∧ b = 0 ⇐⇒ ab = ba. (9)
Thus, the geometric product ab provides a measure of the relative direction of the vectors. Commutativity means that the vectors are collinear. Anticom- mutativity means that they are orthogonal. Multiplication can be reduced to these extreme cases by introducing an orthonormal basis.
B. Basis and Bivectors
For an orthonormal set of vectors {σ 1 , σ 2 , ...}, the multiplicative properties can be summarized by putting (5) in the form
σi · σj = 12 (σiσj + σj σi) = δij (10)
where δij is the usual Kroenecker delta. This relation applies to a Euclidean vector of any dimension, though for the moment we focus on the 2D case. A unit bivector i for the plane containing vectors σ 1 and σ 2 is determined by the product
i = σ 1 σ 2 = σ 1 ∧ σ 2 = −σ 2 σ 1 (11)
The suggestive symbol i has been chosen because by squaring (11) we find that
i^2 = − 1 (12)
Thus, i is a truly geometric
√ −1. We shall see that there are others. From (11) we also find that
σ 2 = σ 1 i = −i σ 1 and σ 1 = iσ 2. (13)
In words, multiplication by i rotates the vectors through a right angle. It follows that i rotates every vector in the plane in the same way. More generally, it follows that every unit bivector i satisfies (12) and determines a unique plane in Euclidean space. Each i has two complementary geometric interpretations: It represents a unique oriented area for the plane, and, as an operator, it represents an oriented right angle rotation in the plane.
C. Vectors and Complex Numbers
Assigning a geometric interpretation to the geometric product is more subtle than interpreting inner and outer products — so subtle, in fact, that the ap- propriate assignment has been generally overlooked to this day. The product of any pair of unit vectors a, b generates a new kind of entity U called a rotor, as expressed by the equation
U = ab. (14)
The relative direction of the two vectors is completely characterized by the directed arc that relates them (Fig. 3), so we can interpret U as representing that arc. The name “rotor” is justified by the fact that U rotates a and b into each other, as shown by multiplying (14) by vectors to get
b = aU and a = U b. (15)
Further insight is obtained by noting that
a · b = cos θ and a ∧ b = i sin θ, (16)
where θ is the angle from a to b. Accordingly, with the angle dependence made explicit, the decomposition (7) enables us to write (14) in the form
Uθ = cos^ θ^ +^ i^ sin^ θ^ =^ ei^ θ^.^ (17)
It follows that multiplication by Uθ , as in (15), will rotate any vector in the i-plane through the angle θ. This tells us that we should interpret Uθ as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vector a as a directed line segment that can be translated at will without changing its length or direction (Fig. 4).
b
a
Fig. 3. A pair of unit vectors a, b determine a directed arc on the unit circle that represents their product U = ab. The length of the arc is (radian measure of) the angle θ between the vectors.
. U
.^
Fig. 6. A complex number z = λU with modulus λ and angle θ can be interpreted as a directed arc on a circle of radius λ. Its conjugate z†^ = λU †^ represents an arc with opposite orientation
Anyone who has worked with complex numbers in applications knows that it is usually best to avoid decomposing them into real and imaginary parts. Likewise, in GA applications it is usually best practice to work directly with the geometric product instead of separating it into inner and outer products. GA gives complex numbers new powers to operate directly on vectors. For example, from (19) and (20) we get
b = a−^1 z = z†a−^1 , (22)
where the multiplicative inverse of vector a is given by
a−^1 = 1 a
= a a^2
= a | a |^2
. (23)
Thus, z rotates and rescales a to get b. This makes it possible to construct and manipulate vectorial transformations and functions without introducing a basis or matrices. This is a good point to pause and note some instructive implications of what we have established so far. Every physicist knows that complex numbers, especially equations (17) and (18), are ideal for dealing with plane trigonometry and 2D rotations. However, students in introductory physics are denied access to this powerful tool, evidently because it has a reputation for being conceptually difficult, and class time would be lost by introducing it. GA removes these barriers to use of complex numbers by linking them to vectors and giving them a clear geometric meaning. GA also makes it possible to formulate and solve 2D physics problems in terms of vectors without introducing coordinates. Conventional vector algebra cannot do this, in part because the vector cross product is defined only in 3D. That is the main reason why coordinate methods dominate introductory physics. The available math tools are too weak to do otherwise. GA changes all that! For example, most of the mechanics problems in introductory physics are 2D problems. Coordinate-free GA solutions for the standard problems are worked out in my mechanics book.^16 Although the treatment there is for a more ad- vanced course, it can easily be adapted to the introductory level. The essential GA concepts for that level have already been presented in this section.
Will comprehensive use of GA significantly enhance student learning in in- troductory physics? We have noted theoretical reasons for believing that it will. To check this out in practice is a job for PER. However, mathematical reform at the introductory level makes little sense unless it is extended to the whole physics curriculum. The following sections provide strong justification for doing just that. We shall see how simplifications at the introductory level get amplified to greater simplifications and surprising insights at the advanced level.
VI. Classical Physics with Geometric Algebra
This section surveys the fundamentals of GA as a mathematical framework for classical physics and demonstrates some of its unique advantages. Detailed applications can be found in the references.
A. Geometric Algebra for Physical Space
The arena for classical physics is a 3D Euclidean vector space P^3 , which serves as a model for “Physical Space.” By multiplication and addition the vectors generate a geometric algebra G 3 = G(P^3 ). In particular, a basis for the whole algebra can be generated from a standard frame {σ 1 , σ 2 , σ 3 }, a righthanded set of orthonormal vectors. With multiplication specified by (10), the standard frame generates a unique trivector (3-vector) or pseudoscalar
i = σ 1 σ 2 σ 3 , (24)
and a bivector (2-vector) basis
σ 1 σ 2 = iσ 3 , σ 2 σ 3 = iσ 1 , σ 3 σ 1 = iσ 2. (25)
Geometric interpretations for the pseudoscalar and bivector basis elements are depicted in Figs. 7 and 8.
Fig. 7. Unit pseudoscalar i represents an oriented unit volume. The volume is said to be righthanded, because i can be generated from a righthanded vector basis by the ordered product σ 1 σ 2 σ 3 = i.
attributed to this structure hinges on the geometric meaning of i. The most important example is the expression of the electromagnetic field F in terms of an electric vector field E and a magnetic vector field B:
F = E + iB. (31)
Geometrically, this is a decomposition of F into vector and bivector parts. In standard vector algebra E is said to be a polar vector while B is an axial vector, the two kinds of vector being distinguished by a difference in sign under space inversion. GA reveals that an axial vector is just a bivector represented by its dual, so the magnetic field in (31) is fully represented by the complete bivector iB, rather than B alone. Thus GA makes the awkward distinction between polar and axial vectors unnecessary. The vectors E and B in (31) have the same behavior under space inversion, but an additional sign change comes from space inversion of the pseudoscalar. To facilitate algebraic manipulations, it is convenient to introduce a special symbol for the operation (called reversion) of reversing the order of multiplica- tion. The reverse of the geometric product is defined by
(ab)†^ = ba. (32)
We noted in (20) that this is equivalent to complex conjugation in 2D. From (24) we find that the reverse of the pseudoscalar is
i†^ = −i. (33)
Hence the reverse of an arbitrary multivector in the expanded form (30) is
M †^ = α + a − ib − iβ, (34)
The convenience of this operation is illustrated by applying it to the electro- magnetic field F in (31) and using (29) to get
1 2 F F^
† (^) = 1 2 (E^ +^ iB)(E^ −^ iB) =^
1 2 (E
(^2) + B (^2) ) + E × B, (35)
which is recognized as an expression for the energy and momentum density of the field. Note how this differs from the field invariant
F 2 = (E + iB)^2 = E^2 − B^2 + 2i(E·B), (36)
which is useful for classifying EM fields into different types. You have probably noticed that the expanded multivector form (30) violates one of the basic math strictures that is drilled into our students, namely, that “it is meaningless to add scalars to vectors,” not to mention bivectors and pseudoscalars. On the contrary, GA tells us that such addition is not only geometrically meaningful, it is essential to simplify and unify the language of physics, as can be seen in many examples that follow. Shall we say that this stricture against addition of scalars to vectors is a misconception or conceptual virus that infects the entire physics community?
At least it is a design flaw in standard vector algebra that has been almost universally overlooked. As we have just seen, elimination of the flaw enables us to combine electric and magnetic fields into a single electromagnetic field. And we shall see below how it enables us to construct spinors from vectors (contrary to the received wisdom that spinors are more basic than vectors)!
B. Reflections and Rotations
Rotations play an essential role in the conceptual foundations of physics as well as in many applications, so our mathematics should be designed to handle them as efficiently as possible. We have noted that conventional treatments employ an awkward mixture of vector, matrix and spinor or quaternion methods. My purpose here is to show how GA provides a unified, coordinate-free treatment of rotations and reflections that leaves nothing to be desired. The main result is that any orthogonal transformation U can be expressed in the canonical form^16
U x = ±U x U †, (37)
where U is a unimodular multivector called a versor, and the sign is the parity of U , positive for a rotation or negative for a reflection. The condition
U †U = 1. (38)
defines unimodularity. The underbar notation serves to distinguish the linear operator U from the versor U that generates it. The great advantage of (37) is that it reduces the study of linear operators to algebraic properties of their versors. This is best understood from specific examples. The simplest example is reflection in a plane with unit normal a (Fig. 9),
x′^ = −axa = −a(x⊥ + x‖)a = x⊥ − x‖. (39)
To show how this function works, the vector x has been decomposed on the right into a parallel component x‖ = (x · a)a that commutes with a and an orthogonal component x⊥ = (x ∧ a)a that anticommutes with a. As can be seen below, it is seldom necessary or even advisable to make this decomposition in applications. The essential point is that the normal vector defining the direc- tion of a plane also represents a reflection in the plane when interpreted as a versor. A simpler representation for reflections is inconceivable, so it must be the optimal representation for reflections in every application, as shown in some important applications below. Incidentally, the term versor was coined in the 19 th^ century for an operator that can re-verse a direction. Likewise, the term is used here to indicate a geometric operational interpretation for a multivector. The reflection (39) is not only the simplest example of an orthogonal trans- formation, but all orthogonal transformations can be generated by reflections of this kind. The main result is expressed by the following theorem: The product of two reflections is a rotation through twice the angle between the normals of
b a
x
x^ '
x^ ''
θ
(^12) θ
Fig. 10. Rotation as double reflection, depicted in the plane containing unit normals a, b of the reflecting planes.
The orthogonal transformations form a mathematical group with (44) as the group composition law. The trouble with (44) is that abstract operator algebra does not provide a way to compute U 3 from given U 1 and U 2. The usual solution to this problem is to represent the operators by matrices and compute by matrix multiplication. A much simpler solution is to represent the operators by versors and compute with the geometric product. We have already seen how the product of reflections represented by U 1 = a and U 2 = b produces a rotation U 3 = ba. Matrix algebra does not provide such a transparent result. As is well known, the rotation group is a subgroup of the orthogonal group. This is expressed by the fact that rotations are represented by unimodular versors of even parity, for which the term rotor was introduced earlier. The composition of 2D rotations is described by the rotor equation (18) and depicted in Fig. 5. Its generalization to composition of 3D rotations in different planes is described algebraically by (45) and depicted geometrically in Fig. 11. This deserves some explanation.
U
a
b c
2
U 3
U 1
Fig. 11. Addition of directed arcs in 3D depicting the product of rotors. Vectors a, b, c all originate at the center of the sphere.
In 3D a rotor is depicted as a directed arc confined to a great circle on the unit sphere. The product of rotors U 1 and U 2 is depicted in Fig. 11 by connecting the corresponding arcs at a point c where the two great circles intersect. This determines points a = cU 1 and b = U 2 c, so the rotors can be expressed as products with a common factor,
U 1 = ca, U 2 = bc. (46)
Hence (44) gives us
U 3 = U 2 U 1 = (bc)(ca) = ba, (47)
with the corresponding arc for U 3 depicted in Fig. 11. It should not be for- gotten that the arcs in Fig. 11 depict half-angles of the rotations. The non- commutativity of rotations is illustrated in Fig. 12, which depicts the construc- tion of arcs for both U 1 U 2 and U 2 U 1.
U 2 U 2
U 1 U 1
U 1
U 2
U 1 U 2
Fig. 12. Noncommutativity of rotations depicted in the construction of directed arcs representing rotor products
Those of you who are familiar with quaternions will have recognized that they are algebraically equivalent to rotors, so we might as well regard the two as one and the same. Advantages of the quaternion theory of rotations have been known for the better part of two centuries, but to this day only a small number of specialists have been able to exploit them. Geometric algebra makes them available to everyone by embedding quaternions in a more comprehen- sive mathematical system. More than that, GA makes a number of significant improvements in quaternion theory — the most important being the integration of reflections with rotations as described above. To make this point more ex- plicit and emphatic, I describe two important practical applications where the generation of rotations by reflections is essential. Multiple reflections. Consider a light wave (or ray) initially propagating with direction k and reflecting off a sequence of plane surfaces with unit normals
