Developing Tools Using Geometric Algebra for Physics Visualization

Brackenridge Fellowship Summer 2017 Student Researcher: Ryan Kaufman Faculty Mentor: Dr. Arthur Kosowsky

I. Introduction

The geometric algebra and the associated geometric Calculus offer a vast and comprehensive approach to many areas of mathematical physics. Not only does this combined mathematical system offer algebraic clarity, it also opens the door to very intuitive visual representations of both mathematical proofs and models of physical phenomena. Such representations have the potential to influence both the physics research community across the discipline as well as students learning physics in intermediate undergraduate courses. In the duration of the fellowship the goal was to develop a visualization method in the form of a scripting language capable of walking through visual proofs of various concepts in geometric algebra as well as to develop visualizations of various physical phenomena. In the process, multiple methods were attempted, some more successful than others. This paper will detail these processes as well as speculate on future continuations of the work done within the fellowship. In the process, some concepts from geometric algebra will be explained and comparisons will be drawn to traditional vector algebra in the context of introductory physics.

II. Fundamentals of geometric algebra

The geometric algebra is composed axiomatically of three operations on fundamental objects called Multivectors. Firstly, Multivectors are composed of a sum of different “grades” of vectors. For example, a grade 0 vector is a regular real number called a “scalar,” a grade 1 vector is the conventional vector (line segment with a point) from linear algebra, and a grade 2 vector is an object dubbed a

“bivector.” In short, a bivector is a “segment” of a plane that is representative of the entire infinite plane

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Ryan Kaufman that it resides within just as a vector line segment is representative of the entire line that it resides within.

Like vectors, bivectors have a sense of orientation much like a rotating wheel does. And just like common language compares rotations to the rotation of a clock so too are or bivectors limited to comparing plane orientations in geometric algebra. That is, orientations are relative, and it is only meaningful to compare the orientation of a bivector to that of another bivector and say the directions are either equal or opposite.1[1] This trend continues throughout an arbitrary number of dimensions forming oriented volumes in 3 dimensions and so on to four dimensions and above.

The three fundamental operators in the geometric algebra are addition, scalar multiplication, and the geometric product. Addition operates just like vector addition in linear algebra, with a few details on how the concept of tip-to-tail vector addition within linear algebra functions within geometric algebra.

For example, matching edges with plane segments [1]. Scalar multiplication acts as an operation that changes the magnitude of a vector (or generally a multivector) that it is acting upon. Thus, multiplying a vector by the scalar 2 doubles its length, multiplying a bivector by the scalar 2 doubles its area, a trivector

(volume element) has its volume doubled, etc.

The geometric product is the core of the applicability of the geometric algebra. The geometric product is an operation that takes in two multivectors and returns a single multivector. It is composed of two separate independent products: the inner and outer product. The inner product has multiple forms [1] but is most often analogous to the dot product from vector algebra. The outer product, on the other hand, is analogous to the outer product from the Grassman algebra. From an intuitive angle, the inner product of two vectors 푎⃗ ∙ 푏⃗⃗ is crucial to projections and can be viewed as an indication of the magnitude of the projection of vector a onto vector b [1]. Instead of “folding” two vectors into one another to form an object of lower grade, the outer product takes two vectors and forms a higher grade vector from them.

This is best illustrated in the picture below.

1 In reality this fact comes from the formation of bivectors from an operation within the geometric algebra called the outer product. However, explaining intricacies of the algebra is better left to a book like [1][MacDonald]

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Figure 1: Results of an Outer Product of two vectors in geometric algebra [A1]

In short, the outer product allows the construction of higher grade vectors from lower grade vectors. These grades can then be summed over to form various multivectors. The geometric product itself is composed of the sum of the inner product and outer product. This is expressed by what is called the fundamental identity below.

풖⃗⃗⃗풗⃗⃗⃗ = 풖⃗⃗⃗ ∙ 풗⃗⃗⃗ + 풖⃗⃗⃗ ⋀ 풗⃗⃗⃗

This knowledge covers the very basics of geometric algebra and this paper will have to defer to a more advanced text like that written by Alan MacDonald [1] for more detail for the sake of brevity.

However, these basics will be enough to sample the visual benefits of the algebra for students and researchers alike.

III. Reflections on Geometric Algebra

Having learned many aspects of the geometric algebra over the course of the fellowship, several characteristics of the algebra can be consequently identified as points of personal note to the researcher and speculation can be made about the possible advantages and difficulties of teaching such an algebra to both introductory students as well as intermediate and advanced students. These observations can be divided up into various categories and subcategories.

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Firstly, it is worth mentioning that the geometric algebra is very pedagogically dense. By this what is meant is that the axiomatic approach to teaching it involves many, many consequences to but one or two theorems at a time. The statements of theorems are not unique in this property, as the algebraic operations in the use of the algebra itself involve at times the manipulation of multivectors that span across multiple dimensions and operations such as rotations manipulate all of them at once without necessarily making the complexities of such an operation obvious.

This, in experience, has led to a very high readability, but very low rates of comprehension per reading. Readability here is used to convey a sense of intuitive understanding that a reader of mathematics gets of what the mathematical statements are conveying. As in, an algebraic statement contains a great deal of meaning that is simplistic in its intent (e.g. rotation) but complex in its execution (rotating a grade

1 vector, grade 2 vector, and so on until the multivector is exhausted) such that it takes multiple readings to comprehend just exactly how the algebraic statement or operation unfolds into step-by-step calculation and what the result of that calculation means for each grade component of a multivector.

An analogy for the quandary here is to take a fast car. A fast car is powerful, and can take you any place you’d like to be. However, to really understand the car’s workings takes time and repairing problems with the car can be time consuming and tedious. The same is true for calculation in geometric algebra. Sticking to higher level representations and symbols goes a long way for proofs of theorems, but unraveling these symbols to get to more traditional calculation can be just as messy as the tensor algebra that it replaces.

This being said, geometric algebra excels at yielding elegant theorems and statements for many fields in mathematics and in physics. As a flagship example, in David Hestenes book Spacetime algebra, he boils Maxwell’s 4 Equations in electromagnetism (which are traditionally expressed in vector calculus) down to one, uniform multivector statement in geometric Calculus with just four symbols [2]

□퐹 = 퐽

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Where, for those familiar with electrodynamics, 퐹 is a multivector analogy to the Lorentz force field, and 퐽 is a multivector combination of traditional current density definitions, and the box operator is a modified vector gradient operator which is also itself a multivector. This simplification occurs in the

Dirac algebra. Said algebra is, to be formal, a conformal model in an indefinite product space with properties analogous to the metrics used in Einstein’s theory of general relativity [1][2]. It is a type of geometric algebra. All of that technical complexity aside, this mathematically elegant statement boils down to the intuitive (but hand-wavy) statement of:

The change in the electric and magnetic fields is equivalent to shifting current densities in different

dimensions in spacetime.

In an admittedly limited experience of mathematics and physics, very rarely can such a simple statement be boiled down to an equally elegant mathematical portrayal. This has great potential to at the very least capture the interest of students without sacrificing mathematical rigor. This is the core of the impetus of the project that took place under the fellowship.

With all of this in mind, the goal of the project emerged, and that goal became developing the foundations of a system which could not only convey the elegance of geometric algebra, but also to convey the visual meanings of its operations and to use this visual tool to convey physical phenomena in an appealing and engaging manner.

IV. Project Overview

Pictured below (Figure 1) is a general outline of the project in terms of the names of various resources used to progress not only in learning the material vital to the goals of the project but to indicate its current state. In addition, the chart highlights difficulties in the progression of the project. The first task, as in any project with such a large size of approachable angles, was to narrow down the topic to a

5 Ryan Kaufman specific goal. The goals settled upon were to develop a method of visualizing geometric algebra in an accessible way that puts symbolic and algebraic notation adjacent to visual representation as to ease the hypothetical students’ linking between the two. Geometric algebra is particularly well suited to this task as the fundamental object within the algebra is a multivector, a sum of grades of vectors. Each individual grade is able to be visualized, with one major caveat. This caveat is that this is only directly feasible in dimensions less than or equal to three. Grades 4 and higher of vectors have to be projected down into 3 dimensions, with some major exceptions detailed in [3] that are worth discussion.

Figure 2: Project Overview Flowchart [A2]

In short, by the methods that Baylis discusses in detail, one can “cut” the size of a geometric algebra, for example the Dirac algebra, in half in size (to be specific, number of basis elements) via allowing the coefficients to be “complex valued.” [3] This involves the addition of a complex component in the form of a scalar multiple of what is called the ‘Pseudoscalar’ of the algebra. This technical trick results in the ability to represent higher dimensional objects with two lower dimensional objects, which

6 Ryan Kaufman once again reopens the possibility of presenting four dimensional volumes with two 2-dimensional volumes one being real and the other complex. Using methods such as this, visualizations in three dimensions can still express higher dimensions. With this clearing out the mathematical problems that were anticipated with higher grade than 3, the project could move to actually creating these representations using computer graphics in parallel with learning more theoretical material.

The creation of these graphics was first attempted using a Python 3.6 coding library called

“MatPlotLib,” although this proved challenging to represent bivectors in, as native support accounted only for grade 1 traditional vectors. Due in part to this complication, the project moved to refocus in

Blender, an open source 3d animation software that also relies on Python 3.6 to function with its own application programming interface (API). This is where the project currently resides, with the animations that have been created thus far being available to download at www.pitt.edu/~rrk26 by August 1st, 2017.

Currently, the project has produced animations in the explanation of angular momentum and torque, the addition of bivectors in terms of the geometric algebra of three dimensions, and the walkthrough of labeling the basis elements of that same algebra of three dimensions. Additionally, more literature is being read in the applications that geometric algebra has in general relativity as well as other areas of physics.

V. Future Directions

The feedback that was received indicated that the animations still needed some refinement in a few key areas, the most notable of which is the pacing of the animations. At times, they were too fast, and did not give enough time for the viewer to comprehend exactly what the animation was saying or demonstrating. This can be resolved by adding more text interludes and explanations into the animation itself, which can be done manually in the editor of Blender. Additionally, plans are in the works for developing a step-by-step rendering (the process used to create the animation from commands given to

Blender) to generate multiple GIFs of the animation instead of one long movie. This method of output

7 Ryan Kaufman would be more conducive to designing a PowerPoint presentation on the topic, as the presenter could just put subsequent steps of the animation on different slides to allow for time to explain what was happening.

Among these small fixes, there is also a framework to allow for the programmatic creation of these animations, which when completed would allow educators as well as students to generate these animations themselves without knowledge of how Blender works as an animating software. The utility of such a scripting language is best emphasized using a picture of the workspace in Blender shown below in figure 3.

Figure 3: Animating Workspace in Blender 2.78 [A3]

In short, it is the belief of the researcher that it is well worth investigating new ways of visualizing geometric algebra and by extension theoretical physics. The work will be continued into the future, to hopefully fruitful gains.

VI. Acknowledgements

I’d like to thank Dr. Arthur Kosowsky, my faculty mentor, for spurring my curiosity for the subject of geometric algebra as well as lending me some of the books required to study it at no cost. I would also like to thank Dr. Peter Koehler and the rest of the Brackenridge faculty staff

8 Ryan Kaufman for excellent feedback on aspects of my project that needed improvement and my peers in the

Brackenridge fellowship for the constructive criticism and ideas throughout the entirety of the summer. Lastly, I’d like to thank my family and friends who have made it feasible for me to do this project and have offered untold amounts of support to me throughout the summer.

VII. References: [1] Macdonald, Alan. Linear and geometric algebra. Charleston, SC: n.p., 2016. Print. [2] Hestenes, David. Space-time algebra. Cham: Birkhä user, 2015. Print. [3] Baylis, William E. Electrodynamics: A Modern geometric Approach. Boston: Birkhä user, 2002. Print.

VIII. Image References: [A1]: https://www.quora.com/What-is-a-brief-explanation-of-the-wedge-product [A2]: Created by the author on LucidChart.com [A3]: Created by the author using Blender 2.78

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