This page is a copy of a paper by Dr Alfred Differ who has kindly given me permission to reproduce it here. He retains the copyright of this paper.
Introduction
to Geometric Algebra
By adiffer
Sat Sep 7th, 2002 at 11:42:05 AM EST
This article will introduce the reader to the basics of what a geometric algebra is and what they may be used to do. Some problems will be provided that the reader may work on their own in order to reinforce the lessons described here. While a few of the solutions to these problems will be provided, readers are encouraged to solve them all and post their answers or additional problems as comments.
The purpose of this article is entirely educational. If the reader works to comprehend the content and also works the problems, they will come away with a basic understanding of what a geometric algebra is and how they work. Specific goals include the following.
Introduction
To a mathematician, an algebra is a very precise thing. The definition they provide is tight and insightful, but it also goes over the heads of most beginning students. Fortunately, it is possible to describe geometric algebras in a manner that builds on a basic understanding of geometry most people acquire in their everyday lives. This informal description leads to the use of the same objects as the mathematicians define, but it does leave out some of the powerful insights that automatically apply if one knew the formal definition. Students are encouraged to learn the formal definition when they feel prepared to do so.
In the most basic sense, a geometric algebra is a set of rules governing objects that permit us to treat geometric things with the tools we learned to use with algebraic tasks. Most students who take both an algebra and geometry class realize that the two subjects are taught differently. In an algebra class, we are taught about functions and rules that enable solving for unknown variables. We are taught to draw graphs that help us find numerical solutions to our problems. In a geometry class, we are taught about line segments, shapes, and a theorem/proof system we use to know that we actually know something. The two column proof encourages deductive thinking while helping to build a geometric intuition.
Algebra is a subject that concerns itself with numbers. Geometry is a subject that concerns itself with magnitudes. In the historical sense, these are distinct concepts even if we blur them in today's classrooms. Whether we are able to link a number to the length to a line segment or the span of an angle doesn't impact purely geometric theorems. Constructing the bisector of an angle requires no numbers. Whether we know that the unknown variable in the formula for the parabolic arc a rock takes when thrown up and forward does not influence the fact that we can solve the equation for the peak height and range on impact.
Numbers are the stuff of algebra. They are the familiar integers augmented with rational and irrational numbers to make up the whole 'real' number line. They are the complex numbers that make use of imaginary numbers if one requires that all polynomials have roots.
Magnitudes are the stuff of geometry. They are the line segments, areas, volumes, and angles of various types. These magnitudes can't be numbers since even the simplest example shows that magnitudes contain more information. Imagine a line drawn at a certain length in the northsouth direction. Draw another one at right angles to the first such that they both share one of their end points. These line segments are different magnitudes if one sticks to the historical definition of the term even though they are of the same length. Today we might be tempted to use a different word like vector, but the difference between the two line segments still stands.
The marriage of algebra with geometry creates a powerful union with many unexpected offspring. Trigonometry is one such wellknown offspring. Knowing numeric values for certain magnitudes associated with right triangles allows us to find numeric values for other magnitudes and construct the triangle if we wish. Less wellknown offspring include studies of tessellation (tiling) and perspective.
The offspring of most interest here, though, is the subject of geometric algebra described by William Kingdon Clifford in 1878 and many others since then. Geometric algebras provide for a mechanism linking magnitudes and numbers and lend themselves neatly to the representation of physical problems and of reality as we know it.
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Section 1: A 3D Euclidean space
Let us start with the geometry of a three dimensional Euclidean space. This is the space most of us already have a wellbuilt intuition to handle problems of shape, size, and location. This is the space a child learns when they play with wooden blocks or those toys with beads on curvy wires. Human brains are wired for it.
The Objects
Describing objects in three dimensions requires both numbers and geometric magnitudes. For our numbers, we will restrict ourselves to real numbers unless otherwise stated. This is the number line we all learned as kids including integers, rationals (one integer/another integer), and irrationals (π, square root of 2, etc). For our geometric magnitudes, we use the symbol e followed by zero to three subscripts and then carefully define what we mean by them.
The first geometric magnitude to define is e with no subscripts. This object is a multiplicative identity. That means it acts like the real number 1 when multiplied by any other geometric magnitude. The reader might occasionally see it written as 1 in other books and articles, but they should not confuse it for the real number 1. To avoid this confusion, e will be used here.
The next geometric magnitudes to define are e_{1}, e_{2}, and e_{3}. These are the magnitudes used as basis vectors along each of the three spatial directions. In a purely geometric sense, they are oriented line segments where one end point defines the start and the other the end.
The next three geometric magnitudes to define are e_{12}, e_{13}, and e_{23}. These represent the basis planes and can be thought of as oriented plane segments. Plane segments require three noncollinear points to define, so oriented plane segments have a start, middle, and end point. Anyone who has programmed with 3D surfaces for games has already encountered oriented planes as triangles in a surface mesh. These surfaces have a front and backside because they have an orientation.
The last geometric magnitude in the space is e_{123}. This one is the basis volume segment. Only one of these magnitudes is needed here since there is one way to make a volume in a three dimensional space. This object does have a sense of orientation, though. Volume segments require four noncoplanar points in their definition, so there have a sense of order among those points if the segment is oriented.
To summarize our list of geometric magnitudes, we have eight objects representing various ranks of geometry and they all have a sense of orientation except the first. Here is the list.
{ e, e_{1}, e_{2}, e_{3}, e_{12}, e_{13}, e_{23}, e_{123}}
A generic member of this list will be labeled E_{i} where 'i' can range from one to eight inclusive.
What to do with them
The next thing to do is define what one may do with these eight geometric magnitudes. The first operation is called scalar multiplication, which is a little unfortunate since the operation looks more like scaling than scalars. The second operation is addition and the third is multiplication. With these three primary operations, most other that people can use can be defined.
Scalar multiplication is an operation that provides a sense of scale to our objects. Take any geometric magnitude and multiply it by a number to get a different sized magnitude. Multiply a 2 against one of the oriented line segments and one moves the end point twice as far from teh start point as it used to be while also keeping the whole arrangement pointing in the same direction. Actually knowing that direction is a little more complex and will be saved for later. One useful thing to remember about scaling is that it is commutative (aA = Aa) and associative (a(bA)=(ab)A) where lower case letters are numbers and upper case letters are geometric magnitudes.
Addition works as most people would expect it. Apples may be added to apples, but not to oranges. So e_{1} + e_{1} may be thought of as 2 e_{1} while e_{1} + e_{2} cannot be further simplified. Addition is commutative (A+B = B+A) and associative ((A+(B+C)) = ((A+B)+C)). Addition is also distributive with respect to scalar multiplication (a(A+B) = aA+aB).
Example 1: Add 5e_{123} and 2e_{13} and 2e_{1} and 14e_{2} and e_{123} and 2e_{23}.
= + 5e_{123} + 2e_{13}  2e_{1} + 14e_{2} e_{123} + 2e_{23}
= + 5e_{123} + 2e_{13}  2e_{1}  e_{123} + 14e_{2} + 2e_{23} {Because
Addition commutes}
= + 5e_{123} + 2e_{13}  e_{123}  2e_{1} + 14e_{2} + 2e_{23} {Because
Addition commutes}
= + 5e_{123}  e_{123} + 2e_{13}  2e_{1} + 14e_{2} + 2e_{23} {Because
Addition commutes}
= + 4e_{123} + 2e_{13}  2e_{1} + 14e_{2} + 2e_{23} {Because Addition and scalar
multiplication are distributive}
No further simplification is possible, but most people will regroup the terms to bring objects with similar rank together and pull out common coefficients. Apply commutativity a few more times and distributivity once to get the following.
= + 2( 2e_{123} + e_{13} + e_{23}  e_{1} + 7e_{2})
Technical Note
With addition and scalar multiplication, we can form linear combinations and span our related vector space. Any reader with more technical knowledge will know what that means, but it isn't very important here.
Multiplication is the last of the three operations. It's definition is best described with a multiplication table. The table must have eight rows and columns to cover all possible E_{i}'s. After a bit of use, the reader will probably wind up accidentally memorizing large parts of the table. It isn't any harder to do than the real number multiplication tables we learned as kids.
e  e_{1}  e_{2}  e_{3}  e_{12}  e_{13}  e_{23}  e_{123} 
e_{1}  e  e_{12}  e_{13}  e_{2}  e_{3}  e_{123}  e_{23} 
e_{2}  e_{12}  e  e_{23}  e_{1}  e_{123}  e_{3}  e_{13} 
e_{3}  e_{13}  e_{23}  e  e_{123}  e_{1}  e_{2}  e_{12} 
e_{12}  e_{2}  e_{1}  e_{123}  e  e_{23}  e_{13}  e_{3} 
e_{13}  e_{3}  e_{123}  e_{1}  e_{23}  e  e_{12}  e_{2} 
e_{23}  e_{123}  e_{3}  e_{2}  e_{13}  e_{12}  e  e_{1} 
e_{123}  e_{23}  e_{13}  e_{12}  e_{3}  e_{2}  e_{1}  e 
A leading '' implies the entry has been scaled by 1.
An 'e' by itself is the additive identity.
An 'e' followed by one number is the generator with that number.
An 'e' followed by more than one number is the product of the generators with
those numbers in the order listed.
A shaded cell implies that the operands happen to commute with each other.
Some geometric meanings
We could stop our description of multiplication by encouraging the reader to look up table entries and not worry about why the entry is what it is. We won't though. The table entries are worth thinking about because they make some sense after a fashion.
If one looks down the main diagonal of the table from the topleft to the bottomright, one will see that any geometric magnitude multiplied by itself results in +e or e. This shows that each of the geometric magnitudes has an inverse, so it is possible to divide by them. Many of the objects we encounter later will not have inverses, so the reader should not assume division can always be done to undo multiplication by geometric magnitudes.
The main diagonal also shows what we mean by the size of a geometric magnitude. The three basis line segments all have squares of +e, so they are said to have a squared length of +1. The three basis plane segments have squares of e, so they are said to have squared lengths of 1. Objects with negative squared lengths behave somewhat like the imaginary magnitude i if one ignores the extra geometric meaning. The fact that these 'imaginaries' are present makes some researchers wonder if we needed to invent i in the first place or if there is deeper physical meaning to the many equations that make strong use of complex numbers to solve physical problems. Those who work with the subject area known as 'Clifford Analysis' are doing more work on this issue.
The last general note about the table to be made here will be about the shading of some cells. The cells that are shaded yellow signify that the two geometric magnitudes happen to commute. Those that do not commute happen to anticommute. In general, then AB = ±BA if A and B are chosen from the list of eight geometric magnitudes. When two of them anticommute, they are said to be perpendicular. When two of them commute, they are said to be parallel. Both meanings are used most often when A and B are of the same geometric rank since they don't make a lot of intuitional sense otherwise.
How multiplication really works
There is an algorithm for determining the entry in each cell of the multiplication table. There are a couple of index rules and a few entries to memorize in the brute force fashion. After memorizing three entries and the two rules, the reader should be able to reproduce the entire multiplication table if they wish to do so.
Multiplication starts with making a long list of all the indices involved with both objects. Make sure to keep the indices from the left side object on the left of the list of indices from the right side object. The memorization part requires the reader to remember that the three geometric magnitudes with one index have squares of +e. The first index rule is one that states that two neighboring indices can have their order swapped if the sign of the cell entry is switched. This means that e_{12} = e_{21}. The second index rule is one that states two neighboring indices may be removed if they are the same and the cell entry does not have to switch signs. This means that e_{112} = e_{2}.
Example 2: Multiply e_{123} and e_{13}.
= + e_{12313} {Just write
long list of indices in rowcolumn order}
=  e_{12133} {Swapping uses index rule 1}
=  e_{121} {Eliminating pair uses index rule 2}
= + e_{112} {Swapping uses index rule 1}
= + e_{2} {Eliminating pair uses index rule 2}
(See problems 1 and 2.)
If the reader works with these rules a bit, they should be able to figure out the entries in the multiplication table. With a little bit of thought, they should also see that knowing how multiplication works for e_{1}, e_{2}, and e_{3} is enough to know how multiplication works for all other objects. These three objects will be referred to as 'generators' because one may start with them and generate all other objects in the R(3,0) geometric algebra through multiplication, addition, and the scaling operations.
Technical Note
These generators are the geometric equivalent of a basis in a vector space. The actual vector space contained within R(3,0) happens to be eight dimensional in the sense of the meaning of 'span'. In the geometric sense, though, there are really only three degrees of freedom and the generators represent them. Multiplication ensures the other five dimensions linear algebra students might expect are not available as degrees of freedom since the other geometric magnitudes are products of the three generators.
The last example for this section shows how two line segments multiplied together work in general. This result will be used later in section 2.
Example 3: Multiply two line segments M and N and show the general result.
M N= ( M ) ( N_{par} + N_{perp} )
where N is decomposed into pieces parallel and perpendicular to M
=( M · direction of M ) (
N cos(θ) · unit direction of M
+ N sin(θ) · a unit direction perpendicular to M )
where M and N are the numeric lengths of M and N and θ is
the angle between them.
=( M N ) [cos(θ) · (direction
of M)^{2}
+sin(θ) · (unit direction of M)(a unit direction perpendicular
to M )]
=( M N ) [cos(θ) · e + sin(θ) · unit plane containing M and then N]
In general, two line segments multiplied together produce a sum of a scalar and a plane segment. If the line segments happen to be parallel (θ=0 or 180) the plane segment vanishes and we are left with the scalar. This is the inner product some students may find familiar from classes involving vectors. If the line segments happen to be perpendicular (θ=±90), the scalar part vanishes and we are left with the plane segment. Some students know this product as a wedge or outer product which is related to the vector cross product.
Summary
In this section, the basic meanings of the objects within a geometric algebra were explained. The eight geometric magnitudes combined with numbers to make sums give us a way to represent our geometric objects with algebraic expressions. Constructing all the objects is done through a combination of addition, multiplication, and scalar multiplication.
With a bit of practice, the basic operations can be added to a student's intuitive toolbox. Simplification of these operations will be as simple as the addition and multiplication we learn as young children. With a bit more practice, the student will discover new things real numbers can't do, though. There are objects whose squares are themselves and they aren't e. There are even objects whose squares are zero and they aren't zero. With enough practice, the tools of geometric algebra will be available to students even when they are employing a different mathematical formalism because these tools bring geometric magnitudes into the powerful language of algebra.
Problems for Section 1
1: Try multiplying both objects from example 2 in the opposite order.
2: What would happen to the multiplication table if the square of e_{1} were e?
3: Add (3e_{1} + 5e_{3}) and (4e_{2} + 7e_{3} 2e_{23}).
4: Multiply (3e_{1} + 5e_{3}) and (4e_{2} + 7e_{3} 2e_{23}).
5: Multiply (4e_{2} + 7e_{3} 2e_{23}) and (3e_{1} + 5e_{3})
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