**A Brief History**
The concept of a geometric algebra can be traced back to 1797 to a Norwegian surveyor named C. Wessel who interpreted the unit imaginary number as a directed line segment perpendicular to the unit real number. Making the complex numbers geometric has proven to be quite fruitful in a number of fields

In 1843, W.R. Hamilton created the algebra of quaternions as an extension of complex numbers with the intent to represent three-dimensional physical objects. Around the same time, H. Grassman invented exterior forms. His algebra would eventually be developed into differential geometry. Both of these tools are powerful, but Grassman's contribution is better known today.

In 1878, W.K. Clifford generalized and reinterpreted the algebras of Hamilton and Grassman. Some will refer to one of the algebras as biquaternions, but in their general form they are named geometric algebras. Clifford's algebras restricted themselves to simpler definitions for multiplication and addition similar to Hamilton's approach but including Grassman's exterior product within the general product. These operations were defined that way in order to obtain geometric meaning. Clifford died of tuberculosis a year later.

Clifford's results have been rediscovered a few times by more recent researchers. Sauter, Sommerfeld, and Eddington all noticed that Dirac's gamma matrices were generators of a four dimensional geometric algebra with a space-time metric. These are the gamma matrices from Dirac's first order relativistic theory for electrons. Several other authors have noted that classical electrodynamics can be easily written in the same geometric algebra generated by a suitably defined set of gamma matrices.

In the early 1880's, J.W. Gibbs wrote about his vector algebra. Gibbs' algebra and his notational approach were a limited form of Grassman's ideas. The system was adopted by those developing the theory of electrodynamics at the time and won the battle for the approach taught to students. The vector algebra taught to students today largely derives from Gibbs' system, though the notation has been changed a little.

Between 1891 and 1894, in the journal Nature, long letters defending various notions can be found between the proponents of both sides of the representational conflict. On one side were Gibbs, and O. Heaviside. P.G. Tait supported the other side. By this time, the main creators of the quaternionic approaches were long dead. The creators and proponents of the vector algebra were alive and well. The actual argument in the journal was a little lop-sided but it went on outside the print media for some time. Anyone interested in what a 19th century flame war looks like is encouraged to do a little digging in the library for volumes 43 through 46. Gibbs first letter was published April 2, 1891 and Tait's first response appeared four weeks later.

**Definitions**

To start, we offer one formal and one informal definition of a geometric algebra. The formal one has a few parts and is meant for those readers with the mathematical background to understand them. In the truest mathematical sense, this definition could be made more formal. This version will suffice for this work. The informal definition, however, is the one most likely to help build an intuitive understanding of how to use these algebras as tools.

*Formal Definition of a Geometric Algebra*

- Binary Operation

A binary operation on a set is a rule that assigns to an ordered pair of elements of the set some other element of the set.
- Group

A group is a set together with a binary operation on that set such that the following holds true.
- The binary operation is associative. (a.b).c = a.(b.c)
- There is an identity element in the set relative to the operation.
- For every element in the set, there is an inverse element such that the operation on both of them always produces the identity element.

(An abelian group is one where the binary operation happens to be commutative. a.b = b.a)
- Ring

A ring is a set together with two binary operations (called + and *) on that set such that the following holds true.
- The set and the + operation form an abelian group.
- The other operation (*) is associative.
- The * operation is distributive over + from the right or the left. a*(b+c)=a*b+a*c and (b+c)*a=b*a+c*a

- Field

A field is a commutative ring under * with a unity element under * where all elements except the + identity have * inverses in the ring.
- Vector Space

A vector space consists of an abelian group under an addition-like operation, a field, and an operation (x) between elements of the field and vector space where the following holds true.
- (x) produces and element of the group.
- (x) is associative
- (x) is distributive in the field AND in the group
- There is an identity element within the field for (x)

- Algebra

An algebra consists of a vector space together with a binary operation (.) on the elements of the group such that the following holds true.
- (.) is associative with elements of the group
- (.) is commutative with the scalar multiplication of the vector space.
- (.) is distributive with respect to the + operation in the group on the right and left.

(Generators of an algebra are a set of elements such that all possible products produce a basis for the associated vector space. Linear combinations of the basis elements cover all possible elements of the algebra.)
- Geometric Algebra

A geometric algebra is a type of algebra with a set of generators where the following holds true.
- products of two different generators under (.) anticommute
- products of two identical generators under (.) are defined

(The generators of a geometric algebra are the elements to which geometric meaning is attached.)

That is a formal enough definition along with the terms that provide the underpinning that makes the definition work. There are two multiplication-like operations, a single addition-like one, a set of elements that starts this all off, and some rules they must all obey regarding commutativity, associativity, distributivity, and a variety of types of operational identity elements and inverse elements.

This cloud of formalism is important in the mathematical sense. For our purposes, however, it is more instructive to approach the tool informally and discover these definitions through usage. Remember the existence of these formal definitions, though. Some of the seemingly magical qualities of geometric algebras discussed later can be shown to be obvious by the fact that they are designed into the tools from the start.

*Informal Definition of a Geometric Algebra*

Start with a set of objects we choose to use to represent directed lines. We will call them vectors later, but for now they are generators. With these generators, we shall construct everything in our geometric algebra.

Note the following.

- The number of generators in our set defines the meaning of 'dimension.'
- The field for our algebra will be assumed to be the Real numbers unless otherwise stated.
- Experience the reader already has regarding the operations + and x from vector spaces will continue to apply here. The concept of 'linear combination' still works here too.
- We shall define (.) by providing the multiplication table involving all generators and their products.
- All elements found on the (.) multiplication table form a set that is sufficient to span the vector space found within the algebra.
- With (.) we introduce the term 'algebraic combination' as an extension of linear combination and allow both (.) and (+) operations.

**A two dimensional example.**

Let's call our generators X and Y. We need to define a multiplication table for the (.) operation to be defined. Here it is written out.

X . X = 1, X . Y = XY, Y . X = -XY, and Y . Y = 1

With this list, we have a two-dimensional algebra where 1, X, Y, and XY span the related vector space. Elements of the geometric algebra can be represented as linear combinations of the set that spans or as algebraic combinations of the generators. Here is an example. (Remember that x is the scalar multiplication with the field.)

M = (A x 1) + (B x X) + (C x Y) + (D x XY)

1 is interpreted as a point. The field element 'A' changes its magnitude.

X and Y are interpreted as directed line segments. (The vectors)

Remember that a directed line is a one-dimensional object with a sense of forwards or backwards.

XY is interpreted as a directed plane segment. (A bivector)

A directed plane is a two-dimensional area with a sense of rotation clockwise or counterclockwise.

**A three dimension example.**

Let's call our generators X, Y and Z. The multiplication table for the operation is as follows. (Assume a.b = -b.a where a, b are any generators.)

1.X=X, 1.Y=Y, and 1.Z=Z

X.X=1, Y.Y=1, and Z.Z=1

X.Y=XY, X.Z=XZ, and Y.Z=YZ

XY.X=-Y, XY.Y=X, and XY.Z=XYZ

XZ.X=-Z, XZ.Y=-XYZ, and XZ.Z=X

YZ.X=XYZ, YZ.Y=-Y, and YZ.Z=Y

XYZ.X=YZ, XYZ.Y=-XZ, and XYZ.Z=XY

XY.XY=-1, XZ.XZ=-1, and YZ.YZ=-1

XY.XZ=-YZ, XY.YZ=XZ, and XZ.YZ=-XY

Elements in this algebra can be written as follows.

M = (A x 1) + (B x X) + (C x Y) + (D x Z) +(E x XY)+ (F x XZ)+ (G x YZ)+ (H x XYZ)

XYZ is interpreted as a directed volume segment. (A trivector)

A directed volume is a three-dimensional object with a sense of rotation (left-handed or right-handed) around a parallelepiped.

(The limitations of HTML or the author's knowledge of HTML in K5 articles makes this table somewhat hard to read. Imagine a square table with the eight elements running across the top and down the left. Fill in the products like one was taught to do with multiplication tables involving real numbers to get the table described above.)

Other examples are possible and worth considering. The reader is encouraged to think about how the multiplication tables would change in both examples if X.X = -1 instead of +1. Such a change makes one of the generators begin to behave like a time-like vector instead of a space-like vector. A later example will show a four-dimensional algebra where one of the generators has a negative square. Such an algebra begins to behave like a Minkowski space, though it is geometrically richer. If both X and Y, in our two-dimensional case, were changed to produce negative squares in the two dimensional example, we would reproduce the algebra of quaternions.

**Why learn another paradigm?**

A reasonable concern for anyone currently using vector algebras and not familiar with geometric algebras is whether this new approach to representing objects is worth learning. The answer to this concern comes in two forms. The first is that an alternate approach to the representation of objects uncovers some of the techniques we use in rendering those representations. The second comes from the fact that some of the elements of our geometric algebras do not have analogs in the vector algebras commonly used to represent physical objects. These unusual elements may be worth exploring in case they help uncover new mathematical structures that imitate unexplained experimental evidence.

Geometric algebras permit the sum of objects of different geometric rank. What is a vector + plane? Readers familiar with vector algebras know that such an object simply can't be constructed without violation of the transformation rules that maintain the concept of identity. Geometric algebras lead to no such violations, so there is a difference.

**Representation Techniques**

The interpretations for object within a geometric algebra depend on their rank and on their position relative to one of the operations. This is very similar to how we do things in other mathematical tools. A simple example will be shown for real numbers and then expanded to cover a three dimensional, Euclidean geometric algebra.

Consider the equation (2 times 3 is 6.) The binary operation is the familiar one of multiplication of real numbers. This same equation could be written as times(2, 3) is 6. Any programmer would recognize the operator and the operands as distinct. However, because the operands of a binary operation are ordered, we could also write the equation with a unary operation as double(3) is 6. If the first operand of the binary operation were held constant, the unary version would be equivalent and more efficient to boot.

This example shows that the operand can also be considered as part of the operation. This ambiguity is the one that leads to two possible interpretations for the same thing. One version treats the object as an input while the second treats it as an operator.

In a three dimensional Euclidean algebra, we have three generators named X, Y, and Z. From these we can create 5 other objects using the multiplication operation alone. The objects are {1, X, Y, Z, XY, XZ, YZ, and XYZ}. Their interpretations as operands are relatively simple. Each is a directed element of rank n where n is equal to the number of generators it takes to represent the object.

- The object '1' is of rank zero and represents points with magnitude. It is not the real number we usually label with a '1' symbol.
- X, Y, and Z are directed line segments similar to basis vectors.
- XY, XZ, and YZ are directed plane segments
- XYZ is a directed volume segment.

Note that no matrices are necessary for these interpretations. Each of these eight objects and their related interpretations is as 'real' as any interpretation of the unit imaginary number.

The interpretation of these eight objects as operands takes a bit more work.

- The object '1' is a scaling operation. By itself, it scales other objects by unity. When it is scaled by a real number from the Field, it transfers that scaling to the operand.
- The generators are reflection operators if the operation is done in a bilinear fashion. Consider a linear combination of the generators. V = a x X + b x Y + c x Z. If you multiply on both sides by X, the result is XVX = a x X - b x Y - c x Z. Notice that the parts of V that were perpendicular to X changed sign after the operation was done from both sides.
- If the generators are reflection operators, then the bivectors must perform two reflections. It is possible to write two reflections about different lines as a single rotation, so the bivectors when operated in a bilinear fashion are rotation operators.
- The last one, XYZ, is a parity operator. All three generators get reflected, but the affect cancels out on the bivectors.

With these two types of interpretation, we can address physical situations. Consider a room full of warm air. We could represent the temperature at every location in the room with a function of scalars (1's) that are scaled to match our readings. We could represent the movement of air molecules with a function of vectors (linear combinations of generators). The size of each directed line is linked to the momentum of the molecule. We could represent the spinning of each molecule with a function of bivectors. The magnitude of each directed plane is linked to the angular momentum of the molecule.

In a vector algebra, we would keep these functions separate in order to avoid violations of the transformation rules that define identity. In a geometric algebra, this distinction is not needed. No matrix representation of the generators and the other objects is needed for the geometric algebra version as long as we keep the multiplication table handy. The matrix form can also be avoided for the vector algebra version, though it is seldom done.

Our freedom to add objects of different ranks in a geometric algebra leads to some unusual constructs one does not normally encounter if one sticks with vector algebras. Consider the following objects.

M = 0.5 x ( 1 + X ) and N = Y ( 1 + X )

The object M has the curious property that when squared you get it back again. The object N has the curious property that when squared you get the additive identity named 'zero.' M is an example of a set of elements of the algebra referred to as idempotents while elements that behave like N are referred to as nilpotents. These constructs are not possible in vector algebras, so if there is anything that could support or demolish the usefulness of geometric algebras as a representational tool in physics theories, it will probably lie with these constructs.

**Examples**

An example or three will demonstrate just how similar geometric algebra can be to the current vector algebra. The rendering of objects in both systems is very close and easily confused. The difference all rests in how the reference frame is written when rendering real objects.

**Example 1**

Imagine our room full of warm air again. Each molecule in the room is moving around at some velocity and, therefore, with some momentum. Imagine tagging each molecule with a directed line segment representing that momentum. With a little sleight of hand we could pretend your tags actually referred to a small volume of some continuous fluid and avoid the discrete nature of the air. This step isn't strictly necessary, but it does make writing a function for the momentum of the fluid easier. That function would appear as follows.

Momentum density P (position) = sum [coefficient of P x a basis element]

where the basis elements are the eight objects that can be generated from the basis vectors. Obviously, most of the coefficients will be chosen to be zero in order to ensure that P is a vector density. That works just fine.
In a vector algebra we would write the same equation as follows.

Momentum density P (position) = {Px, Py, Pz}

where the coefficients are part of a 3-space vector. The reference frame to which those coefficients apply is implied in the equations and usually gets described somewhere nearby in the text explanation.
Some people will start with an equation that looks like the first one but limit it to a sum over only the vector coefficients. Then, when a calculation must be performed, they switch to the matrix notation for vectors and proceed. These people are living halfway between the two systems and probably find some intuitive advantage to the geometric approach whether they know they do or not.

**Example 2**

For another example, imagine a planet orbiting the sun. That planet is at a distance R and moves with a momentum P. We shall write the angular momentum of the planet using both approaches.

In the vector algebra, we would use a cross product defined by the right hand rule to get the following.

L = R X P (R cross P)

L is a vector defined as perpendicular to R and P at the same time. Since there are two possible directions for this 'vector', the right hand rule is used to choose the right-handed one. The magnitude of L is the product of the magnitudes of R and P times the sine of the angle between them. It turns out that the cross product defines a different kind of vector from R and P. It is important for users of this system to distinguish polar vectors from axial vectors, as they are not really the same thing even if they appear to be.
In the geometric algebra, we would use the general product defined in the algebra to get the following.

L = R ^ P = 0.5(R.P - P.R) (^ is the antisymmetric part of (.) in this case)

The geometric nature of L is dependent on the (.) product and not defined to be a vector. As it turns out, L is a bivector. In a three dimensional, Euclidean space, L has the same number of coefficients whether it is described as a bivector or as a vector, but its nature under parity transformations is quite different. A little more exploration would show that all the axial vectors from the other system are bivectors in a geometric algebra. Keeping the polar vectors separate from the axial vectors is done automatically since they are of different ranks.
**Example 3**

The last example involves an operator that rotates other objects. This example will be restricted to two dimensions in order to make it easier to understand. There is only one plane in two dimensions, so there is only one kind of rotation operator.

Imagine a two dimensional vector described as V = A x X + B x Y. Suppose we want to rotate it by 45 degrees swinging the X direction up toward the Y direction a bit. The left sided operator that does this is as follows.

R = cos(45) x 1 + sin(45) x XY

Multiply them as follows.
Vprime= R.V =

- = (cos(45)1+sin(45)XY)(AX+BY)
- = cos(45)AX + sin(45)AXY.X + cos(45)BY + sin(45)BXY.Y
- = cos(45)AX - sin(45)AY + cos(45)BY + sin(45)BX
- = [cos(45)A+sin(45)B]X +[cos(45)B-sin(45)A]Y

Thus far, we have shown enough material to demonstrate how to use the tool for classical theories of physics. With a bit of practice, the reader could use geometric algebras to see why special relativity is built into the fabric of their solution space if they pick the right algebra and the right metric.
For the physicists, though, this is not enough. A better mathematical tool to do what we already know how to do is only of pedagogical interest. Whether geometric algebras provide engineers and software developers more efficient approaches to writing solutions remains to be seen.

For the physicists and physics students, then, we shall provide one more example that needs more work on a theoretical and experimental level. To do this, we must provide one more definition.

8. Left Ideal

A left ideal is a set of elements from a ring that behave as follows. If M is an arbitrary element of the ring and L is an element of the left ideal, then M.L is an element of the left ideal.

Our geometric algebras are rings, so a left ideal in one of them has the property that any element multiplied from the left with an element of the ideal is still in the ideal. This has the curious property of ensuring that operators that can be written in the same algebra as the operands will not take you outside the left ideal if you start there. Why this is interesting to the physicists is best shown by an example in a four dimensional algebra with a space-time metric.

**Example 4**

Consider an algebra whose generators are CT, X, Y, and Z. Let the first generator have a negative square while the others have positive squares. Also let the algebra be defined over the complex numbers for the field.

From this algebra it is possible to write an idempotent that looks as follows.

I = 0.25 (1 + iCT)(1 + CTXY) or = 0.25 (1 + ip/m)(1 + E5.S) where p is a time-like four-vector, S is a space-like four-vector and E5 is the quadvector.

It turns out that if you try left multiplying any element of the algebra against I you will not cover all the possible elements in the algebra. Only a subset can be reached through left multiplication, so I can be said to generate a left ideal.

In a four dimensional algebra, we start with four generators and create sixteen basis elements through multiplication. This sixteen-element set is the basis of the more familiar vector space associated with the algebra. If we use our idempotent described above, we can create a smaller vector space spanned by four elements that can be written as follows.

Basis elements of the left ideal are 0.25(1 +/- ip/m)(1 +/- E5.S)

Anyone with any exposure to early relativistic quantum theory will recognize this ideal as having the same 'states' as a four-component bispinor. Since these elements span a left ideal, left-sided operations on them from other elements of the algebra ensure the resulting element is still in the ideal. This is very much like saying that an electron is an electron no matter what state you happen to find it in. Therefore, the left ideals ARE renderings of the particles that an algebra can describe. No appended spinor space is required to get spinor-like behavior from these algebras. The behavior is already there.
The really cute thing is that spinor-like behavior can also be found in the classical theories written with these algebras. There is no way to get it out, really.

**Future Work**

The bulk of the work that is occurring today using geometric algebras is being done at the post-graduate level by academic researchers. Their results get discussed at conferences and published in peer-reviewed journals. As a consequence, it is not that easy for someone outside the immediate field to break in as a beginner and learn to be productive. This is mostly due to the fact that the physics problems being research are complex, post-graduate problems. The usage of the geometric algebras is not all that complex.

There are two open source projects managed by the author that make use of geometric algebras. The first supports the creation of a java library to be used much like a library that supports matrix and tensor objects and their operations. The second project is a solar sail simulation application that makes use of the library from the first project. Anyone curious about the topic of geometric algebra is invited to check out these projects and see geometric algebras in action.

Clados Project = http://clados.sourceforge.net

SailAway Project = http://sailaway.sourceforge.net

**References**

Representation of Objects: http://www.kuro5hin.org/story/2002/5/7/1345/25152

Clifford Algebra Society: http://www.clifford.org

D. Hestenes Book: New Foundations for Classical Mechanics. ISBN 90-277-2090-8

D. Hestenes website at ASU: http://modelingnts.la.asu.edu/GC_R&D.html