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 Usage of Geometric Algebra as a representational tool By adiffer in ScienceSun Jul 21, 2002 at 10:24:08 PM EST Tags: Science (all tags) In this essay we shall explore an alternative to the use of vector algebra for representation of physical objects in the sciences. It is not the purpose of this essay to make converts of the readers. It is enough to demonstrate how geometric algebras can be used to do many of the same tasks as the vector algebras along with a few additional ones usually reserved for other mathematical tools. In a previous work, the topic of how we choose to represent physical objects in our theories was discussed. Three postulates were laid out defining the concepts of completeness, rendition, and identity. This work will focus mostly upon the second term by offering geometric algebra as an alternative to vector algebra and matrix algebra.

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

1. 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.
2. 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)
3. 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
4. Field
A field is a commutative ring under * with a unity element under * where all elements except the + identity have * inverses in the ring.
5. 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)
6. 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.)
7. 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.

1. The number of generators in our set defines the meaning of 'dimension.'
2. The field for our algebra will be assumed to be the Real numbers unless otherwise stated.
3. 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.
4. We shall define (.) by providing the multiplication table involving all generators and their products.
5. All elements found on the (.) multiplication table form a set that is sufficient to span the vector space found within the algebra.
6. 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.

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.

1. 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.
2. X, Y, and Z are directed line segments similar to basis vectors.
3. XY, XZ, and YZ are directed plane segments
4. 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.

1. 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.
2. 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.
3. 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.
4. 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.

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

Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/

 Poll
 I stopped reading further 25% I skimmed ahead and looked at comments 45% I read the whole thing 13% I read the whole thing and followed links 4% I read the whole thing and looked to how I can use geometric algebras 4% I gave up my current life to evangelize about geometric algebras 7%

 Votes: 140 Results | Other Polls

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 Usage of Geometric Algebra as a representational tool | 78 comments (31 topical, 47 editorial, 1 hidden)
 The problem with clifford/geometric algebras (4.50 / 4) (#6) by Basalisk on Sat Jul 20, 2002 at 09:49:41 PM EST

 There appear to be some inherent contradictions SIGILL, took two, doctor will see it in the morning.
 Lounesto's work (5.00 / 2) (#8) by adiffer on Sat Jul 20, 2002 at 09:55:47 PM EST

 He was very good at catching modern researchers who were being too loose with their claims.  He excelled at punching holes in other research because he had an excellent eye for detail. Every field of research needs someone like him to keep everyone else honest. His work does not undermine the topic though.  He keeps us on the straight and narrow. -Dream Big. --Grow Up.[ Parent ]
 Sigh. (2.75 / 8) (#16) by valeko on Sat Jul 20, 2002 at 11:31:42 PM EST

 Well, rusty did say to vote -1 to articles you don't care about. Sorry, but I don't care. -1. At least I'm honest and didn't try to critique your article on unfounded lies, US centrism, false editorial objections, and other nonsensical turd. "Hey, what's sanity got going for it anyways?" -- infinitera, on matters of the heart
 quite all right (4.00 / 1) (#19) by adiffer on Sun Jul 21, 2002 at 01:03:52 AM EST

 If I thought everyone should care about physics and math, I'd be in a rubber room or prison by now.  8) -Dream Big. --Grow Up.[ Parent ]
 or (none / 0) (#57) by jjayson on Mon Jul 22, 2002 at 12:39:01 AM EST

 you would be on slashdot. -j "Even I can do poler co-ordinates and i can't even spell my own name." - nodsmasher You better take care of me, Lord. If you don't[ Parent ]
 origins (5.00 / 1) (#65) by adiffer on Mon Jul 22, 2002 at 01:47:08 PM EST

 I came from slashdot.  I never felt the urge to comment or attempt to post anything there.  I just felt like the intent for the place was too narrow. I like K5 no matter what any of you think (good, bad, or indifferent) about physics or math.  You all are much more interesting. -Dream Big. --Grow Up.[ Parent ]
 How much I care about mathematics (none / 0) (#75) by jjayson on Tue Jul 23, 2002 at 04:42:14 AM EST

 I was a math major at Cal. My non-lower division classes were mostly focused on topology. I suck at anything applied and refuse to voluntarily go near it. -j "Even I can do poler co-ordinates and i can't even spell my own name." - nodsmasher You better take care of me, Lord. If you don't[ Parent ]
 People are voting oddly (4.33 / 3) (#25) by godix on Sun Jul 21, 2002 at 03:10:47 AM EST

 At this point 39 people have voted to have this article posted. NO ONE has voted in the poll that they read the whole article. I think this goes a long way to explaining how some of the articles I've ever seen end up on FP....
 I agree (5.00 / 2) (#28) by adiffer on Sun Jul 21, 2002 at 03:21:17 AM EST

 That's part of why I was interested in putting this poll on this article. I appreciate the +1 votes, but I was curious if people read through it or just wanted the coffee table version.  I'm learning a lot here. -Dream Big. --Grow Up.[ Parent ]
 +1 section (none / 0) (#39) by khallow on Sun Jul 21, 2002 at 02:06:56 PM EST

 I skimmed through the article. Looked sufficiently neat. Voted for it. Stating the obvious since 1969.[ Parent ]
 Polls are usually stupid (5.00 / 1) (#58) by PresJPolk on Mon Jul 22, 2002 at 01:48:47 AM EST

 I don't notice them most of the time.  I read the article but paid no attention to the poll. [ Parent ]
 for those who think that it is too much for them (5.00 / 1) (#49) by notAcoolNick on Sun Jul 21, 2002 at 07:51:34 PM EST

 +1 FP. Very nice write up. And for those who think that it is a bit too much to to grasp. Guys, that's ok. I suggest you to give it a try. And if you do not understand something just ask questions right here. I am sure that author and other people who know this stuff will be glad to answer them. Just remember there are no "stupid" questions.
 I got lost somewhere around (4.00 / 2) (#51) by JChen on Sun Jul 21, 2002 at 09:36:12 PM EST

 here: "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." What's the unit imaginary number as a directed line segment perpendicular to the unit real number? They definlately did not talk about that in Calc AB as far as my notes copied from more productive classmates show. Let us do as we say.[ Parent ]
 Scroll finger cramp (2.00 / 1) (#52) by HypoLuxa on Sun Jul 21, 2002 at 09:53:04 PM EST

 My scroll finger cramped up, so I couldn't finish. I'll give it a zero since I really have no clue what the article is about, and don't particularly feel like learning. -- I'm guided by the beauty of our weapons. - Leonard Cohen[ Parent ]
 too long (none / 0) (#63) by adiffer on Mon Jul 22, 2002 at 01:35:31 PM EST

 I agree that it went too long.  If I were doing it again, I would drop about 1000 words.  It is all the editorial comments that helped me decide that.  Unfortunately, they get dropped once something posts. -Dream Big. --Grow Up.[ Parent ]
 Editorial (none / 0) (#68) by Scrymarch on Mon Jul 22, 2002 at 04:03:05 PM EST

 Unfortunately, they get dropped once something posts. They're not dropped, just hidden, viewing preferences can be set at the bottom of the article just before the comments start. [ Parent ]
 oops... I feel like a newbie (nt) (none / 0) (#71) by adiffer on Mon Jul 22, 2002 at 11:29:52 PM EST

 -Dream Big. --Grow Up.[ Parent ]
 Complex Plane (4.50 / 2) (#55) by pexatus on Sun Jul 21, 2002 at 10:15:10 PM EST

 A complex number can be represented as a two-dimensional vector in what is called the "complex plane", with one axis real and the other imaginary. The complex number a + b*i, where i = sqrt(-1), is when written in vector form. The unit imaginary number is i, which written as a vector is <0,1>. The unit real number is 1, which written as a vector is <1,0>. <1,0> (unit real) and <0,1> (unit imaginary) are perpendicular to each other. The "number line" you studied in grade school (the 1-dimensional line with 0 in the middle, the negative numbers to the left, and the positive numbers to the right) was the section of the complex plane where i = 0. [ Parent ]
 complex numbers and analysis (4.00 / 1) (#64) by adiffer on Mon Jul 22, 2002 at 01:41:06 PM EST

 As pexatus has said, this little idea lead to the complex plane using real numbers and imaginary numbers as 'coordinates'. If you follow the complex numbers further into the future, you will eventually run into complex analysis which is really powerful stuff.  Problems that seemingly have no method for finding a solution can be attacked and solved.  Powerful tool! While the physicists are a little slow to catch on to new math tools, especially ones that make us get up off our lazy behinds and relearn things, the mathematicians have been working on Clifford analysis.  I'm sure there is equally revolutionary stuff in there if I could just get my mind around the mathematicians language for it all.  8) -Dream Big. --Grow Up.[ Parent ]
 Unit imaginary number (none / 0) (#73) by phliar on Tue Jul 23, 2002 at 01:33:45 AM EST

 ... C. Wessel who interpreted the unit imaginary number as a directed line segment perpendicular to the unit real number." What's the unit imaginary number as a directed line segment perpendicular to the unit real number? You may know the unit imaginary number as ι -- where ι2 = -1. You must have also encountered the complex plane -- the complex number a + bι is represented as the point (a, b). Another way to think about the imaginary numbers and the complex plane: you know the number line. The unit real number is where +1 is on that line, or the line segment (0, 1). Now construct a line segment starting from 0 that is perpendicular to the line segment (0, +1) and of the same length -- in other words, if the number line is the x-axis, the imaginary line is the y-axis-- voila! A unit imaginary number. Faster, faster, until the thrill of...[ Parent ]
 Interesting (4.50 / 2) (#56) by cdgod on Mon Jul 22, 2002 at 12:11:06 AM EST

 This is my first comment on K5.  I have been lurking for about 3 years.  Been through the blackouts and fund raisers. This article intrigued me since I am currently taking linear algebra 2. Linear algebra is mainly focused on solving sets of linear equations.  Ax=b Where A is a matric, x is a column vector, and b is the solution set.  Ax=b, only when x is 0, that is called the trivial solution.  Meaning that A is invertible, the vectors of A are linearly independent, and more...  Basically everything is fine and dandy. What is making linear algebra even more fun is the fact that even when there is no solution to that equation (Ax=b), you can find a solution that is the CLOSEST to Ax=b.  You find an x-hat that would make it the optimal solution to the set of linear equations.  (Gram-Schmidt process and some eigen values and vectors then you're done!  Easy.  (hint: det(A-lambdaI) = 0)) Now (back to the link with geometric algebra), I was wondering if it could do some of these things as well.  It seems to be great at representing the geometric addition and representations.  (The vector * a plane thing is cool, but I am sure linear algebra could handle this trivial problem.)  I multiple matrices with vectors all the time.... they just have to have the same dimension (mxn matrix * nxw matrix), although the result does not always have the same row space as the original matrix.... So.  How does geometric alegra solve sets of linear equations and find optimal solutions to a set of inequalities?
 linear algebra (3.00 / 1) (#60) by superflex on Mon Jul 22, 2002 at 11:06:45 AM EST

 heh, it'll get even more fun later on, assuming you keep taking math courses. after my introductory linear algebra courses, i did a course on numerical analysis. have you ever used MATLAB? have you ever used it for like, 12 hours straight? arrrggghhhh.... OTOH, it generates some absolutely kickass graphs in postscript... Anyways, just wanted to comment that I did in fact read the whole article, but I didn't really take the time to understand the entire piece based on the initial axioms of the formal system laid out by the author. After all, I'm just a dirty engineering student and we like numbers so much more than letters. I leave material like this to my buddy in pure math. I thought I recognized some of the terms used in this piece from the course descriptions. Scary stuff... [ Parent ]
 linear algebra (none / 0) (#61) by adiffer on Mon Jul 22, 2002 at 01:10:20 PM EST

 If you are curious, it is possible to write generators of an algebra as square matrices.  Because the generators start out square, all the elements of the algebra are square.  All you have to do is find a set that satisfy the basis definitions of the generators involving anticommutivity and a generator squared is plus or minus unity.  The trap some people fall into is to restrict their way of thinking about vectors to column matrices.  The number of components does not have to equal the number of degrees of freedom. Linear algebra is where you come face to face with the meaning of Vector Space.  It's good stuff and worth every bit of effort to get comfortable with the material.  Do it now and it will save you grief later. -Dream Big. --Grow Up.[ Parent ]
 Linear Algebra (none / 0) (#74) by phliar on Tue Jul 23, 2002 at 01:52:42 AM EST

 What is making linear algebra even more fun is the fact that even when there is no solution to that equation (Ax=b), you can find a solution that is the CLOSEST to Ax=b. Closest? We haven't talked about distance metrics! We only know whether or not a = b! You know, of course, that linear algebra is all about vectors? In n dimensions, a column (or row) matrix of size n is a vector; a non-singular square matrix of size nxn is an operator (which happens to be affine). In linear algebra, the field is the set of reals with multiplication etc. Thus linear algebra is a special case of geometric algebras. Faster, faster, until the thrill of...[ Parent ]
 Some random thoughts... (5.00 / 1) (#59) by Pseudonym on Mon Jul 22, 2002 at 03:28:13 AM EST

 I'm going to use * as geometric product just because I'm used to it. The quantity XYZ (in 3 dimensions; that would be XY in 2 dimensions) is given a special name: the pseudoscalar, and is usually denoted by I. It is usually identified with i in complex arithmetic because I*I=-1. In the examples above, X*X = Y*Y = Z*Z = 1. This need not be the case with all generators. It's possible to have a mixed arrangement, say, where X*X = Y*Y = Z*Z = 1 but W*W = -1. This example the Minkowski spacetime algebra, where X,Y,Z are "space-like" and W is "time-like". Does anyone know how geometric algebra is related (or not related at all, as the case may be) to dual arithmetic? sub f{(\$f)=@_;print"\$f(q{\$f});";}f(q{sub f{(\$f)=@_;print"\$f(q{\$f});";}f});
 imaginary number and the pseudoscalar (none / 0) (#62) by adiffer on Mon Jul 22, 2002 at 01:29:10 PM EST

 I've seen people do that before.  (I tend not to, but that is just a personal choice.)  Hestenes does it in his book while working in a 3-D spatial algebra.  It works fine as long as you are careful about a couple things.  First, the 'i' should commute with everything in the algebra.  Pseudoscalars from odd-dimensional algebras fit the bill just fine.  Second, don't forget that your 'i' is not algebraically independent of other elements in the algebra.  It is possible to construct a pseudoscalar 'i' from other stuff. The Minkowski-like space you describe is an algebra some of us refer to as R(3,1) if the related field is real numbers.  There is a neat property of that algebra that really captured me as a proponent of geometric algebras.  If you start with R(3,1) and assume a vector current that obeys a continuity equation, you can derive an electromagnetism theory.  There is nothing in the derivation that sets the strength of the coupling constant, but all the structure of E&M falls out by default without any room to make other choices.  The U(1) symmetry, the relatistic relationships, and a lot of other things look right back up at you from the page and say 'Aren't I obvious?'   I struggled with E&M as a graduate student until I saw this approach.  Everything clicked after that. I don't know much about dual arithmetic.  We do have a dual operator.  It's just the pseudoscalar. -Dream Big. --Grow Up.[ Parent ]
 Dual arithmetic (none / 0) (#69) by Pseudonym on Mon Jul 22, 2002 at 09:05:00 PM EST

 Dual arithmetic is constructed from an existing algebra by adding an element (which we'll call e) with the property that e*e=0. A dual number looks like a+b*e where a and b can be real numbers, complex numbers, quaternions or pretty much anything, really. Dual numbers have some interesting properties. For example, if f is an analytic function, for example, then: f(a+b*e) = f(a) + e*f'(b) Dual numbers have found a few uses in computational geometry, for example, you can embed 3D affine algebra with rotations in dual quaternion algebra, and the embedding is quite computationally efficient. I was wondering if geometric algebra could give a similarly efficient representation. sub f{(\$f)=@_;print"\$f(q{\$f});";}f(q{sub f{(\$f)=@_;print"\$f(q{\$f});";}f});[ Parent ]
 don't know (none / 0) (#70) by adiffer on Mon Jul 22, 2002 at 11:28:33 PM EST

 I haven't seen anything like this, but that is probably because of my physics background.  Our interest in geometric algebra is admittedly biased. Most of these algebras are not division algebras, so there are elements where e*e=0.  We tend to refer to them as nilpotents.  They are already present in the algebras, though, so you don't have to add anything new. -Dream Big. --Grow Up.[ Parent ]
 neat! (5.00 / 1) (#66) by bitmask on Mon Jul 22, 2002 at 02:11:39 PM EST

 Thanks for a good article. I know objects like M as differential forms, so it's interesting to see them presented from a purely geometric perspective. What text are you learning from?
 sources for learning (4.00 / 1) (#72) by adiffer on Tue Jul 23, 2002 at 12:06:19 AM EST