Talk:Geometric algebra/Archive 1

Clarification?
I've never heard of a geometric algebra before, but your remark about Grassmann algebras giving a more natural treatment of physics without complex numbers piqued my interest, although I don't quite see how. Would you care to clarify what you meant? Phys 17:33, 20 Aug 2003 (UTC)


 * Hmm, I just read a bit of it, but I still don't see anything new geometric algebra has to say that can't be said already in the language of differential geometry and "dot products", linear representations, etc.. Phys 10:31, 21 Aug 2003 (UTC)


 * Also, I'm a bit suspicious of defining the wedge product as 1/2(ab-ba) because unless both a and b have an odd grading, $$a\wedge b+b\wedge a\neq 0$$ in general. Phys 11:01, 21 Aug 2003 (UTC)


 * Though it wasn't stated on the article, small case refer to vectors, so that definition is for vectors only. I added a little wider definition of both inner and outer product with higher grade elements. In any case, even with mixed grade elements, the wedge product is 0 when the elements involved are "parallel" or "containded" or "linearly dependent". With vectors it is clear that wedge product cancels if the vectors are linearly dependent (don't know if the concept can be expanded to multivectors). --Xavier 18:48, 2005 Mar 30 (UTC)


 * Replacing vector spaces and algebras over the complex numbers with algebras over real Clifford algebras achieves just exactly what? Sure, any quantity whose square is -1 and commutes with everything can be thought of as for all intents and purposes as i, but choosing the n-vector for an n-dimensional space as i doesn't really make any difference. It doesn't really matter what i "really" is. Insisting it's a certain element of a Clifford algebra doesn't really matter. It's already well-known that in many fields like quantum mechanics, we could simply deal with real algebras and real vector spaces provided we define an element in the center whose square is -1. Phys 14:00, 21 Aug 2003 (UTC)
 * in my humble opinion, and as a non-mathemathitian, one of the strong points of geometric algebra is that is connected to geometry (something more or less real), and so it is much more easy to understand/learn. Things like "polar" vectors got sense if you see them as bivectors; or to me it's easy to see that the scare of a unitary bivector is -1, while the imaginary unit always was kind of too imaginary. --Xavier 18:48, 2005 Mar 30 (UTC)


 * I'm not too sure, but it sure seems like the treatment of Maxwell's equations using real Clifford algebras sure does look quite a bit like the stuff Reddi and some other contributors are adding everywhere about quarternions... Phys 15:28, 21 Aug 2003 (UTC)

Least Squares Application/ Mason's Rule
i am a not fluent in GA, but i have noticed that the section of matrix inversions might be tied somehow into the method of least squares.

this bit especially,

http://en.wikipedia.org/wiki/Linear_least_squares#Inverting_the_normal_equations

they even refer to the quantity, which appears here as the Moore–Penrose pseudoinverse. is there a connection?

Another article which seemed related, but i havent pinned it down is the article on Mason's rule. It is my understanding, that this is an implementation of Cramers rule, by making use of the topology of directed graphs. It is used frequency in Controls, and microwaves engineering.

http://en.wikipedia.org/wiki/Mason%27s_rule#Equivalent_matrix_form

it may be a well known result, but i wished the article gave proof for the matrix equation, because the formulation it escapes me.

this comment may be a bit dis-ordered but it seemed nice bridge, linear equations -> matrix algebra -> topolog -> GA  —Preceding unsigned comment added by Afrodocter (talk • contribs) 14:26, 17 February 2010 (UTC)

POV
This article is extremely POV, to say the least. David Hestenes should be mentioned, maybe after two or three long paragraphs, and Emil Artin should have higher billing than Hestenes. Michael Hardy 22:20, 13 Jan 2004 (UTC)


 * The history of GA including the people involved should be moved down to a separate section, and an explanation be at the top, yes.


 * Who should have higher billing may depend on your POV -- Hestenes' wrote for people with a mathematical background like engineers and natural scientists. As far as I can see, he was the first to do so.  Presenting a subject in an accessible way is a highly valuable thing to do (especially in this case, as far as I am concerned).  Artin on the other hand seems to have written for mathematicians.  Quote from the description of Emil Artin's book "Geometric Algebra" (supposedly from the publisher):


 * Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.


 * It would be interesting to know who contributed what to the development, understanding and application of GA.


 * RainerBlome 21:58, 19 Sep 2004 (UTC)

The article is not POV. Emil Artin has nothing to do with geometric algebra except that he wrote a book by the same name. Putting Emil Artin in this article would absolutely make no sense. Geometric Algebra is a book written by Artin. But it is also a mathematical system proposed by Hestenes. However the book and the mathematical system having nothing to do with each other and discussing them in the same article would only confuse things. If you want to write about Emil Artin's book then it should be written about in a seperate article.

The mathematical system that Hestenes proposed is however a special kind of Clifford algebra. So the people who contributed to it are exactly the people who formulated clifford algebra. The main advantage of Hestenes work is that it is much easier to understand than clifford algebras and it has a very nice geometric interpretation. Some might say that Hestenes has not come up with anything new because all of the formalism is already present in Clifford algebras. However the geometric interpretation is more important than the formalisms IMHO because people understand visual ideas but they don't understand formalisms.

—Preceding unsigned comment added by Special:Contributions/ (talk)

Definition?
This article lacks a definition. Is a geometric algebra nothing but a Clifford algebra over the reals? If so, which quadratic form is being used to define the Clifford algebra? Is there one geometric algebra for every quadratic form on the reals?

Also, the relation to exterior powers remains unclear. Are the geometric algebra and the exterior power the same as vector spaces? AxelBoldt 01:04, 28 Sep 2004 (UTC)


 * The definition is where the three axioms for the geometric product are given. Maybe it should be more prominent.  One axiom may be missing, along $$1 A = A$$.  And only an implicit definition of what a multivector is is given.  I have read elsewhere that a geometric algebra is indeed a Clifford algebra over the reals.  There is a degree of freedom with regard to the choice of contraction.  That may be equivalent to freedom in choice of quadratic form, but I don't know.
 * The $$1 A = A$$ axiom was missing. --Xavier 18:48, 2005 Mar 30 (UTC)


 * What do you mean by "exterior power"? The exterior algebra?  A geometric algebra is a vector space, but the converse is not true.  Same for an exterior algebra.  For example, with GA, you get inverses for vectors.  RainerBlome 19:31, 22 Oct 2004 (UTC)

Some text
Here are notes I haven't had time to add TEX to: [edits, anyone?]


 * Geometric algebra is Clifford algebra given a geometric interpretation which makes it useful in an exceptionally wide range of physics problems, particularly those that involve rotations, phases or imaginary numbers. Proponents of geometric algebra say that it more compactly and intuitively describes classical mechanics, quantum mechanics, electromagnetic theory and relativity than standard methods do.


 * The elements of geometric algebra are called blades. In two dimensional geometric algebra there are scalars (grade-0 blades), vectors (directed line segments, grade-1 blades), and bivectors (directed areas, grade-2 blades). The direction of an area is defined by the direction taken around its perimeter, usually in a right-handed sense so that counterclockwise is considered positive. Similarly, in higher dimensional algebras there are directed 3D volumes called trivectors. The highest grade blade in an algebra is called a pseudoscalar.


 * Just as real and imaginary parts are combined to make a single complex number, so all different grades of blades in an algebra are combined to make a single entity called a multivector. In an algebra describing a geometric space of dimension d, there will be d^2 independent blades in a single multivector. For example a basis for a 2D geometric algebra has a scalar, 2 orthogonal vectors and 1 bivector. A 3D space has a basis of 1 scalar, 3 orthogonal vectors, 3 orthogonal bivectors (planes of rotation) and 1 trivector. Algebras of dimension n have the number blades of each grade given by the binomial coefficients in the nth row of Pascal's triangle.


 * Addition of multivectors is performed by simply adding elements with corresponding blades - scalar parts to scalar parts, vectors in a given direction to vectors in the same direction, and so on.


 * Multiplication of vectors a and b is defined by the geometric product, ab = a.b + a^b . The inner (dot) product part is the product of the lengths of the two vectors scaled by the extent to which the vectors are going in the same direction. It is a scalar given by the multiplying the lengths of a and b and scaling by the cosine of the angle between them. It is commutative: a.b = b.a.


 * The outer (wedge) product of two vectors on the other hand is a bivector with a magnitude given by the product of the lengths of the two vectors scaled by the extent to which they are perpendicular to each other. (|a||b|sin(theta)) The outer product is anticommutative: a^b = -b^a.


 * If one pictures the two vectors tail to tail, the area of the bivector a^b is given by the parallelogram swept out by sliding a down b. If on the other hand one slides b down a the direction indicated by the vectors around the perimeter of the area will go in the opposite direction.


 * The outer product is analogous to the vector cross product in ordinary vector algebra, but while the cross product only works in 3D, the outer product works in any dimension. This is because the cross product gives an "axial vector" perpendicular to a plane of rotation (which is not unique in dimensions higher than 3), while the outer product gives the plane of rotation itself.


 * In general the inner product of two blades yields a result one grade lower than the lower of the original two, while the outer product yields a result one grade higher than the higher of the original two.

unwritten ideas: [More about bivector multiplication yielding rotations, unit bivector square = -1, similarities an differences between bivectors, (& pseudoscalars in odd dimensions) and imaginary numbers. Rotors. Compositon of succesive rotations by embedded rotor expressions. Brief EM discussion / relativistic Maxwell's Eqns condensed to "delF = J" - spacetime vector derivative of the EM field strength equals the spacetime current. 4D with ---+ signature -> boosts as hyperbolic rotations. Derivation of quaternions and octonions from the mother algebra, deficiencies of q and o. Descriptions of links.]

Some previous comments miss the point of physics, as do most theoretical types and mathemeticians - it's not mere math where you can assert nonsense like i with no physical interpretation, its a description of reality which has to explain, and has value only insofar as its application can be understood physically, geometrically. Also it's no good having a different mathematical dialect for every little subspecialty or using unnecesarily general and abstract structures such as matrices for physics with constraints not naturally modelled by such general methods. GA goes a long way to solving all these problems.

Mathemeticians who want to explain the obvious in terms of the obscure have the Clifford algebra page to do their thing on - GA is for physics. Physical intuitions were the basis for geometry, set theory, calculus and so forth - most mathematical symbol shuffling is just imitation physics underneath the bafflegab. It won't do to mess up a good physical description like GA with the obscurantist pseudorigor of mathematicians' style.

Enon 04:23, 22 April 2005 (UTC)

Geometric product of multivectors ?
It is not clear from the article what is the geometric product of multivectors.Serg3d2 20:28, 29 November 2005 (UTC)

I agree. the geometric product is shown for two vectors, but I dont see where the geometric product is defined for two multivectectors. —Preceding unsigned comment added by 67.233.107.148 (talk) 16:35, 24 January 2010 (UTC)


 * It follows from the associativity and distributivity of the product. So e.g. if you have multivectors M1 and M2 each can be written as the sum and product of vectors. You can then use the associative law and distributive law to expand the product. You can use the generalised inner and outer product to deal with products between elements other than just vectors.


 * And you rarely have to worry about this anyway. In practice you only have to deal with products between vectors, bivectors, the pseudoscalar, etc., in low i.e. up to four dimensions. The general multivector has no uses that I know of in geometric algebra, except to exist as the general element of the algebra.-- JohnBlackburne wordsdeeds 17:01, 24 January 2010 (UTC)


 * I would argue that one does have to worry about this. One of the most important equations in physics is a multivector equation (Maxwell's).  That is $$\nabla F = J/\epsilon_0 c$$.  The LHS has vector and trivector parts. Peeter.joot (talk) 14:49, 15 February 2010 (UTC)


 * That's an interesting example, which I added something about at Bivector, as it comes down to a relationship between a bivector B and vector J. There should be higher grade i.e. trivector parts but they are eliminated. And that's often how it works. E.g. when applying a rotor to a vector x′ = RxR-1 the rotor R is an even element with scalar, bivector, 4-vector parts. But applied to a vector x it results in a vector, and as you know this you can often avoid working out the full multivector intermediate products. Rotor are about the most complex things you encounter, but they can be written as eB for some bivector. It's one of the attractions of GA that most things reduce to or can be described in terms of single grade elements with a natural geometric interpretation.-- JohnBlackburne wordsdeeds 15:42, 15 February 2010 (UTC)


 * Your blurb about this on the bivector page is using a different convention than I am used to (I've learned it with the STA formalism of Doran/Lasenby's book, which basically uses the Dirac basis as vectors). What author uses the notation you've used on that page? Peeter.joot (talk) 02:59, 18 March 2010 (UTC)
 * That section is largely based on Lounesto's Clifford algebra and spinors (see the refs). Notation in GA is tricky as so many different sorts of quantities (scalars, vectors, bivectors, rotors) are part of the same algebra, and it also at times borrows notation from other areas of mathematics such as electromagnetism. In that article I followed what I thought was the most familiar notation for vectors, bivectors, etc., but with exceptions in electromagnetism as mentioned at the top of that section matching Lounesto.-- JohnBlackburne wordsdeeds 09:05, 18 March 2010 (UTC)

cf Clifford algebras
Have a look at the recent article on Clifford algebras, largely written by Richard Borcherds. This is some of the most clearly written mathematics I have seen on wikipedia (assuming others have not overedited it). Readers should compare this article with the one under discussion.

-Jenny Harrison 02:38, 3 June 2006 (UTC)

A geometric algebra bibliography
This article and its talk page seem to suggest that geometric algebra was invented by either Emil Artin or somebody named Hestenes in the 1980s. But geometric algebra as such was invented by Hermann Grassmann in his Ausdehnungslehre of 1844. He discussed the exterior and interior products, as well as the inversion of a vector &mdash; and did all of this using almost pure geometry. His work remained largely obscure, until taken up by Giuseppe Peano in 1888, and Alfred North Whitehead in 1898. I would place the climax of geometric algebras in the 1940s with Forder's book Calculus of Extension, although he only treats the exterior and interior product (if I recall).

Clifford too interpreted his (and Grassmann's) algebras geometrically (in 1878), and at least the article points this out (though not the year or reference). In fact, this was the original conception of an algebra: as a geometrical version of the propositional calculus. Algebras in the sense of abstract algebra weren't invented until much later. But the article doesn't go far enough to say how its geometric algebra is any different from a Clifford algebra or exterior algebra. How are they geometrical, precisely? Even Clifford does a better job (in an 8 page review article, by the way). He introduces the "rotors" of Hestenes' algebra, explains how they are formed in Grassmann's algebra, interprets them geometrically, etc.

Hestenes' only meaningful contribution to the subject seems to be to slap a snazzy new name Geometric algebra onto a subject that had been known to some for over 100 years, and understood thoroughly by many for almost 80 years. (Actually, he didn't even do that much. I just checked Clifford's paper, and he calls the new algebras geometric algebras.)  Perhaps he also made it more accessible for the masses (neither Grassmann, Peano, Whitehead, nor Foder are known for their easy reading), but that hardly seems notable. It seems to me that the real import of geometric algebra isn't just a "re" interpretation of Clifford's algebras (and its rather dubious rechristening), but its flexibility as applied to many different problems in geometry (vis-à-vis Emil Artin's applications to many other sorts of situations). Consider, for instance, the circle algebra (see Pedoe, 1970), which can be used to prove many classical theorems in the geometry of circles (such as Descartes theorem).

I think a much more encyclopedic approach would be to define a geometric algebra, following Artin, and derive the Clifford algebra as a special case. Some attention must also be paid to other geometric algebras: Grassmann algebras, circle algebras, etc. Silly rabbit 11:34, 12 May 2007 (UTC)

There is a mainstream in Clifford Algebra's led by theoretical oriented pure mathematicians, and a minority stream led by Hestenes and his followers. The work that Hestenes did almost alone during the 60's and 70's was essential for the revival of a fantastic mathematical tool which before was known only in a small circle of specialists. For example the reinterpretation, initiated by Hestenes, of the Pauli and Dirac matrices as vectors, in respectively ordinary space and spacetime, shed's a new light on a lot of subjects in quantum mechanics. Chessfan (talk) 13:24, 7 February 2008 (UTC)

The inner product in GA is more than a standard dot product
About this paragraph: The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as
 * $$ \mathbf a \, \mathbf b = \mathbf a \cdot \mathbf b + \mathbf a \wedge \mathbf b $$

Difference from standard dot product and inner product in GA. The conventional definition of the dot product as given in dot product, valid only for standard (grade 1) Euclidean space, is different from the more general definition of the inner, or interior (or dot?) product in geometric algebra, which generalizes the conventional "dot product" to an operation involving any kind of multivector (e.g. a 2-blade by a 3-blade, or a mixed-grade multivector by a 2-vector, or a mixed-grade multivector by another). Paolo.dL 18:11, 28 May 2007 (UTC)

Definition of inner product in geometric algebra. Nowhere in WikiPedia I can find an explicit definition of this generalized innner product (between multivectors of different kind), although I guess it might become equivalent to a standard dot product if we represent the general multivectors as vectors in a 2N-dimensional space of grade 1 (for instance, any multivector in a 3-D space can be represented by 8 ordered real numbers, i.e. as a conventional vector in an 8-D space: a scalar, 3 coefficients for the e1, e2, e3, 3 coefficients for e12, e23, e31, and 1 coefficient for e123). Is this correct? Paolo.dL 18:11, 28 May 2007 (UTC)

Terminology. I am not even sure that the name "dot product" is correct, as a synonim of the inner product in GA. Notice that Grassmann originally called the latter "inner product". I don't know if Grassmann or Clifford also called their inner product "dot product", and if they used the dot to indicate it. As for the outer (or exterior) product, it seems plausible that Grassmann or Clifford called it "wedge" product as well, referring to the notation they adopted. Thus, it is also plausible they called "dot product" their inner (interior) product. But I am not sure. If they did, then we have to highlight the distinction between "dot product" in GA and the "standard dot product" (see above). Paolo.dL 18:11, 28 May 2007 (UTC)


 * From my understanding of the article, the geometric product is given by the sum of the exterior product and the dot product for vectors, i.e., elements of degree one (or 1-blades, or whatever). The interior product (at least as it is conventionally used today) applies for the higher degree blades, and the simple $$ab = a\wedge b + a\cdot b$$ does not hold in higher degree.  So the nomenclature "dot product" for the inner product in degree one is probably to avoid potential confusion.  Anyway, I agree that the article does suffer from some serious clarity issues.  For instance, it doesn't bother to tell us what the geometric product of more general elements is (even though it follows from the degree one multiplication given by the dot and wedge products, once associativity and distributivity have been applied).  Silly rabbit 06:35, 29 May 2007 (UTC)

I see. I have moved upward a sentence saying that small case bold letters indicate just 1-vectors. About "avoiding potential confusion", what of the two choices below creates more confusion? I believe that case 1 should be avoided, because it is the only one which for sure creates confusion. Case 2 is just incomplete, but in my opinion does not create confusion. However, explaining both concepts would be advisable. Paolo.dL 07:48, 29 May 2007 (UTC)
 * 1) Not specifying that the dot product is not valild for multivectors (in an article which is about multivectors), or
 * 2) not saying that, when both operands are vectors, the inner product happens to coincide with the dot product?


 * I think 2 is the way to go: introduce the idea of an interior/inner product, generalizing the dot product for arbitrary multivectors. This would seem to be consistent with the way the framers of the article viewed the geometric algebra, as an algebra carrying several different products: the exterior product, and the geometric product (and now, perhaps, the interior product as well).  References would need to be checked at this point, but I'm certain somebody somewhere must use an interior product.  Silly rabbit 12:28, 29 May 2007 (UTC)


 * Some relevant material along these lines can maybe be found at User:Jheald/sandbox/Geometric_algebra -- an rewrite/expansion of the article I started a few weeks ago. I put it on hold, because it seemed to me I was developing more and more stodge, and in effect burying, rather than bringing out the things that make GA exciting - eg the way it naturally embeds exterior algebra; the simple way it expresses reflections, and even more simply rotations; the way this carries over to pseudoeuclidean spaces, essentially unchanged; the way "analytic" properties and manipulations of complex analysis naturally extend to higher dimensional GAs, and can be related to Stokes theorem etc;  the simple multivector / geometric calculus consolidation of several laws of physics; the usefulness of multivectors in projective geometry, representing in a single object hyperplanes through the origin, and the interpretation of meets and joins in such algebras ... etc.


 * My previous attempt wasn't really working, which is why my progress with it rather stalled; though it does discuss some of the different product. But maybe I can try again, with a more top-down, visioned approach, and perhaps get closer to the sort of article the subject deserves. (Especially now some rather capable other editors seem ready to get involved!) Jheald 23:51, 29 May 2007 (UTC)

Please do not count on my help. I am not using geometric algebra. Dot products, cross products and cross divisions (yes, they exist, although of course they are pseudo-inverses) are enough for me. On this topic, I can only express doubts, highlight evident weaknesses or correct evident typos. Paolo.dL 14:56, 30 May 2007 (UTC)

Introduction
Relevance. I believe it is extremely important to be clear in Wikipedia about this point, because lack of knowledge in this case means high likelyhood of mistakes when data describing k-vectors or multivectors (in the form of their "scalar components") are exchanged between different scientists or processed by different computer programs. Let's start with a simple example:


 * Example A) This formula is published in the exterior algebra article, and refers to vectors in R3:
 * $$ \mathbf{A} = \mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{i} \wedge \mathbf{j}) + (u_1 v_3 - u_3 v_1) (\mathbf{i} \wedge \mathbf{k}) + (u_2 v_3 - u_3 v_2) (\mathbf{j} \wedge \mathbf{k}) $$
 * and since it is also valid as a simple example of geometric product, namely
 * $$ \mathbf{u} \mathbf{v} = \mathbf{u} \wedge \mathbf{v} $$, if $$ \mathbf{u} \perp \mathbf{v} $$
 * it is sufficient to point out a more general and complex problem, regarding the convention for representing any kind of multivector constructed over RN as a vector with 2N elements.

1) "Internal" ordering. Is it advisable to use $$\mathbf{i} \wedge \mathbf{k}$$? I have always seen $$\mathbf{k} \wedge \mathbf{i}$$ in books and other web pages (mainly concerning geometric algebra, which has the same historical roots as exterior algebra). Of course, $$\mathbf{k} \wedge \mathbf{i}$$ = - $$\mathbf{i} \wedge \mathbf{k}$$.

2) Scalar component ordering. The three components appear to be ordered according to an unusual criterion. I am not sure about the correct (or conventional, or most frequently used) order. I guess there are other two possibilities:
 * Example B) This is my best guess about how Grassmann would order it, and it is also the order adopted by several contemporary authors who wrote about geometric algebra:
 * $$ \mathbf{A} = \mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{i} \wedge \mathbf{j}) + (u_2 v_3 - u_3 v_2) (\mathbf{j} \wedge \mathbf{k}) + (u_3 v_1 - u_1 v_3) (\mathbf{k} \wedge \mathbf{i})$$
 * Example C) And this is also a nice criterion, which is also compatible with the numerically equivalent cross product:
 * $$ \mathbf{A} = \mathbf{u} \wedge \mathbf{v} = (u_2 v_3 - u_3 v_2) (\mathbf{j} \wedge \mathbf{k}) + (u_3 v_1 - u_1 v_3) (\mathbf{k} \wedge \mathbf{i}) + (u_1 v_2 - u_2 v_1) (\mathbf{i} \wedge \mathbf{j})$$
 * These geometric products (where the three basis vectors i, j, k, are indicated with a different notation) show the rationale for this example:
 * e1 e123 = e23
 * e2 e123 = e31
 * e3 e123 = e12


 * Example D) Cross product:
 * $$ \mathbf{a} = \mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2) \mathbf{i} + (u_3 v_1 - u_1 v_3) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}$$

Notice that, on a geometrical standpoint, I am actually referring to the order and name of three Cartesian planes:
 * A) xy, xz, yz (using xz)
 * B) xy, yz, zx (using zx)
 * C) yz, zx, xy (using zx)

Main questions:
 * Is there a conventional order, used in exterior algebra, Clifford algebra and geometric algebra?


 * Is there a different conventional order in each of these algebras?


 * What was the order suggested by Grassmann, who created the exterior product in the 19th century? ...

Paolo.dL 11:07, 30 May 2007 (UTC)

More about relevance. This point is very important as far as compatibility between different computer programming implementations of the above-mentioned family of algebras is concerned. Notice that the right-hand member of equations A, B, C is a simple bivector in 3-D, but the criterion that we chose for ordering the components of that bivector might be extended, probably with little effort, to any general mixed grade N-D multivector (see geometric algebra), including k-vectors. Paolo.dL 14:12, 30 May 2007 (UTC)

Discussion
"Pragmatic" ordering in computer science. I am posting this enlightening comment that I received by e-mail by Ian C.G. Bell, the author of "Maths for Programmers", which includes several interesting pages on geometric algebra:
 * While one can argue mathematically on the "most proper" ordering and signing (internal ordering) of 2-blade basis elements e12, e23, and e31 (aka i^j, j^k, k^i) my own preference is for the "pragmatic" computer science ordering e12, e13, e23 within a full basis ordering:


 * Example of full basis ordering. 1, e1, e2, e12, e3, e13, e23, e123, e4, e14, ..., e1234, e5, ...


 * Bitsflag indexing. This correspends to a bitsflag indexing with ek being present if the (k-1)th bit (ie 2^(k-1)) of the ennumerating index is set, being "read" from least significant bit upwards, and is the system geometric algebra programs tend to exploit. While it can seem natural to work with e31 = e2e123 = k^i rather than e13, and indeed I have tabulated geometric products with regard to such a basis on my website, this increasingly feels more "obtuse correctness" than "practical choice" to me.
 * Ian C.G. Bell

Human mind preference. The list of scalar components proposed above by Ian Bell ("example of full basis ordering") makes sense for a computer. However, blades of different grade are mixed together. Perhaps, when humans read a scalar component list, they need a more human ordering, to facilitate the interpretation of the numbers. I mean, we need an ordering by homogeneous groups, such as, for instance:
 * {| class="wikitable"

! Table 1 - Components for multivector in Cl4,0
 * scalar
 * A
 * scalar components of 1-blades
 * A1
 * A2
 * A3
 * A4
 * scalar components of 2-blades
 * A12
 * A23
 * A34
 * A41
 * A31?
 * A42?
 * scalar components of 3-blades
 * A123
 * A234
 * A341
 * A412
 * scalar component of 4-blade
 * A1234
 * }
 * A412
 * scalar component of 4-blade
 * A1234
 * }
 * }


 * {| class="wikitable"

! Table 2 - Components for multivector in Cl3,0
 * scalar
 * A
 * scalar components of 1-blades
 * A1
 * A2
 * A3
 * scalar components of 2-blades
 * A12
 * A23
 * A31?
 * scalar component of 3-blade
 * A123
 * }
 * scalar component of 3-blade
 * A123
 * }
 * }

Self-criticism. Notice that the last two columns (A31 and A42) are not easy to place, and I am not sure about their "internal signature" (should they be A13 and A24?). Also, a human mind is probably not able to understand multivectors in spaces with larger N, but as far as I know, most applications in physics of geometric algebra are just constructed over R3 or R3,1. However, this is just a preference. What about the above listed "main questions"? Paolo.dL 18:52, 31 May 2007 (UTC)


 * 5-D space. GAs up to R5 are sometimes used for projective geometry -- so that points in R3 get mapped to planes through the origin in R5 and one can use the "sandwich" rotation formalism in R5 to represent translations in R3 (and also do nice things with the "meet" and "join" operators, and geometric calculus). See eg the Dutch group's tutorials for Siggraph.


 * Other ordering methods. On the component ordering question, well, so long as all the components are there, I don't think it much matters. In R3 where you have a simple cyclic symmetry, it is probably most common to bring this out, so A12, A23, A31.  In R4 and higher it may be simplest just to go for phonebook order, so so A12, A13, A14, A23, A24, A34; although for R1,3, it's probably most common to go for A01, A02, A03, A12, A23, A31, emphasising the cyclic symmetry of the 3 space-like directions.


 * Bitsflag indexing. Ian's bitsflag indexing is very systematic and neat, particularly if you want to allow multiplication tables for smaller dimensional algebras to be read off as simply the top left corner of a larger table.


 * Pseudovectors and pseudobivectors. For yet another twist, note that the top half of the algebra may sometimes be written in terms of pseudovectors and pseudobivectors, using the pseudoscalar ω: so for R4 one might have: ..., A12, A23, A31, ωA12, ωA23, ωA31,  ωA1, ωA2, ωA3, ωA4, ω.


 * However, apart from setting out multiplication tables, or for actually implementing code, I'm slightly puzzled as to why you're needing to enumerate the basis elements of the algebra at all? Jheald 08:01, 1 June 2007 (UTC)


 * I'm glad you stepped in. Actually, I find the question interesting.  Does the lexicographical "bit ordering" of the product actually lead to faster multiplication, or is this a naive assumption based on coding other sorts of data structures in the "obvious" way?  This got me thinking about whether there is a Fast Fourier transform for the Clifford algebra, some kind of non-commutative analog of the Schonhage-Strassen algorithm (or even Karatsuba multiplication.)  Any thoughts? Silly rabbit 11:01, 1 June 2007 (UTC)


 * Key point about bitsflag indexing. The key point about the bitsflag ordering is that, if you're using a 1-D array to store all the components of a multivector A, it's a nightmare without it to work out where a component like A124 actually stores in the sequence. But with the bitsflag ordering, you know immediately that it will store in the binary 1011th = decimal 11th element.  You also know, for an orthogonal basis, that the product of two components will be a component indexed by the XOR of the bitsflag indexes of the two factors.


 * WRT the FFT, I'm a little dubious. Don't see it (yet!).  But maybe. Jheald 11:32, 1 June 2007 (UTC)


 * FFT for real Clifford algebras. A Fast Fourier Transform can be defined for the real Clifford algebras. It is a fast algorithm for the the real matrix representation: a mapping between a vector space model of the algebra and a real matrix model. There is also a fast inverse algorithm. See my paper "A generalized FFT for Clifford algebras,Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 11, Number 5, 2005, pp. 663-688. or the corresponding preprint: UNSW Applied Maths Report AMR04/17, March 2004. The FFT is related to the Fast Fourier Transforms for finite groups in a way which is explained in detail in my paper.


 * These algorithms are implemented in the GluCat C++ library, a generic library of universal Clifford algebra templates. GluCat uses index sets (sets of integers) as a basis for a real Clifford algebra of arbitrarily large size. Negative integers yield negative squares; positive integers yield positive squares. The signed index sets form a group.


 * There is a natural lexicographic ordering of the index sets. This ordering is used to define the order of rows and columns of the real matrix model, leading to the FFT. Internally and for output, the vector space (framed) model orders terms of a multivector by grade and then lexicographically within grade. Penguian 02:31, 1 September 2007 (UTC) Penguian 22:19, 12 September 2007 (UTC)

It seems that there are no widely adopted conventions. Anybody knows what was Grassmann's and/or Clifford's opinion? Jheald, I already explained the reasons why I think an ordering convention is needed. See "Relevance" and "More about relevance". Paolo.dL 13:18, 1 June 2007 (UTC)

Bitsflag indexing. I agree about Jheald's "key point", but I cannot see the reason why you may "want to allow multiplication tables for smaller dimensional algebras to be read off as simply the top left corner of a larger table". Or, at least, I cannot see why this preference may be regarded as more important than others (e.g. "human mind preference") Paolo.dL 13:18, 1 June 2007 (UTC)


 * Well, if you're publishing such a table in a book, or on a website, it means you can publish the table for Cl4,0, and that will include a nice, contiguous sub-table for Cl3,0; and within that a nice contiguous sub-table for Cl2,0. So you only need to put up one table instead of three.  That's all I was suggesting. Jheald 13:31, 1 June 2007 (UTC)

I see, but I believe that's not really a problem. Whatever is the method you use, you obtain "subtables" by just deleting the very cleary identifiable elements belonging to higher dimensions. For instance, if you delete all elements containing 4, in table 1, you obtain table 2. Paolo.dL 13:42, 1 June 2007 (UTC)


 * No, my point was that if you're making an explicit 16x16 multiplication table for Cl4,0, to show the pattern of plus and minus signs, this will automatically include the relevant 8x8 table for Cl3,0 in the top left corner, the 4x4 table for Cl2,0 in the top left corner of that, and the 2x2 table for Cl1,0 in the top left corner of that. (And a 1x1 table of the identity element in the top left corner of all!)  Jheald 13:56, 1 June 2007 (UTC)

What's more important?. Ok, I am not saying your criterion is not useful. But don't you think it is also useful to group by "grade"? If you see all the 2-blades together, that facilitates your understanding, your interpretation of the numbers, doesn't it? Any ordering method has its pros. Two problems remain unsolved: Paolo.dL 14:01, 1 June 2007 (UTC)
 * 1) is there a "preferable" method?
 * 2) is there a conventional method, at least adopted by groups of authors (e.g. physicists, computer scientists)?

Comparison between methods A and B. Let's compare example A (bitsflag indexing method) and B ("human ordering" method) with the cross product in example D. Let me use x, y, z rather than 1, 2, 3. If Axy, Axz , etc. were numbers returned by a computer program, and if these numbers were computed and ordered with method A, I would fail to understand the approximate direction of vector a, while I would interpret the numbers easily with methods B and D. Does this clarify my point about the "human mind preference"? Paolo.dL 15:53, 1 June 2007 (UTC)
 * By method A, [Axy, Axz , Ayz] = [az , -ay , ax].
 * By method B, multivector A and vector a are just permutations of each other:[Axy, Ayz, Azx] = [az, ax, ay].

Multiplication tables. Notice also that the point here is about scalar coefficient vectors with 2N elements, not about multiplication tables. Multiplication tables are used for a different purpose and may have a different format, although I agree that it is desiderable to have scalar coefficient vectors built with the same method as the row or column labels of multiplication tables.

For instance, Ian Bell did not use his favourite method (bitsflag indexing) for writing his multiplication tables. He actually used method C, and this choice appears perfectly reasonable to me (more than it does to him). Paolo.dL 14:01, 1 June 2007 (UTC)

Conventions and very authoritative preferences. I encourage other people to contribute. I think it is important, in this context, not only to write about our preferences, but also to collect informations about conventions adopted by authoritative groups of authors, or about very authoritative preferences (e.g. by Grassmann and Clifford). Ian's contribution was interesting in this respect. He maintained that bitsflag indexing is the method that computer scientists "tend to" adopt. Paolo.dL 14:07, 1 June 2007 (UTC)


 * Oh for goodness' sakes. Paulo, use whatever convention you want to.  It's just housekeeping.  It's just detail.  It doesn't matter.  Leave it to your computer software to worry about.  The whole style of geometric algebra is to express objects without using co-ordinates; to achieve proofs and results without working in terms of any particular axes; to think of a bivector as a bivector, not as a triple of numbers.  You ask, what do the authoritative books do?  The modern authoritative books very very seldom use components at all.  And even where they do use components, they don't use an anonymous triple of numbers [p ,q, r ].  Instead, if they want the bivector part of A, they simple write 2.  The beauty of GA is the breadth and depth of results that can be achieved at this level of abstraction, without ever dirtying the hands with components.  But if they really really do want to break 2 into components, it will be as something like  2 = p e12 + q e23 + r e31 -- with the bases e12, e23, e31 explicitly specified, so if somebody else wants to work with e12, e13, e23 the translation is obvious.


 * Now I've written above what I'd probably use -- cyclic ordering in Cl3, otherwise probably phonebook ordering for basis elements of each successive grade. But you do whatever you like, because it doesn't matter. Jheald 17:32, 1 June 2007 (UTC)

Not looking for advices. Well, I am not participating to this discussion to solve a personal problem. I am just trying to draw the attention of expert editors on a particular point. Actually, I am not going to use geometric algebra. In my field, conventional vector algebra is enough. My only personal problem was to know how to transform a 2-vector into a vector, and I solved it much before starting this discussion. I didn't know geometric algebra and was surprised to discover that there were different methods to order multivector components. The method A used in WikiPedia, for instance, was different from that used by Ian in his web site, and different from that used by the Cambridge group (Gull, Lasenby, Doran). Since you seemed to be interested in defending a particular preference, I thought it was interesting to show that any preference has its pros, and to stress the difference between a preference and a convention. Paolo.dL 18:23, 1 June 2007 (UTC)

Again about relevance. Do you mean you never look at numbers to understand if your calculations are correct? This section is not about elegant symbols such as 2, but about lists of numbers (see title and first paragraph). Pure mathematicians might not be interested in numbers and, if true, this is perfectly understandable. But there exist some people, I guess and hope, who use geometric algebra for practical purposes. People who express measured or estimated physical or geometrical quantities using multivectors. And if they do, for sure they are going to look at the numbers once in a while. Results must be interpreted, discussed, understood. Programs must be tested. Then results might be shared, sent to other people, or the output of a function might be used as input for another function. And this is why I believe a convention might be useful (see "Relevance"), or at least that there is a need to publish on Wikipedia information about the existance of different conventions, or no convention at all. Paolo.dL 18:23, 1 June 2007 (UTC)

Relevance of people with dirty hands. Without people "dirtying their hands" with these numbers geometric algebra would be useless! We are discussing today about geometric albebra only because it has interesting applications. I really hope that those using it are much more than those just studying it. Paolo.dL 19:06, 1 June 2007 (UTC)


 * As someone once said, Mathematics isn't about calculation, "Mathematics is the art of avoiding calculation". The power of GA is its compactness for developing mathematical results and equations and representing physics by working and manipulating at an object level of a blade or a multivector, without having to go down to the level of components.  Books like "New Foundations for Classical Mechanics" and "Clifford Algebra to Geometric Calculus" are full of powerful physical and mathematical proofs and results, achieved without using components.  Above all, GA helps express geometric intuitions you simply don't get from blocks of numbers and horrid expressions full of subscripted components.


 * But since you want a convention, all right then.


 * For binary communication, the API is likely to be a format like Ian's, or perhaps a sparse equivalent like a linked list. For human communication, the convention is to express the objects as a sum of components, explicitly including the basis elements, eg: 2 = p e12 + q e23 + r e31, and to trust readers to be able to translate that e31 = -e13 or whatever if they so wish. And that is it.  There endeth the convention. Jheald 19:37, 1 June 2007 (UTC)

Art of mathematics. Well, I see your point and I love the artists who created the mathematical tools I use. Notice that computer programmers, engineers and scientists share the same aim: they write programs to avoid manual calculations. But then again, when I do my experiments and have final results automatically computed by a specifically designed computer program (or when I am debugging that program), I need to read the numbers anyway.

Creating a convention. Notice that a convention is, etymologically, something on which many people agree. In my field, there is an international scientific society which, when needed, nominates a committee composed of a dozen of very authoritative scientists who writes a standardization proposal which is then published on the international journal in the hope that other people will try to comply (and this often happens). I basically agree with your suggestion, but unfortunately it is not a convention (by the way, a true convention typically does not include a "perhaps" :-). Paolo.dL 11:27, 2 June 2007 (UTC)


 * Ring order in quaternion. Being a novice, I prefer the form in "Example C" above because it is displayed in the same form as the vector product. Also the ring order is maintained in the i, j & k product's, as it would - had they been considered as the quaternionic elements. Scot.parker 21:26, 8 June 2007 (UTC)

Tentative conclusions
Perhaps we may conclude that there are no general, widely adopted and official conventions, but some authors agree about example B, while others (among which many computer scientists) agree about example A (bitsflag indexing). Example C is also used (e.g. by Ian Bell in his multiplication tables), but as far as I know less frequently than A and B. Plain phonebook ordering is also possible, and differs from bitsflag indexing for mixed grade multivectors. The method used for multivectors built over R3 and R1,3 may be different than that used for R4 and N-dimensional spaces with N>4. When pseudovectors are incuded, a special ordering method is advisable.

Notice that not only the order of the elements, but also the "internal order" and sign of each element may vary (e.g. A31 rather than A13, with A31 = -A13). Thus, we are actually discussing here not only about component ordering, but about (1) selection and (2) ordering of the nonunique ordered set of independent basis blades of which a multiproduct is a linear combination.

Do you agree?

However, we still don't know about Grassmann's and Clifford's preferences. Paolo.dL 10:54, 3 June 2007 (UTC)

The reason why this is an important issue
I agree that it's best to define the bivector basis so that the outer (wedge) product has the same multiplication table as the vector cross product. I struggled with this issue here.

The problem is what to do in algebras based on higher dimensional vectors (see here). I can't find any way to make this multiplication table as symmetrical as the 3D case and can't really find a way to choose one basis over another. It's an important issue because it's going to cause a lot of confusion if everyone does something different and I don't want to have to refactor this part of my site at a later stage.

What do you think? I would welcome any ideas or inspiration on this. (Posted by Paolo.dL on behalf of) Martin John Baker 15:37, 15 June 2007 (UTC).

Explicit example of geometric product
The article doesn't make it clear how to find the geometric product of any two multivectors. There's a definition for the product of two vectors, but not for higher grades.

For example, what is the answer to:

$$\mathbf{A} = x_1 + x_2 \mathbf{i} + x_3 \mathbf{j} + x_4 (\mathbf{i}\wedge\mathbf{j})$$

$$\mathbf{B} = y_1 + y_2 \mathbf{i} + y_3 \mathbf{j} + y_4 (\mathbf{i}\wedge\mathbf{j})$$

$$\mathbf{A}\mathbf{B} = ?$$

And similarly for higher dimensions. —Preceding unsigned comment added by 82.103.112.60 (talk) 12:42, August 27, 2007 (UTC)

--

This is one of my pet peeves with the development of this fascinating subject. The construction of the geometric algebra is not particularly obvious from the special cases given for the geometric product of grade 1 vectors. In fact it is easier to work out what the geometric product is from first principles, than what the somewhat elusive  wedge product is,  whose definition changes depending on the 'grade' of the elements involved.

In particular you can use the fact that the product is distributive over addition and associative and then use the fact that all the grade 1 basis vectors anti-commute  to obtain the product. Thus assuming $$ \mathbf{e}_1 = \mathbf{i} $$ and  $$ \mathbf{e}_2 = \mathbf{j} $$,  one can exploit the fact

$$ \mathbf{i} \wedge \mathbf{j} = \mathbf{i} ~\mathbf{j} $$

and by associativity one can quickly find all possible products of the basis elements in your expression. e.g.

$$ ( \mathbf{i} \wedge \mathbf{j} ) \mathbf{j} = \mathbf{i} ~ (\mathbf{j} ~ \mathbf{j}) = \mathbf{i} $$

$$ (\mathbf{i} \wedge \mathbf{j}) \mathbf{i} = - (\mathbf{i} ~ \mathbf{i}) \mathbf{j} = -\mathbf{j} $$

etc.

However my question is how is the wedge product and for that matter the dot product defined for arbitrary elements of the geometric algebra?

The answer changes depending on the grade, which is somewhat annoying and IMHO it starts to lose some of it's geometric intuition. The relationship

$$ \mathbf{a} ~ \mathbf{b} = \mathbf{a} \cdot \mathbf{b}  + \mathbf{a} \wedge \mathbf{b}  $$

is preserved at some cost. This is touted as a definition for the geometric product, but it fails to be a definition if we don't even have a decent constructive definition for dot product and inner product for arbitrary elements of the algebra. The somewhat stale abstract algebra construction of Clifford algebras using the quadratic form and the quotient ring etc. can make things a bit more precise, but it would be nice if the operations can be extended using pure geometric reasoning to higher dimensions and can be more precisely defined.

When I first saw this I wondered to myself,  how do I know this product exists and that it's even associative as defined? There are algebras that extend the complex numbers that are not associative (e.g. the Cayley-Dickson construction). Another complaint I have is that the author touts geometric algebra as being inherently superior to quaternions in part because quaternions are not right-handed in some sense. Last time I checked all these geometric algebras are isomorphic to Cartesian products of quaternions and complex numbers. Moreover one can define all the relevant quantities of interest, including plane,  2 and 3 dimensional rotations using similar operations in the quaternion algebra. Ultimately the geometric algebra generated by 3 vectors is isomorphic to the biquaternions and hence to 2 x 2 complex matrices.

Quaternions have one big advantage over geometric algebra. They form a field and thus every quaternion has an inverse. There are many ways to use quaternions to define various important relations in mathematical physics. I'd like to see some more motivation why geometric algebra is better. It does tend to proliferate the dimensionality of spaces in question. —Preceding unsigned comment added by Mattcbro (talk • contribs) 07:44, 1 January 2010 (UTC)


 * I don't think


 * $$ \mathbf{a} ~ \mathbf{b} = \mathbf{a} \cdot \mathbf{b}  + \mathbf{a} \wedge \mathbf{b}  $$


 * is preserved for products between elements of higher grade: the simplest example I can think of is the product of two bivectors, in e.g. four dimensions. This in general generates a multivector with scalar, bivector and pseudoscalar (or 4-vector in dimensions > 4) parts, e.g.


 * $$ \mathbf{a} ~ \mathbf{b} = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \times \mathbf{b}  + \mathbf{a} \wedge \mathbf{b}  $$


 * where a × b is the commutator product. For higher dimensional blades and multivectors with mixed grade I'm not even sure this holds. But I think the reason why it's better can be shown without reference to this. Its "better" first as it generalises vector, complex, quaternion algebra so they all have the same theoretical underpinnings, second as it extends into higher dimensions so rotations etc. can be better understood in them. It has also been applied to some surprising areas of physics, like electromagnetism and quantum mechanics, simplifying and unifying them. This is all mentioned, though it could perhaps be made clearer. Maybe we need an Introduction to geometric algebra article that omits much of the algebra and focusses on concepts? -- John Blackburne (words ‡ deeds) 09:45, 1 January 2010 (UTC)

I skipped over to chapter 4 of Doran and Lasenby's book and indeed that relationship does not necessarily hold for general multivectors. However there appears to be many ways to approach the same subject. Patrick Girard's book, "Quaternions, Clifford Algebras and Relativistic Physics", for example makes Quaternions the centerpiece of the theory and derives the exterior (wedge) and interior products entirely using Quaternions and the result due to Clifford that his algebra can be decomposed into a tensor product of quaternions. I'd have to say though that Doran's book is an easier read. Girard gives me a headache after a while. Also I'd like to point out, that although there is no doubt many topics in physics and engineering can be reformulated using geometric algebra, the same holds true of pure quaternions over the reals. In fact the concise representation of electromagnetics in the biquaternions (CL(0,3)) can be replicated using just the real quaternions and differential operators that apply as either a left or a RIGHT operator multiply. There are also commutative hypercomplex numbers that can be used for the same purpose http://home.comcast.net/~cmdaven/nhyprcpx.htm. The latter formulation can not be encapsulated within geometric algebra. Perhaps geometric algebra is the preferred or natural algebraic approach for formulating mathematical physics, but the bewildering alternatives and differing viewpoints does not make it completely obvious. —Preceding unsigned comment added by Mattcbro (talk • contribs) 02:47, 2 January 2010 (UTC)


 * I'd mention another approach which is the one in "Clifford Algrbras and Spinors" by Lounesto, which introduces a new product, a "contraction", an easier to generalise alternative the interior product. But in general although it's well defined working with higher grade elements of a geometric algebra is difficult. My approach is to mostly ignore it, i.e. stick with operations on vectors, or decompose things into vectors. It helps that in general I never use something more complex than a bivector or rotor — I don't think I've ever needed to do calculations with multivectors much more complex than that. -- John Blackburne (words ‡ deeds) 10:22, 2 January 2010 (UTC)


 * Dorst's paper makes the interpretation and importance of the contraction a little clearer. It also defines all the products needed explicitly.--Leon (talk) 10:26, 2 January 2010 (UTC)

Historians' use of the term
According to Margaret E. Baron, in her 1969 book Origins of the Infinitesimal Calculus, p. 16, the term “geometric algebra” is scholarly jargon for the contents of Euclid’s Books II & VI:
 * It was Zeuthen who first drew attention to the algebraic nature of the contents of Euclid (II and VI) and gave to it the title geometric algebra by which it has subsequently been known: Neugebauer [The Exact Sciences in Antiquity, pp. 143-4] was able to demonstrate the close relationship it bears to the Babylonian rules for the solution of what are now termed quadratic equations.

On the next page Baron sees geometric algebra as a toolbox in use:
 * Apollonius of Perga (ca. 230 B.C.) applied the whole complex apparatus and terminology of geometric algebra to the systematic study of conic sections.

Baron also refers to the doubling of the cube as geometric algebra:
 * An example of the extension of geometric algebra to three dimensions is the Delphian problem (or duplication of the cube) which Hippocarates (ca. 440 B.C.) reduced to the finding of two mean proportionals between given magnitudes.

Given that historians of the caliber of Baron, Otto E. Neugebauer, and Hieronymus Georg Zeuthen all use the term, albeit in an historic context, there is the question of acknowledgement in the WP article.207.102.64.90 (talk) 19:13, 18 January 2008 (UTC)


 * Thanks. At one time or another, I had meant to include some discussion of the Alexandrian and Mesopotamian origins of algebra qua geometric algebra.  Unfortunately, I got into a bit of an argument about the extent to which it is applicable to this article.  Further work is clearly needed on the history section. Silly rabbit (talk) 20:20, 18 January 2008 (UTC)

There is in fact a derogatory tone to the term as used by Carl B. Boyer in his History of Analytic Geometry (1956):
 * An enlightening example of the Greek attitude toward arithmetic and geometry is seen in the classical treatment of quadratic equations. …solutions show that Greek algebra – as distinct from arithmetic and logic – was wholly dependant upon geometry. Probably one of the chief reasons that Greece did not develop algebraic geometry is that they were bound by a geometrical algebra.  After all, one cannot raise himself by his own bootstraps. (pp.8,9)

Nevertheless, Boyer points out (pp. 12,13) that the duplication of the cube was solved by Archytas using geometric algebra.Rgdboer (talk) 06:43, 17 February 2008 (UTC)

Remark on the history section

 * Moved from the article by Silly rabbit (talk) 15:25, 7 February 2008 (UTC)

Additional remark : Less theoretical inclined readers, engineers and physicists searching for an efficient, easy to learn mathematical tool, might be puzzled by the above historical description, at the end of an article mainly inspired by Hestene's and his follower's work. It seems that the difficulty arises from the fact that the mainstream of Clifford algebras is logically in the hands of a relatively small circle of specialists, whereas the minority stream developped by Hestenes is more practically oriented and even doesn't use the same tools. A decisive step taken by Hestenes - and certainly not Clifford ... - whose scientific implications are still not clear, is the reinterpretation of Pauli and Dirac matrices as vectors respectively in ordinary space and in spacetime. 82.124.211.217 (talk) 15:22, 7 February 2008 (UTC)


 * Could you or Chessfan please clarify for me what you mean when you say that Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space/spacetime? This strikes me as wrong, the way I read it, since the usual vector-contraction approach to Pauli spin matrices goes back at least to the 1930s (see Weyl-Brauer matrices). Silly rabbit (talk) 15:53, 7 February 2008 (UTC)

I am not a specialist ! Of course I do not know the Weyl-Brauer matrices. Hestenes explains it all in one of his papers which you will find on his site http://modelingnts.la.asu.edu/html/Impl_QM.html "clifford algebra and the interpretation of quantum mechanics". Chessfan (talk) 16:07, 7 February 2008 (UTC)


 * The idea of rotors and other geometric-algebraic tools as somehow "living in the usual space" goes back to Clifford as well if you look at his papers. All the early geometric algebraists were keen to keep the new objects as close to the geometry as possible.  The 20th century certainly moved away from this view towards greater abstraction, but the geometric approach was always available.  If Hestenes actual contribution can be clarified, then that would certainly be a welcome improvement to the article.  However, it currently reads like unnecessary fancruft.  Silly rabbit (talk) 16:26, 7 February 2008 (UTC)

Thanks !! To appreciate Hestenes and by the way the Cambridge group you must read them ! As a true amateur (71 old) I am fascinated by the passions in scientists discussions. That's war ! See you later perhaps. 82.124.211.217 (talk) 17:14, 7 February 2008 (UTC)

I would like to make a last remark. It is impossible for me to clarify Hestenes contribution in only a few lines. But what I know is the fact that, had he not done his work, almost alone in the 60's and 70's, there would be no article on "geometric algebra" today in Wikipedia. Every mathematical relation, except the strictly Grassmannian ones, of that article, was first written by him. You know very well that Clifford had no time to develop his ideas, as he prematurely died. If recognizing that is an act of "fancruft" (?!), then, well, I declare that I am a fan of Hestenes who gave me the opportunity to gain access to physicat theories I would have never dreamed of. Thank you David ! Chessfan (talk) 14:08, 8 February 2008 (UTC)

I see nothing changes in the history of GA. I am a Frenchman. Understandably Hestenes is not well known by French scientists. But it is sad to see how undervalued he is in the English speaking World. To me the historical section of the geometric algebra article remains a shame. Artin and Chevalley were certainly high level mathematicians, but as I see it they made no contribution to Clifford's geometric algebra as a useful and efficient mathematical tool for "ordinary people". Chessfan (talk) 14:54, 4 November 2009 (UTC)

For an appreciation of Chevalley's position take a look at http://members.fortunecity.com/jonhays/clifhistory.htm. Chessfan (talk) 23:15, 18 November 2009 (UTC)

Some thoughts on direction for this article.
When learning the algebra initially I dumped a bunch of notes in this page as I went along (there wasn't much in the way of hard examples before that). I've since stopped putting my GA notes in wiki since its much harder to edit than plain latex, and have collected them for myself on a regular web page. Wiki is kind of nice in that it makes material available to everybody (whereas nobody knows about my blog like http://sites.google.com/site/peeterjoot/ webpage), but my notes were starting to get very un-encyclopedia-article like, and I didn't think it made sense to keep adding them to wikipedia.

Although I've stopped adding to this article for now, I'm not entirely pleased how I left it, and have some general thoughts about future direction:

1) Some of what I put in, in particular the torque section and what is now the matrix inversion and determinants (Cramers rule) section, I think don't neccessarily belong in this "intro page". The torque section is incomplete, and should really introduce and motivate bivector torque and compare to the 3D cross product varient (I expanded my private latex version a bit from what's in the current article, but didn't feel it was appropriate to keep adding to it).  The numerical Cramer's rule example would probably be better on a wedge product or exterior algebra page if anywhere (the use of GA division could be replaced by the notion of similarity of wedge products. ie: solution exists when similar,is the equivalent of when the GA quotient of the two multivectors is a scalar).

2) I think the contraction rule section is a bit out of place way at the end, but am not sure where the best place to put it is.

3) Should be some content on Rotor definition and use. I described rotations of vectors constrained to a specific plane in a way that I thought was easy to understand (because it is not really any different than complex numbers).  However, being able to rotate arbitrary multivectors regardless of whether they are constrained to the subspace, is one of the most GA features, and it is a shame to omit it.  But it's also hard to cover it compactly and retain understandability.

4) I was careless with my grammar and spelling, so I've probably got lots more to fix (although Paulo has done some nice cleanup of at least some of that).

5) It would be kind of nice to also see an axiomatic driven definition of the GA product using just contraction, linearity and associativity, in addition to the definition in terms of dot product and wedge product. I think this can be motivated nicely by considering a number line and analogy to one dimensional vectors (consider the set of all numbers as isomorphic to all the scalar multiples of a unit vector).

6) It is a bit of a shame to omit an explicit description of how maxwell's equations are so naturally expressed using GA, or how STA can so naturally express the Minkowski metric. For accessability a mention o f maxwell's equation that omits the explcit STA formulation is probably best (ie: leave the equations in the explicit multivector form instead: $$(\nabla + \partial_{ct}) F = \rho - J/c$$ ).

I think a wikibook is perhaps a more natural place for more detailed content on the subject. If anybody feels inclined to figure out how to create one, I would add to it;)

Peeter.joot (talk) 15:41, 31 July 2008 (UTC)


 * I think a wikibook is a fantastic idea. You may want to post a note to the Wikiproject mathematics page (WT:WPM), and I am sure someone who knows the ropes will enthusiastically help out.  Cheers,  siℓℓy rabbit  (  talk  ) 19:02, 31 July 2008 (UTC)


 * I found there's already a geometric algebra link (empty) in the clifford-algebra section of the abstract algebra wiki book. I've posted a question to the talk page of that to see what the Author's intent for that link was: http://en.wikibooks.org/wiki/Talk:Abstract_algebra/Clifford_Algebras Peeter.joot (talk) 17:34, 1 August 2008 (UTC)

"outer" vs. "wedge"?
It isn't clear from the text what the difference is among "wedge", "outer", and Kroneker products. The text says "The outer product (the exterior product, or the wedge product) "$$\wedge$$" is defined..." but "outer pro duct" is about the Kroneker (aka tensor) product not about the wedge product. How is the wedge product related to the tensor product, or are they just two things that both sometimes get called an outer product? —Ben FrantzDale (talk) 03:12, 21 January 2009 (UTC)


 * Unfortunately, yes, they are "just two things that both sometimes get called an outer product". Dependent Variable (talk) 16:28, 26 October 2010 (UTC)

I would like to suggest that the terminology, with respect to "outer product", "wedge product" and "exterior product" be changed for greater consistency, in the sense that only one of these should be the primary term used throughout the text. The introduction makes reference to the "exterior product". As I see it, the "wedge product" is predominantly used in the field (in my rather limited reading), and would perhaps be the term of preference on the body of the article, with secondary reference to "exterior product" where clarity is needed (or vice versa?). The "exterior product" also seems to be the term used with Grassman Algebras. I would strongly move for the removal of the term "outer product" aside from a single mention to note that it may be used synonymously in some contexts, but does not correspond to the more common use of the term in physics and methematics. Would others like to venture an opinion on this? "Inner", "interior" and "dot" will have similar issues, but I make no suggestion here on those. Quondum (talk) 09:51, 13 November 2010 (UTC)

You may have seen that I have been heavily editing this article for a number of reasons, not least its length. Hestenes uses outer and so do many other sources (when they mean exterior); wedge is often used in a fairly casual way. Inner is also frequently used although scalar would be more accurate. Again, dot is used casually even though that's wrong above 3D. There are many uses of words in physics that strictly are not correct in a purely mathematical sense and of course, Hestenes is a physicist. Anyway, I went with inner and outer as these are most often used and cause no confusion in context( I have linked outer to "exterior").88.82.206.110 (talk) 18:39, 17 November 2010 (UTC)


 * I've never heard anyone say "dot" is wrong above 3D. Can you explain/cite? Similarly, why is "scalar" more accurate than "inner"? —Ben FrantzDale (talk) 21:29, 17 November 2010 (UTC)


 * Up to 3D we have the usual Euclidean inner product, this may not be case in higher dimensions, when different definitions of inner product may come into play (eg the left contraction).

Apart from emphasizing that the result is grade zero ie a scalar this is something of a hangover from Clifford algebra (Clifford Scalar Product).


 * Anyway I have avoided the use of both of these terms in the article in favour of customary terms and people can then simply follow the definitions for the avoidance of doubt as to their meaning.Selfstudier (talk) 23:54, 17 November 2010 (UTC)
 * Sorry, you asked for a cite, there is a full discussion of the precise differences between dot, scalar, inner and contraction in Ch 3 Metric Products of Subspaces and Appendix B Contractions and Other Inner Products in Geometric Algebra for Computer Science (its in the refs).Selfstudier (talk) 00:50, 18 November 2010 (UTC)


 * Thanks. I should get that book. Chapter 3 is on Google Books. In the intro:
 * "We clearly have a need to compare lengths on different lines and areas in different planes. The nonmetrical outer product cannot do that, so in this chapter we extend our subspace algebra with a real-valued scalar product to serve this (geo)metric need. It generalizes the familiar dot product between vectors to act between blades of the same grade."
 * That's a start, at least. Thanks. —Ben FrantzDale (talk) 13:33, 18 November 2010 (UTC)

Examples?: Traction? Stress?
It seems like constructs of geometric algebra should work with continuum mechanics. For example, geometric algebra seems to naturally describe a surface traction—conceptually a force (vector) divided by an area (bivector). Could someone take a stab at explaining it? Here's what I get: Given a force, f, and an area, a, the traction is
 * $$fa^{-1}=f\frac{a}{\|a\|^2} = \frac{f\cdot a + f \wedge a}{\|a\|^2}$$.

It seems like there aught to be a sane way to integrate the surface tractions across the entire surface of a differential element and wind up with the geometric-algebra equivalent of a stress tensor. What grade would that object be? Would it be of mixed grade? —Ben FrantzDale (talk) 04:22, 11 February 2009 (UTC)


 * To get an answer to this you probably have to provide enough detail about the topic that somebody else can understand without being an expert. Does surface traction have an orientation that is different than the force?  How is the stress tensor defined?  There is a GA construction for the stress energy tensor of electromagnetism that is suprisingly compact, and perhaps the mechanics stress tensor is similar.  A discussion of this in the wikipedia talk page would probably be awkward.  I'd suggest instead


 * http://groups.google.com/group/geometric_algebra


 * This discussion group has pretty low membership right now, but perhaps one of the subscribers has considered this. Peeter.joot (talk) 14:17, 13 February 2009 (UTC)


 * Thanks for the link. I'll look into the GA description of the stress-energy tensor; that sounds related enough to get the gist.
 * Unfortunately Traction (engineering) isn't as good as I remember it. The quick version is that a traction (when describing a boundary condition) is a force vector per unit area, so e.g., the bottom of a car's tires, with the car at rest, would have an average traction of the car's weight (in the up direction) divided by the contact area. If the car were accelerating and the area were the same, the force vector would have a component in the acceleration direction. I'm using traction for this question because it's the simplest physical example I can think of of a vector divided by an area yielding something other than a scalar. —Ben FrantzDale (talk) 14:04, 16 February 2009 (UTC)
 * The energy momentum tensor in GA formalism actually has a four vector representation. It is $$T(a) = \frac{-\epsilon_0}{2}F a F$$ where F is the bivector field (E + icB).  I've got some notes on this that may or may not be helpful here
 * http://sites.google.com/site/peeterjoot/electrodynamics
 * under 'Energy density, Poynting vector, energy momentum conservation, and stress energy momentum tensor.' Peeter.joot (talk) 17:12, 22 February 2009 (UTC)

Disadvantages of Geometric Algebra?
I'd be interested in reasons why vector algebra is (or is not) better suited to applied or numerical problems. —Preceding unsigned comment added by 118.90.25.89 (talk) 02:58, 30 April 2009 (UTC)
 * Try here if you're interested for some discussion.--Leon (talk) 17:03, 1 January 2010 (UTC)

'''Disadvantages ? Well, let us hear what the multiple opponents of Hestenes Geometric Algebra (GA) have to say, and why they say it ?''' (I apologize in advance for all errors and perhaps injust affirmations I will introduce in the following text. I do not pretend to cover entirely the subject in a few lines, but perhaps induce some people to change their mind about GA).

-- First of all a lot of people try to minimize the role David Hestenes played in the revival and development of the geometric algebra as Clifford could have created it, had he lived longer. Jealousy exists everywhere ...

-- Second, GA gives access to a very efficient mathematical tool, a new representation of Clifford algebras. That fact seems not appreciated by theoretically oriented specialists of that domain. They perhaps dont like that amateurs or application oriented physicists, who would never have read more than three pages of abstract Clifford algebras, are able now to play on their reserved greens.

-- Quaternionists ! That seems to be a very specific tribe, which I do not know well. Of course they all are specialists : video game developpers, ray tracers, robotics developpers, spaceflight programmers, ... Understandably they do not want to change their programs and codes, etc ...  And they are right ! Until the revival of GA quaternions were the best tools for them.

But what they dont seem to know, is the fact that Hamilton and his followers lost the battle against Gibbs algebra, probably because quaternions were to difficult to interprete for "ordinary people", and so could not pretend to be a "universal algebra". I think that originates in the very simple fact that Hamilton did not - could not, as he knew nothing about Grassmann's work - recognize that the "imaginary" vector in his quaternion was in fact a real bivector.

If now we adopt an attitude towards GA trying to reduce it to quaternions, the sad result will be that GA also will remain confined to some algebraic speciality. Then GA will never become what truly it is, a "unified mathematical language for the whole of physics" (Hestenes).

Let me just give one example of the "plasticity" of Geometric Algebra. Suppose we know almost nothing about a rotation operator. We want to rotate a unit vector a to a vector b by an angle $$\theta$$. Obviously we can write, with c representing the unit vector of the inner bissectrix and I the pseudoscalar of the plane (a,b):


 * $$b=baa=bccaa=bcaac=bcacb=RaR^{\dagger} $$


 * $$ R=bc=b.c-c\wedge b=\cos(\theta/2)-\sin(\theta/2)I $$


 * $$ R^{\dagger}=\cos(\theta/2)+\sin(\theta/2)I $$

When we add to $$\lambda a$$ a vector $$\mu d $$ perpendicular to the rotation plane, we get :


 * $$R(\lambda a+\mu d)R^{\dagger}=R\lambda a R^{\dagger}+RR^{\dagger}\mu d=\lambda b+\mu d$$

Thus in two lines we have demonstrated one of the most basic formulas of GA.

- Last but not least what about the quantum physicists. Since 1920 they have accumulated an immense knowledge based on Hilbert space, Clifford algebra, Copenhague interpretation, etc ... In that very abstract domain the hestenian GA is a refreshing, fascinating new tool. I am not qualified enough to express an opinion on the possibility to obtain with it new results, but as I see it the interpretation should be very different from the prevailing one. To put it simple that seems very near to the Bohmian interpretation of QM. A particularly sensitive question is the interpretation of quantum spin, which in GA is much easier, relating to a simple rotation (plus dilatation). There intervenes the highly abstract notion of spinor, which in GA can be simply represented by a rotor.

It is obvious that physicists will not be enthousiastic supporters of GA. But they cannot evacuate the fact that GA is strictly isomorphic with the Cliffford algebra they already practice. Then the question of interpretation remains open ...

May I add finally that I was surprised that a GA article could be written without ever mentioning the notion of rotor ! That means perhaps that the editors still did not fully grasp the true importance of GA in the whole mathematical and physical world. I was tempted to modify the article, but having more than once been greeted "with baseball bats" I will do no more attempt.

Chessfan (talk) 15:08, 16 June 2010 (UTC)


 * Are there really opponents to GA? There is a learning curve to the subject, and if the reasons to learn the formalism are not understood or deemed (perhaps erroneously) to not be worth the time, then that time won't be invested.  People are lazy, and will use what they think will be easiest, and a tool that one doesn't know how to use is not easy.  If you had somebody who was equally comfortable with GA, tensors, differential forms, quaternions, ... then I'd expect that person would pick the best tool for the job.  I'd expect that to depend on the problem (plus how well each of the respective tools are understood by the person attempting to apply one of them.)  Peeter.joot (talk) 00:08, 6 July 2010 (UTC)

Well I hope you are right. But I am still under the impression I had : 1/ By reading Hestenes comments on the subject, 2/ corresponding with a French physicist , Roger Boudet who worked with him, and wrote many interesting articles in "annales de la fondation Louis de Broglie" , 3/my own minor disappointments here (see my historical controversy with "Silly rabbit") on spinors. By the way congrats for your very interesting internet page ! I have one, but in French ... Chessfan (talk) 21:51, 7 July 2010 (UTC)

APS ?
APS is a variant of Clifford algebra strictly associated to Pauli matrix representation ; I would call it a disguised matrix representation (paravectors).

Hestenes Geometric Algebra is a real Clifford Algebra, associated with Spacetime Algebra by a projective spacetime split.

So, as it is true that we can do physics with APS, it is a misleading idea to associate APS with STA.

Chessfan (talk) 08:42, 18 June 2010 (UTC)

Banish determinants and coordinates.
We want to demonstrate :


 * $$a(b \wedge c)=a.(b \wedge c)+a \wedge (b \wedge c)$$

We can decompose $$\mathbf {a}$$ into $$a_\perp$$ and $$a_\parallel$$.

Then it is easy to show that :


 * $$[a_\parallel(b \wedge c)]=[a_\parallel(b \wedge c)]^\dagger \qquad [a_\perp(b \wedge c)]=-[a_\perp(b \wedge c)]^\dagger$$

Thus the first expression cannot be else than a vector, and the second a trivector. And as :


 * $$a_\perp .(b \wedge c)=0 \qquad \qquad a_\parallel \wedge (b \wedge c)=0$$

we get :


 * $$a_\parallel (b \wedge c)=a_\parallel .(b \wedge c)=a.(b \wedge c)$$


 * $$a_\perp (b \wedge c)=a_\perp \wedge (b \wedge c)=a \wedge (b \wedge c)$$

and finally :


 * $$a(b \wedge c)=a.(b \wedge c)+a \wedge (b \wedge c)$$

Now, how can we interpret the contraction $$a.(b \wedge c)$$ ? More generally let us look at $${a.B}$$. We can write :


 * $$a.B=a. \hat B |B|=a_\parallel .\hat B |B|=a_\parallel \hat B |B|$$

Let us define :


 * $$a_\parallel =|a_ \parallel| e_1 \qquad \qquad \hat B=e_1 e_2$$

Then :


 * $$a.B=|B||a_\parallel|e_1 e_1 e_2=|B||a_\parallel| e_2$$

The interpretation is : we project the vector $$a$$ on the plane defined by $$B$$, dilate it by $$|B|$$, and rotate it by 90 degrees from $$e_1$$ to $$e_2$$, that is in the rotation direction defined by $$B$$.

Chessfan (talk) 22:22, 18 June 2010 (UTC)