Talk:Geometric algebra

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1 Emil Artin
2 Clarification?
3 POV
4 Definition?
5 Some text
6 Geometric product of multivectors ?
7 cf Clifford algebras
8 A geometric algebra bibliography
9 The inner product in GA is more than a standard dot product
10 Is there a convention about the order of multivector components?
11 How comes that geometric algebra and Clifford algebra are not synonyms?
12 Explicit example of geometric product
13 Historians' use of the term
14 Remark on the history section

[edit] Emil Artin

Not that I have anything against Hestenes-worshippers (I met Hestenes once and he's a nice guy, if a little bit ..... well, let me put it this way: I think he said Clifford algebras will settle all questions of physics, or something like that) .... OK, where was I? Oh: Well doesn't Emil Artin also warrant some attention on this page? -- Mike Hardy - 131.183.73.24 (contributions) 02:13, 25 February 2003 (UTC)

[edit] 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)

[edit] 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 70.27.25.227 (talk) 17:24, 22 November 2005

[edit] 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)

[edit] 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.]

<rant> 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. </rant>

Enon 04:23, 22 April 2005 (UTC)

[edit] 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)

[edit] 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)

[edit] 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 — 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)

[edit] 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 2^N-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?

Not specifying that the dot product is not valild for multivectors (in an article which is about multivectors), or
not saying that, when both operands are vectors, the inner product happens to coincide with the dot product?

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)

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)

[edit] Is there a convention about the order of multivector components?

[edit] 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 R^N as a vector with 2^N 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)

[edit] 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:

Table 1 - Components for multivector in Cl_4,0
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

Table 2 - Components for multivector in Cl_3,0
scalar	A
scalar components of 1-blades	A₁	A₂	A₃
scalar components of 2-blades	A₁₂	A₂₃	A₃₁?
scalar component of 3-blade	A₁₂₃

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 R_3,1. 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 R^1,3, 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)

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 Cl_4,0, and that will include a nice, contiguous sub-table for Cl_3,0; and within that a nice contiguous sub-table for Cl_2,0. 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 Cl_4,0, to show the pattern of plus and minus signs, this will automatically include the relevant 8x8 table for Cl_3,0 in the top left corner, the 4x4 table for Cl_2,0 in the top left corner of that, and the 2x2 table for Cl_1,0 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:

is there a "preferable" method?
is there a conventional method, at least adopted by groups of authors (e.g. physicists, computer scientists)?

Paolo.dL 14:01, 1 June 2007 (UTC)

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.

By method A, [A_xy , A_xz , A_yz] = [a_z , -a_y , a_x].
By method B, multivector A and vector a are just permutations of each other:[A_xy, A_yz, A_zx] = [a_z, a_x, a_y].

If A_xy , A_xz , 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)

Multiplication tables. Notice also that the point here is about scalar coefficient vectors with 2^N 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 <A>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 <A>₂ into components, it will be as something like <A>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 <A>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: <A>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)

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)

[edit] 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 R_1,3 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)

[edit] 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).

[edit] How comes that geometric algebra and Clifford algebra are not synonyms?

Here's what is written in the History section:

(Clifford's) "contribution was to define a new product — the geometric product — on an existing Grassmann algebra, which realized the quaternions as living within that algebra"

This statement generates a doubt. Here's how I interpret it:

(Original) Clifford algebras. Clifford introduced only a subset of what are now called the "Clifford algebras". This subset is currently called "geometric algebras".
Generalized Clifford algebras. Later, someone else generalized Clifford algebras, and the resulting set was called "Clifford algebras".

Is my interpretation correct? If it is correct, how comes that geometric algebra and Clifford algebra are not synonyms? Why did that happen? Isn't there some better terminology in the literature? Isn't it desirable to clarify this point in the history section?

Paolo.dL 18:14, 26 July 2007 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)