User talk:Maschen/Geometric algebra

Please post substantially different versions as new drafts just above the "critiquing notes" section, and when revising an existing draft in place, please add your signature to the list of signatures following the draft to make it easier to find intermediate versions in the history. Enon (talk) 02:12, 11 May 2013 (UTC)

And please be concise! Thank you ^_^

Enon's proposed 1st draft

Geometric algebra applies Clifford algebra to geometric problems in physics and computation. Intuitively, Clifford algebra expands on the idea of vector spaces by using as its basis not just a set of n vectors to represent an n-dimensional space, but all the possible combinations of those vectors as well (using the "outer product", see below), thus creating a space of 2ⁿ degrees of freedom or "blades".

The additional degrees of freedom are interpreted in geometric algebra as oriented areas, volumes, and higher-dimensional subspaces. Bivectors (the plane areas formed by combining pairs of vectors) in particular are frequently used, for instance to simplify and generalize the representation of rotations and to replace axial vectors in magnetism. A weighted sum of different blades can be constructed in a way similar to the separate real and imaginary parts of a complex number, but with up to 2ⁿ parts. Such a hypercomplex number is called a "multivector".

Although the weighting coefficients in a multivector are real numbers, other fields including complex numbers and quaternions can be represented by certain types of multivectors. The square of a vector in geometric algebra is always a real number, but as with imaginary numbers, the square may be negative. A wide range of spaces is representable using geometric algebra, including euclidean and non-euclidean spaces of any dimension. Euclidean spaces have basis vectors which all have the same sign when squared, while non-euclidean spaces such as those used in relativity have a mixed signature.Enon (talk) 01:31, 11 May 2013 (UTC) (posted earlier on the GA talk page and the 2nd version in this page's history)

Rschwieb's draft 2.1

In pure and applied mathematics, the field of geometric algebra is the application of Clifford algebra to geometric problems in physics and computation. The main algebraic structures of study are called geometric algebras which are Clifford algebras over the real numbers. Results from geometric algebra are applied in physics, graphics, robotics, and computational science, among other fields.

Technically, a geometric algebra for an n dimensional real vector space V is an algebra containing V whose multiplication operation encodes a bilinear form on V. Intuitively, the bilinear form determines a geometry for V, and the geometric algebra carries geometric information about V in its algebraic operations: for instance, things like distance, angles and length. A basis of n elements for V can be used to produce a basis of 2ⁿ elements for the algebra. The bilinear form dictates the multiplication of basis elements, and hence influences the character of the whole algebra. Many geometric operations in the vector space V (such as rotation, projection, reflection, projection etc.) can be translated into simple operations (addition, multiplication, coordinate projection) of the algebra.

In this larger space, two or more vectors from V can be multiplied together: these products are called blades. These elements can be interpreted as oriented areas, volumes, and higher-dimensional subspaces. Bivectors (the plane areas formed by combining pairs of vectors) in particular are frequently used, for instance to simplify and generalize the representation of rotations and to replace axial vectors in electromagnetism. A linear combination of different blades can be constructed in a way similar to the separate real and imaginary parts of a complex number, but with up to 2ⁿ parts.

A wide range of spaces can be studied using geometric algebra, including Euclidean and non-Euclidean spaces of any dimension. Two non-Euclidean examples include that of the spacetime algebra for Minkowski spacetime and the universal geometric algebra. Although the algebra is created using real number scalars, the complex numbers and quaternions can still appear as other elements of the geometric algebra. For example, there can exist elements that have square -1, as with imaginary numbers. In fact, the real numbers and the complex numbers can both be considered as special cases of geometric algebras. Other physically important structures appear in geometric algebras: for example, the gamma matrices introduced by the Dirac equation appear as elements in a certain geometric algebra.

In the late 19th century, William Kingdon Clifford originally called these types of algebras "geometric algebras." Later, they were called "Clifford algebras" in his honor, but now that term spans algebras over other fields than the real numbers. In the 1960s, David Hestenes repopularized the term "geometric algebra" for the real Clifford algebras with applications to geometry.

Markup, changeable version starting from Enon's 1st draft

Geometric algebra applies Clifford algebra to geometric problems in physics and computation. Intuitively, Clifford algebra expands on the idea of vector spaces by using as its basis not just a set of n vectors to represent an n-dimensional space, but all the possible combinations of those vectors as well (using the "outer product", see below), thus creating a space of 2ⁿ degrees of freedom or "blades".

The additional degrees of freedom are interpreted in geometric algebra as oriented areas, volumes, and higher-dimensional subspaces. Bivectors (the plane areas formed by combining pairs of vectors) in particular are frequently used, for instance to simplify and generalize the representation of rotations and to replace axial vectors in magnetism. A weighted sum of different basis blades can be constructed in a way similar to the separate real and imaginary parts of a complex number, but with up to 2ⁿ parts. Such a hypercomplex number is called a "multivector".

Although the coefficients of the blades in a multivector are real numbers, other fields including complex numbers and quaternions can be represented by several different types of multivectors. The square of a vector in geometric algebra is always a real number, but as with imaginary numbers, the square may be negative. A wide range of spaces is representable using geometric algebra, including euclidean and non-euclidean spaces of any dimension. Euclidean spaces have basis vectors which all have the same sign when squared, while non-euclidean spaces such as those used in relativity have a mixed signature.Enon (talk) 02:12, 11 May 2013 (UTC)

Maschen's combined version of Enon's and Rschwieb's drafts

Fused with some additions/tweaks by M∧Ŝc²ħεИ_τlk 17:12, 11 May 2013 (UTC).

Additional lead in by Rising Eagle01 1 Aug 2013: (I welcome you to scrutinize, clean up, add hyperlinks, meta-formatting, etc (I'm new to wiki - this is my first contribution). These paragraphs welcome the HS and college level readers with some softer, motivating, generalized explanation before jumping in cold (as per comments by enon). It leads to a good jumping off point for technical explanations. One more thing, technical explanations are extremely difficult to craft. I know GA and am unsure of the clarity of the first two paragraphs that I have seen here (below the two that I added). I would recommend starting with an example of a 2-D vector space, show how it is geometrically interpreted and extended into a multi-graded geometric algebra. This leads naturally into why complex components are not needed as the 2-D ga will serve as a complex field. Then rotation naturally follows as the product of vectors in the 2-D space. Lots of pictures and diagrams.)

GA is a field of mathematics where geometric objects (e.g., lines, planes, volumes, etc.) are represented as algebraic variables and their interactions and behaviors (e.g., intersection, complement, reflection, rotation, etc.) are encoded as algebraic operations and equations. GA is a union of geometry and algebra which maintains the intuition of geometric visualization while adding to it the precision and power of symbolic analysis (best of both worlds). GA emphasizes its visualization and geometric interpretability of algebraic expressions as its primary asset, which makes it a widely useful mathematical tool for problem solving and analysis in the applied sciences and engineering. GA is so named for its central mathematical structure: the geometric algebra.

Technically, a geometric algebra is a particular kind of clifford algebra, which is, itself an extension of a vector space. As there are many vector spaces of various dimension, there are also many clifford algebras of various dimension. Special properties of these particular clifford algebras make them especially well suited to geometric interpretation and so be aptly renamed as geometric algebras. GA is the field of study of these geometric algebras and the development of their use as a practical analytical tool. — Preceding unsigned comment added by Rising Eagle01 (talk • contribs) 03:04, 2 August 2013 (UTC)

In pure and applied mathematics, geometric algebra applies Clifford algebra to geometric problems in physics and computation. Intuitively, Clifford algebra expands on the idea of vector spaces by using as its basis not just a set of n vectors to represent an n-dimensional space, but all the possible combinations of those vectors as well (using the "outer product", see below), thus creating a space of 2ⁿ degrees of freedom or "blades".

The additional degrees of freedom are interpreted in geometric algebra as oriented areas, volumes, and higher-dimensional subspaces. Bivectors (the plane areas formed by combining pairs of vectors) in particular are frequently used, for instance to simplify and generalize the representation of rotations and to replace axial vectors in electromagnetism. A weighted sum of different blades can be constructed in a way similar to the separate real and imaginary parts of a complex number, but with up to 2ⁿ parts. Such a hypercomplex number is called a "multivector".

Although the weighting coefficients in a multivector are real numbers, other fields including complex numbers and quaternions can be represented by certain types of multivectors. The square of a vector in geometric algebra is always a real number, but as with imaginary numbers, the square may be negative. In this respect, the real numbers and the complex numbers can both be considered as special cases of geometric algebras.

A wide range of spaces is representable using geometric algebra, including euclidean and non-euclidean spaces of any dimension, including infinite dimensional spaces, see universal geometric algebra. Euclidean spaces have basis vectors which all have the same sign when squared, while non-Euclidean spaces such as Minkowski spacetime occurring in special relativity have a mixed signature. Enon (talk).

More abstractly, a geometric algebra for an n dimensional real vector space V is an algebra containing V whose multiplication operation encodes a bilinear form on V. Since the bilinear form determines a geometry for V, the geometric algebra encodes geometric information about V: for instance, things like distance, angles and length. A basis of n elements for V can be used to produce a basis of 2ⁿ elements for the algebra. The bilinear form dictates the multiplication of basis elements, and hence influences the character of the whole algebra. Many geometric operations in the vector space V (such as rotation, projection, reflection, projection etc.) can be translated into simple operations (addition, multiplication, coordinate projection) of the algebra.Rschwieb (talk)

Historically, in the late 19th century William Kingdon Clifford originally called these types of algebras "geometric algebras," who made contributions in the field building on the work of Hermann Grassmann. Later, they were called "Clifford algebras," but now that term spans algebras over many fields other than the real numbers. The Dirac equation (1928) introduced the gamma matrices which can have application in Clifford algebras. In the 1960s, David Hestenes repopularized the term "geometric algebra" for the real Clifford algebras with applications to geometry.

Critiquing notes

Archived notes

Weaknesses in draft 1: The original suggestion contained avoidable jargon (degrees of freedom/axial vectors/weighting coefficients/hypercomplex number). It also tried to get at the connection with complex numbers, but only with a superficial fact about squaring vectors. Strengths in draft 1: This intro gets at the geometric interpretations of algebra elements quickly and clearly. It relates to euclidean and hyperbolic geometry clearly. Draft 2 OK, I've tried to address my above notes in the new draft, and I've tried to bring over some of the best parts of the existing lead. I figured I would leave out wlinks for now until the contents settle. Rschwieb (talk) 15:00, 10 May 2013 (UTC) Your notes raise some fair points, though I don't see your version as correcting them, or really having much connection with my version or as better meeting what I believe should be the criterion for assessing the introduction: "rate of increase of actual, intuitive understanding of GA (as distinguished from CA by its applied nature) in a target audience with high school/undergraduate-level knowledge of math and physics". Your version has even more of the problems I mentioned recently in the talk page than the current active GA article, and it wouldn't serve any purpose to address them point by point since the new version simply runs completely against the goals of the first draft. It's not really a revision but a different proposal which seems to have completely different goals and a completely different audience in mind. No doubt you meant to add your proposal to the talk page rather than substituting it, so I will add the first draft back. On jargon in the old version: "(degrees of freedom/axial vectors/weighting coefficients/hypercomplex number)". I can't think of a better concise term than "degree of freedom" - a wikilink should allow seeing what is meant if it isn't clear in context, which I think it would be for most as: "something that it takes a real number coefficient to describe". What would "Dimension" would give the wrong idea entirely. "Coefficient" would confuse the thing and its representation, and be too coordinate-linked. What concise term do you think be clearer? "Hypercomplex number" was also, I thought, quite clear from context as "something like a complex number but with more parts", which is good enough for a first understanding of multivectors, an analogy that turns out to be more than an analogy. "Coefficients" is scarcely jargon, certain to be known by anybody who has done algebra in school, though "weighting" could be taken out. "Axial vector" could easily be left out, but in context the naive reader will see that it has something to do with rotation and magnetism and that GA uses bivectors instead. Anyone who has had freshman physics in college, or even most good high school classes will have come across axial vs. displacement vectors, and it's a good opportunity to bring it to the attention of the rest, because using bivectors instead of axial vectors is perhaps the biggest single advantage of GA. There are two ways of making the draft better - changes and additions. Changes that simplify and clarify are the best changes, but additions are better if the point can't be made in a revision without going into a digression or other interruption of the flow. One can't say everything at once, so an order of presentation is needed, and getting into details or using more technical and more difficult terms should be deferred to later parts of the article, so that an clear initial overview of GA can first be presented to give the reader a schema for mentally organizing later, progressively elaborated parts of the article. That's why I didn't get into all the various equivalents to complex numbers, instead giving as an example negative signature basis elements. Signatures are explained so early primarily in order to introduce the quadratic form / space in its concrete application rather than as just a term, and to introduce the idea not only that there can be many different varieties of CA/GA, but even conveys a pretty good implicit idea of how they are classified and distinguished. These initial understandings can be refined or generalized later. I'm not opposed to the mention of related mathematical concepts and terminology in later parts of the article, but they should preferably be explained in terms of GA concepts (rather than the other way around), and the GA concepts in turn need to first be explained and understood by the reader using physical and geometric intuition, if possible, or the most widely-known levels of academic knowledge - e.g. high school algebra, not college-math-major abstract algebra.Enon (talk) 01:11, 11 May 2013 (UTC) I would like to hear everything you want to talk about, but it would really help (me at least) if we could limit the scope of the discussion. (Or, more succinctly, please write shorter answers!) I hope to also hear some third opinions from the other editors here, and there is only going to be so much bandwidth. Some fast feedback on your comments: "coefficients" is at the right level. My main problem was "weighting": that's truly is jargon borrowed from a different context. I agree that I used a bit too much detail, and some could be moved out of the lead. I will stare at what's there and see if I can think of more ways to kick out technical details and focus on what geometric algebra "is about" (but I don't plan on editing it again until Maschen (and Quondum and hopefully the other editors there) give some feedback. Rschwieb (talk) 01:39, 11 May 2013 (UTC) Sorry, I couldn't resist making a few more changes, so my promise about not editing is out the window. I wound up trying to reduce my draft 2 a little bit. I hope the changes I made were harmless/productive enough. Rschwieb (talk) 01:53, 11 May 2013 (UTC) With respect Rschwieb (a pure mathematician!), I think Enon's opening with a slight physical bias is much nicer. Perhaps Rschwieb's could be worked into the lead after Enon's like I did above. Due to a lack of time (and knowledge) I can't comment/edit much yet, maybe in a few weeks. I only know about some of the operations in GA, some things GA can allow where other formalisms cannot, and the implications GA has in applied mathematics. You two have made a good start though, so hope everyone enjoys a fruitful transfer of ideas. M∧Ŝc2ħεИτlk 17:12, 11 May 2013 (UTC) I like Enon's angle of explaining GA without building on concepts from competing disciplines, but only to show their equivalence later. But that can be quite a challenge. I guess I'm not being particularly helpful here. — Quondum 01:43, 14 May 2013 (UTC)

Draft 3

Don't try to play down your draft, Maschen: it's every bit as helpful to the process as the others. I will freely admit that my draft overlooked a lot of physics that should be there, my physics vision not being up to par with you two. I expect that you two will help me improve that vision as we go here. Here are notes for the draft:

• I notice you and Enon both prefer stating that "geometric algebra" is the umbrella term for the field of study, and allow the fact it is also a name for an algebra object remain muddled. I think this is confusing for those outside of math and physics. I do think it would be an improvement to my version to make the object aspect secondary (I had formerly made it primary.)
• I still object to the jargon I listed before in the second and third paragraphs. It is not really helpful unless you already know about the topic (COIK).
• I like what you did in the last paragraph. It has a lot of good examples of physics things that I'm oblivious to.

I'm going to go ahead and make my draft a 2.x version to adjust for the thoughts your draft provoked. Rschwieb (talk) 19:49, 12 May 2013 (UTC)

K, I adopted several changes inspired by Maschen's version. This might be a convenient way to keep revisions going without too much confusion. We keep refining our versions and eventually decide on one (or a fusion of them, somehow.) Rschwieb (talk) 21:12, 12 May 2013 (UTC)

Thanks for feedback, but I don't understand your first point just above, in particular "umbrella term for the field of study, and allow the fact it is also a name for an algebra object remain muddled", what do you mean?

I like your version 2.1. also, however you have introduced abstract jargon sooner than we did already in the second paragraph, indeed in the first sentence of it - "the multiplication operation encodes a bilinear form on V". Not sure if worrying too much about jargon would help... since it's inevitable sooner or later in any maths/phys article.

Please don't let me hold anyone behind for feedback, as I can't really touch WP for the next week... Hopefully Quondum, Selfstudier, and Chris Howard will join.. Keep up the good work. M∧Ŝc²ħεИ_τlk 22:06, 12 May 2013 (UTC)

I largely agree with Rschwieb that geometric algebras (as objects: real Clifford algebras) should be the primary meaning being addressed. Geometric algebra as a discipline or field of study ("geometric algebra applies Clifford algebra to geometric problems") is quite a different thing. At the very least, we should make the distinction very clear, and Rschwieb's first para is a good attempt at that. I don't know how helpful I'll be chipping in like this though – I barely look at WP once a week at this stage. — Quondum 01:33, 14 May 2013 (UTC)

@Maschen's question: Check out the start of algebra. It explains that the word has two senses in mathematics. Both sense are used in the sentence "Commutative algebras (objects) are studied in commutative algebra (field of study)." The same distinction is here with "geometric" swapped for "commutative." Initially I wanted to make the object primary here, as Quondum seems to prefer. However, I later came to the conclusion that "geometric algebras" are already the star of "Clifford algebra" (pretty much!), so it makes sense to let the "field of study" aspect to be primary here. @Maschen's comment: Am I understanding right that "bilinear form" is the jargon in the spotlight? (I can't see anything else I would consider jargon! If I picked the wrong thing(s), just let me know.) I agree that the term is a little above high school level, but I think it is a critical piece of information. ("inner product" might be more recognizable but for most authors that would preclude negative signatures.) The presence of the bilinear form is what distinguishes a geometric algebra from a run-of-the-mill algebra. I am very confident in the usefulness of this summary: "Here is why a geometric algebra deserves to be called "geometric". Since the vector space V has a bilinear form, it has a geometry due to that form. Since the geometric algebra's multiplication incorporates the bilinear form, the algebra has the geometry of V encoded inside it." Rschwieb (talk) 11:10, 15 May 2013 (UTC)

Sorry for not replying sooner. So you're just considering what terminology and stance is best for the lead, which is fine. M∧Ŝc²ħεИ_τlk 15:27, 19 May 2013 (UTC)

"Since the vector space V has a bilinear form, it has a geometry due to that form." Isn't this statement really only true when qualified as "real bilinear form"? Oh, and I'm not too fussy about the field of study vs. object debate, but we do need to realize that it is the focus of the article (because this is its name) that we are dealing with here. How often is the term used in each sense (outside of WP)? — Quondum 12:07, 15 May 2013 (UTC)

@Q Technically it still counts as geometry for any field! That said, I think you're right, we should add "real" once we have convinced our friends that bilinear form belongs in the lead. This reminds me of another point we might pick up later. Strictly speaking, I think "geometric algebra" only uses symmetric bilinear forms. (I've seen alternating forms used in Clifford algebra, but I don't know if they have gotten into geometric algebra that much.) We could bypass "symmetric bilinear form" with "quadratic form," but I think the latter term is slightly more jargony than the first. It is open for debate.

I seem to remember "geometry" having this abstract nature from somewhere. I was leaving the "symmetric" qualification for when the other aspects were resolved. In this instance, orthogonal Clifford algebras are meant, hence only symmetric bilinear forms. I doubt whether symplectic Clifford algebras are candidates for being called geometric algebras. A quibble: specifying only a quadratic form usually carries the implicit assumption that the associated bilinear form is symmetric. I prefer a minimum of implicit assumptions, but of course the most readily interpreted would be the way to go. — Quondum 01:34, 16 May 2013 (UTC)

As for the subject-object distinction, it is implicitly expected of mathematics readers... I don't recall any explicit explanations of it. Books come with "Basic algebra" on the front, and inside you see "Definition: an algebra is a set with...". Maschen/Enon: what do you think about the subject-object aspect? Rschwieb (talk) 13:45, 15 May 2013 (UTC)

I don't have much of an opinion on the subject-object difference, but in any case I would prefer we take Enon's applied mathematical approach at the outset with the pure maths immediately following. M∧Ŝc²ħεИ_τlk 15:27, 19 May 2013 (UTC)

@Quondum Oh wow :) I wasn't even aware of the Weyl-symplectic connection. Then I think you're right that we should safely ignore the symplectic version here. As I mentioned, this was the only thing giving me doubts about whether or not we should add "symmetric." But now I have no doubts. Thanks for bringing that to my attention. Rschwieb (talk) 13:27, 16 May 2013 (UTC)

The current process is slow?

The above process of each editor writing their draft in separate sections, then discussing almost each and every edit in detail, appears to be rather inefficient and slow. So far we (Enon and Rschwieb, I don't count as such) have only completed the lead, never mind the other sections. It's OK as far as it goes, but it would be faster and more effective to write the entire draft article in one place Geometric algebra/Sandbox, then everyone edits it, and can discuss here. When ready the draft article can overwrite the real one. If need be, we can write our own versions and store them in show/hide boxes at the top of this page. This would also reduce the amount of material on this page making it easier to scroll and navigate. But if everyone else like the current way of doing things, that's fine... M∧Ŝc²ħεИ_τlk 15:27, 19 May 2013 (UTC)

I must agree with Maschen here. Working on a common draft, if need be squabbling about details, is much more efficient. The minute I have to collate different drafts, my ability to focus is compromised. Working on the live article might even make more sense – that, after all, is probably behind the exhortation to WP:be bold. — Quondum 17:56, 19 May 2013 (UTC)

Yes. All we need to do is treat this talk page like an ordinary talk page and the draft article like an ordinary article, relaxing any irritating/fussy WP "not here - take it to the ref desk" policy by an informal and enjoyable collaboration, no more. That was the initial intention of this sandbox.

I'm going to take the liberty of inserting the Enon–Rschwieb (not mine) version there, and others are more than welcome to rewrite or replace that. M∧Ŝc²ħεИ_τlk 18:37, 19 May 2013 (UTC)

Hm, I was actually under the impression that the lead was the main thing in question (probably because it's what Enon first brought up as a draft.) Since the conversation simmered down, I had assumed we had made progress and were going to post a version soon. I don't know about the rest of the article. Perhaps a good look at the "big picture" of the article would help? I don't know enough about the subject as a whole, yet. Rschwieb (talk) 19:13, 19 May 2013 (UTC)

Actually, Maschen, I would like to request we begin with my draft. There are a few intolerable things in the draft you are posting, and several good things which I hope can be counted as a part of my draft too. I'm not saying my draft is final, but I think it's had a longer evolution than that particular draft. I'll go ahead and make the change. Rschwieb (talk) 19:22, 19 May 2013 (UTC)

Feel free to overwrite it by all means. But Enon clearly stated "the entire article needs to be scrapped and started over" so why not write a full draft? M∧Ŝc²ħεИ_τlk 19:26, 19 May 2013 (UTC)

There is no consensus on starting the article over. I think even Enon should acknowledge that such a statement is potentially an overreaction. Enon makes some good points, but I (and I suspect Rschwieb) do not agree with this one. What should be (and is not adequately) expressed is that GA primarily exploits the algebraic structure of a real CA to represent geometric objects and operations. Because of the correspondence and the versatile algebraic structure of CA, this is a particularly powerful and convenient algebraic formalism of geometry. It should be obvious (when properly phrased) that a useful GA is expressible in CA, but does not necessarily utilize the full CA – so, for example, vectors are restricted to one subspace, rotations to another, and so on. Anyway, before and in addition to the more formal approach, Enon's idea of an intuitive description in the lead really does make sense, but the words that are there at the moment do not achieve this (in either the original or Enon's suggestion). — Quondum 21:19, 19 May 2013 (UTC)

I think I agree with Q's analysis. I don't really know if a major (complete?) rewrite is in order. I never said I was against rewriting more than the lead: I just said I thought that only the lead was on the menu :) Concrete gripes will be the most useful motivators of change. So: what's the worst feature of the article? Rschwieb (talk) 21:27, 21 May 2013 (UTC)

Trying to answer this, I think the question should be "What are significant problems with the article?", since hunting for the worst leaves me wondering how to rank the issues. Currently the main article tries to address primarily the algebraic nature of GA, whereas that is really the province of the CA article. I think that the focus should be more on interpretation and useful operations (it actually does address this to some extent). Perhaps we could start with the general concepts to be dealt with (vectors, subspaces, scaling, projections, a metric, rotations, spinors, multilinear products in geometry and physics (areas, volumes, torque)), and only later move to the way GA handles them (the algebraic equivalents)? Producing an outline, and then moving whole sections across from the existing article might work - the existing content has a lot of good stuff in it. The algebraic aspects (how well CA does the job of representing the concepts, and the algebraic underpinnings and things such as invertability) deserves some mention, but perhaps not at the beginning. The typical reader is going to be more a student/physicist than a mathematician. So, in a nutshell, the article should be more "concrete" in its focus. — Quondum 23:04, 24 May 2013 (UTC)

Yeah, if it's too difficult to rank, by all means pick the bottom several :) I like the idea of leaving as much algebra as possible to the Clifford algebra article and focusing on the geometric interpretations here. There will still be algebra, but hopefully it will be "more targeted." Rschwieb (talk) 23:58, 24 May 2013 (UTC)

Main namespace

Judging by the fact that the sandbox page in the main space attracted attention and was tagged, I suspect we've overstepped a line. Sandboxes probably do not belong in main space. Subpages technically are part of the main article space (and previously were used as a way of splitting an article into many pages). Talk space and user space would probably have been fine... Anyway, let's bring it to a conclusion, then we can delete the subpage. We should probably not let the page hang about indefinitely. — Quondum 23:15, 19 May 2013 (UTC)

You're probably right, though I haven't encountered any WP policy or even any reason which forbids it...

Maybe we should just pack up here and take all the discussion to the talk page in a new section, and perhaps box the drafts at the top of that section? Leaving the draft talk and draft article pages blank until someone deletes them? M∧Ŝc²ħεИ_τlk 18:54, 20 May 2013 (UTC)

No, I wouldn't bring it back unless we're going to park it for ages or someone complains; it's already created and doing no harm. But I'd think twice before doing it again, just to keep things shipshape. — Quondum 00:32, 21 May 2013 (UTC)

Outlining the rest of the article

Anyone have an idea for a general outline of the rest of the article? One of you might want to start fresh and not duplicate the old article's outline... Rschwieb (talk) 21:22, 21 May 2013 (UTC)

Not yet, though I look forward to thinking about this ^_^ M∧Ŝc²ħεИ_τlk 23:07, 21 May 2013 (UTC)