Talk:Geometric algebra/Archive 4

Terminology: "outer product" and "inner product"
Hestenes appears to have introduced some unnecessary new terms into GA, with conflicting meaning to closely related branches of mathematics, not in universal use in GA texts. I propose replacing throughout the article, while retaining the mention of equivalent terms: Opinions? —Quondum 07:32, 30 October 2016 (UTC)
 * outer product with exterior product
 * inner product with scalar product (unsuitable due to existing use as an extension as in the article  —Quondum 21:24, 27 August 2017 (UTC))


 * Since there does not seem to be overwhelming engagement, I dare to deposit my strong support for this idea, even when it might be considered near OR, or if it should go slightly beyond bureaucatic limits of Wikipedia. -Purgy (talk) 09:56, 1 November 2016 (UTC)


 * I see that modern text
 * makes no mention of Hestenes, uses the term "exterior product" (which is also used by many earlier authors as far as I know). They seem to be very careful/precise with terminology and rigorous in their development of the algebra.  They do not cover everything, but at least use of the term "exterior product" cannot be considered to be OR.  They refer to the scalar/inner product of vectors as the symmetric bilinear form.  They are also careful to separate the algebra and its dual, which makes some refinements to the contractions etc.  —Quondum 15:46, 1 November 2016 (UTC)
 * makes no mention of Hestenes, uses the term "exterior product" (which is also used by many earlier authors as far as I know). They seem to be very careful/precise with terminology and rigorous in their development of the algebra.  They do not cover everything, but at least use of the term "exterior product" cannot be considered to be OR.  They refer to the scalar/inner product of vectors as the symmetric bilinear form.  They are also careful to separate the algebra and its dual, which makes some refinements to the contractions etc.  —Quondum 15:46, 1 November 2016 (UTC)


 * One reason why exterior product may have been avoided is because mathematicians (as is their habit) have generalized the idea of exterior product beyond where GA wants to go which is the Grassmann exterior product (ie on a vector space, although it should be evident from context in this article).Selfstudier (talk) 18:13, 6 September 2017 (UTC)


 * To be honest, it does not take me by big surprise that a math oriented, rigorous, careful text might not mention Hestenes and his pedagogic roots, although I pity this a bit. In any case, if you find noteable sources for your intentions, just the better; may I assure my unimportant support for all your efforts of reducing parallelity in notion, especially, since you referred to mentioning the equivalent terms. Thanks! -Purgy (talk) 07:19, 4 November 2016 (UTC)


 * As the person who originally fleshed out this article back in the day (and who understands abstract algebra) I was very careful to avoid turning it into a page about Clifford algebra and specifically warned against doing so and yet...Selfstudier (talk) 17:55, 11 February 2017 (UTC)


 * I understand a desire to keep an article on geometric algebra different from a more abstract one on Clifford algebra. If one thinks of 'geometric algebra' as an area of study as defined in the second sentence of the lead, then such a separation totally makes sense.  However, 'a geometric algebra' is nothing other than a real Clifford algebra, so here the distinction is only in what is of interest about it within a field of application.  It might make sense to make the first paragraph of the lead less abstract.  Yet, I would resist turning the article into the handwavy thing it once was, which omitted a lot of clarity of definition.  I remember the article lacking the answer the question: What is the difference between a Clifford algebra and a geometric algebra?  What is it you want the article to be about if it isn't to be about a class of Clifford algebras and their applications?  —Quondum 20:22, 11 February 2017 (UTC)

Hah. When I get some time, I will make a page about applied GA for people who actually want to use it as oppose to theorizing about it.Selfstudier (talk) 13:24, 19 August 2017 (UTC)


 * I, for one, would support this: a cluster of articles that serve as reference for different purposes makes sense, e.g. Clifford algebra as the most mathematical perspective, Geometric algebra as a high-level outline of its geometric expressibility and applicability to geometry, Geometric calculus for its extension to manifolds, and then several articles such as Conformal geometric algebra and Applied geometric algebra (the latter at present being nonexistent) for more specific operations and techniques of interest to practitioners. Some of the current content in Geometric algebra might belong in the latter.  —Quondum 18:29, 19 August 2017 (UTC)

Refactoring Content
I made a new section to hold the principal models of geometry that are encountered in GA (this is typical way of presentation in literature). This picks up parts of the article that were otherwise just "sitting there" in an unrelated manner, hopefully its better than before. Selfstudier (talk) 05:09, 8 September 2017 (UTC)

Quality scale
If I were more bold, I'd lower the quality, currently assessed as "B", of this article. The recent edits added a plethora of interesting facts and points of views and names of algebras, especially their acronyms (GA, CGA, AST, STA; even the well known quaternions are introduced as Hamilton algebra), thereby lowering the mathematical quality of this article, which only recently has dramatically improved. While the recent improvement targeted tidying up of diverging nomenclature, and sorting of concepts, developing the article to a resource for a transparent concept formation, the newly introduced information lacks mathematical rigor and transparent sorting to an extent, which makes this article inappropriate for me to acquire a resilient picture, covering the mathematical notion of Geometric algebra. Imho, this article is on the verge to turn to a fan site: The best algebra ever, you'll never want an other. Purgy (talk) 13:32, 8 September 2017 (UTC)
 * The quaternion algebra is a generalization of the Hamilton quaternions (see Quaternion algebra) in just the same way as Exterior algebra is a generalization of Grassmann algebra. I don't know why you are complaining about the alphabet soup, its standard in the literature (except I have no idea what AST is, you made that one up). I am adding to the article material on GA that is missing, homogeneous and 1-up algebras, spinors in GA and relation to physics,etc. I don't know what "transparent concept formation" or "transparent sorting to an extent" even means. Your comments simply sound like an unhelpful, unconstructive and non-specific rant to no useful purpose. If you have specific edits to make, make them, if you have specific complaints or issues, there are procedures you can follow. Selfstudier (talk) 15:30, 8 September 2017 (UTC)
 * Since it has been mentioned, I'm downgrading this to quality rating C. This is because IMO its quality has never been at the B level, and not because of any recent changes.  As to the changes, if structure and worthwhile information are being added, a period of flux prior to making it more rigorous again may be reasonable.  I have my own reservations (as I've indicated already), intend to remain out of edits and commentary for a while, hoping that since minor reorganization might address such reservations at a later point.  I think we're all interested in a quality article with really useful information on the topic, but will need to find a way of reasonably accommodating multiple readership.  A spirit of cooperation usually helps.  —Quondum 17:18, 8 September 2017 (UTC)


 * @Selfstudier: I followed the procedures to complain about the development of this article in a way I considered appropriate. I did not want to oppose to adding material, which you consider missing, but I tried to express that my efforts to retrace your amendments to rigorously defined notions are unsuccessful ("intransparency"). Of course, this may be caused fully by my personal incompetence. In general, I do not want to discuss each of your amendments, but, e.g., please, why do do you link Hamilton algebra (obviously inexistent) to quaternions, if you want to establish a "just-the-same-way generalization" between "quaternions" (a skew-field) generalized to "quaternion algebra" and "exterior algebra" generalized from Grassmann algebra? In my opinion there is not much wrong with having just an other fan page in WP, but this property is disconnected to mathematical quality. Purgy (talk) 09:11, 9 September 2017 (UTC)
 * As for whether the Hamilton algebra can be said to exist, try Google, its a much used expression. Ditto Grassmann algebra. That it doesn't exist on Wikipedia is not my concern, I provided what I consider a helpful link for users (quaternion algebra would do just as well). If you don't like the link, edit it out. If you really are interested in GA, I suggest that you read some of the references,it has nothing to do with "fan pages", it is standard material for the subject.I don't see any future for this conversation, so that's my last word.Selfstudier (talk) 12:26, 9 September 2017 (UTC)
 * Selfstudier, you recently expressed a dislike for the way that the page had changed since you had substantially edited it, but the same is likely to recur if you remain so disrespectfully dismissive of the considered input from others. Purgy, I understand (and to some extent agree in general with) your general point about a "fan page", but perhaps it will be more effective to present specifics.  Generalities (such as your "fan page" or Selfstudier's "try Google") are usually counterproductive in forums such as this.  —Quondum 14:33, 9 September 2017 (UTC)

Spinors
I see you recently added a section on spinors, which is a good thing. However, we perhaps need to indicate that the usage of the word "spinor" by GA people does not exactly correspond to its traditional meaning in mathematical physics. It can cause a lot of confusion if people meet the GA meaning first, then think that this is how the word is generally understood. Jheald (talk) 10:45, 11 September 2017 (UTC)
 * Hi, I did think I had done that, the main article on spinors is linked out at section top and the first sentence contains a specific caveat as well as the statement that this is the GA definition. If you think it needs further specifying, go ahead, I am not going to object. Selfstudier (talk) 10:53, 11 September 2017 (UTC)
 * I think one could make a case that the phrase "desirable to view spinors from multiple perspectives" is actually positively misleading. Spinors in GA are not a "different view" of the same object. What GA calls spinors are actually fundamentally different objects, with - for example - fundamentally different numbers of degrees of freedom.
 * They're so different, it's hard to say what relation (if any) there is between the two concepts. Jheald (talk) 11:40, 11 September 2017 (UTC)
 * These seem like additional objections. If we are going to go beyond the introductory, I think I would be more comfortable handling this in terms of cites so let me ask you first whether it is your contention that the material https://arxiv.org/pdf/math-ph/0403040.pdf or https://arxiv.org/abs/1703.01244 is wrong? Selfstudier (talk) 12:14, 11 September 2017 (UTC)
 * The statement could be modified to state that GA models only one type of several different types of object called 'spinors' in various contexts can be made; this should not be too difficult or contentious. Vaz and da Rocha construct a table of classification by isomorphism of three general classes of "spinor" in low dimension, and the clear difference kind of brings this home.  They further make the point that even with a given type of spinor, what some might call a spinor, others might call a pinor, and what some call a spinor, others call a semi-spinor.  So the language here would have to be fairly specific.
 * As a separate point, "The above identifies spinors with the even subalgebra (a subalgebra under the geometric product), in other words spinors are general combinations of the even elements of $$\mathcal G(p,q)$$, yields the GA definition, a multivector $$\psi$$ in $$\mathcal G(p,q)$$ such that $$\psi v \psi^{\dagger}$$ is in $$\mathcal G(p,q)$$" is significantly incorrect.  Where was this sourced?  —Quondum 12:25, 11 September 2017 (UTC)
 * On your second point, that is cited in the article already, Bromborsky p.28.Selfstudier (talk) 12:36, 11 September 2017 (UTC)
 * Only the second part is there that I see, and it was not cited in a footnote that is visibly attached to the statement. I've fixed your error, but the first part of the statement is still in question.  Please exercise more care; others will reasonably delete rather than correct obviously incorrect stuff.  —Quondum 15:53, 11 September 2017 (UTC)


 * So, taking language from the Francis & Kosowsky (2008) paper, here are some points that I think it would be useful to make explicitly:
 * "A spinor is an object that transforms under one-sided multiplication by an element of a spin group."
 * However, the conventional notion of a spinor is more restrictive than this:
 * "Typically a ... spinor is regarded as a ... complex column vector [of appropriate dimension] (from a physicist’s standpoint), or the left-minimal ideal of a real or complex geometric algebra (from the mathematician’s point of view)."
 * (This is important to physicists, because they want objects that reflect the freedoms induced by the irreducible sub-groups, per Wigner's theorem).
 * In a Clifford algebra, "the most obvious choice" (for a space that is closed under the one-sided action of the spin group) "is the set of even multivectors Cℓ+p,q".
 * The restriction to a minimal left ideal can be achieved in a many-to-one way by right multiplication by an idempotent projector, ½ (1 + r), where r2 = 1. (Or, in higher dimensions, by a series of such projections).
 * There is also a further twist: physicists' spinors are usually constructed from the complex Clifford algebra for a particular number of spatial dimensions. This can be identified with a real Clifford algebra -- but this is a bigger algebra than that usually associated with the Geometric algebra for those spatial dimensions.  (Though things can be worked out, for Pauli and Dirac spinors at least).
 * What GA people call spinors correspond to what would conventionally be called an "element of the spin group" or a "spin matrix".


 * One could go on to discuss in more detail to what extent the absence (or presence) of the idempotent projector in what GA people call a spinor changes things, and alters the physical notions a physicist would have about the meaning of the word "spinor". Jheald (talk) 13:45, 11 September 2017 (UTC)
 * In terms of amending the article, having laid out what GA physicists do (I assume this is a minority) with GA and their version of a spinor, what I think you want to see is some set of exceptions to this representing what the majority of physicists do instead, is that right? Also, would you know of any sources that could usefully be cited for that purpose? Selfstudier (talk) 15:37, 11 September 2017 (UTC)
 * No, I would present right up front what the accepted understanding of "spinor" is in Physics; and then talk about how the GA use of the term is different and not as specific, and then discuss to what extent that lack of specificity is (or isn't) a problem, if GA people want to talk about something that corresponds to what physics people would understand as a spinor. Jheald (talk) 16:16, 11 September 2017 (UTC)
 * I have deleted the section since there is no consensus. Selfstudier (talk) 17:14, 11 September 2017 (UTC)
 * That's a pity, because I think some material on spinors (in physics and in GA) would be useful. Would something following the flow of the six points above make sense, probably after the "Hypervolumes" section (being a more advanced topic), rather than before?  Jheald (talk) 19:10, 11 September 2017 (UTC)
 * I've tried to present at least the skeleton of what makes sense in the start. I think one can alert the idea that spinors are disparate (as I've done in the first sentence of the section) and present the GA definition of a spinor.  I still need to sort out detail (whether it the versors or only the rotors, whether they are required to be normalized).  Vaz & da Rocha actually cover this quite well, including which are defined as a left minimal ideal, but I do not have the book with me right at the moment.  Going into other types of object also called spinors might not be appropriate here, but at least we should identify what they are useful for.  This is an area in which Clifford algebra improved understanding substantially, an observation I can probably find a cite for.  The rest of the section is more like examples for illustration, which will need review but is important.  Once I'm done, feel free to strip out whatever does not fit or make sense.  —Quondum 20:37, 11 September 2017 (UTC)

/, after trying to get to grips with this a bit, I've had to conclude that (aside from me being out of my depth): My inclination is thus now to limit my earlier impulse for the article to include spinors to little more than a mention that versors give a way of expressing spinors (of any definition), but cannot be regarded as a definition. This is only now bringing home to me how oversimplified the idea of a spinor is in much of the GA literature. Any feelings on how you would like to proceed on this? —Quondum 03:49, 12 September 2017 (UTC)
 * It will be essentially impossible to do the subject of spinors any justice here, because the concept is too subtle by an order of magnitude; essentially all your bullets above might be of interest to an algebraist and to a physicist, but to even define any of the spinors with any degree of rigour requires building blocks not covered here, starting with some involutions and two types of spinor norm. This goes into abstract algebraic properties in a way that is a bit out of character for the article.
 * The concept of a spinor as defined by Bromborsky seems to be equivalent to an element of the Clifford–Lipschitz group (a.k.a. Clifford group) contained in a Clifford algebra, and is nothing other than the versors. This seems to be a bigger group than any of the three definitions of spinor described by Vaz & da Rocha.  It thus has the character of a name being used differently from its standard usage (which seems so typical of the GA community).  One could define the Clifford–Lipschitz group, the Pin group and the Spin group, ans how their relation to the versors and rotors.
 * The articles Clifford algebra and Spinor cover actual spinors in quite a bit of detail, so it seems out of place to address it here. The topic, in a sense, seems at home in Clifford algebra rather than here.
 * As Heald pointed out there is no real discrepancy as such until you get beyond Pauli and Dirac. Debate (aside from interpretations) is in fully translating spinor theory a la Penrose into GA (I think this will get done in the end even if it might take a while). The GA approach has the merit of simplicity and one can get quite far quite quickly with it(Francis & Kosowsky explanation is the clearest exposition that I know of, there might be something later, I don't know); one can say that the GA definition is "wrong" if one wants to, however since it's a definition the objection can only be to the name "spinor" (some people have taken to calling them g-spinors as in geometric), if you think the terminology is troublesome right now, its as nothing compared to subsequent material (that I purposefully avoided at the outset). I don't mind whether the section is in or out, if it will cause trouble it may be better to just leave it out.Selfstudier (talk) 11:27, 12 September 2017 (UTC)
 * Okay, I'm going with that: I see that I will not be able to get this into a reasonable shape, and have reversed my own re-insertion. In fact, I see no difference between "rotor" and this GA version of "spinor".  One has to be awfully careful to define terminology and to relate it to existing terms (and the uses of the same terms used differently elsewhere) if one is to avoid confusing a reader.  This is of course complicated by that one also has to contend with a lot of authors writing about highly restricted contexts (e.g. Euclidean space), so their statements are not general to GA and must be qualified by such context here.  —Quondum 12:06, 12 September 2017 (UTC)
 * I just noticed there is a page Spinors in three dimensions, not sure why they wanted to hive that one off in particular, anything that mentions cross product in the first sentence is begging for a GA treatment haha Selfstudier (talk) 11:48, 13 September 2017 (UTC)
 * Who knows. It is clearly based on GA, but seems to be WP:OR, and thus presumably does not belong in WP.  —Quondum 19:23, 13 September 2017 (UTC)
 * That was a joke right? Complex numbers, cross products, matrices...No GA there Selfstudier (talk) 21:33, 13 September 2017 (UTC)
 * No, it was me being hasty and getting it wrong, through mistaking matrices for multivectors in the expressions early in the article. —Quondum 00:43, 14 September 2017 (UTC)

Terminology specific to geometric algebra Section
This section is coming after first use of some of the mentioned terms so I will copy edit the material to each first use(as has already been done with vector) and eliminate this section altogether. Selfstudier (talk) 11:35, 25 September 2017 (UTC)

CGA as a unifying framework
Hongbo Li's claim that CGA provides a general framework, including for projective, is a bit over the top. The Grassmann algebra with 2 dimensions lower provides the incidence structure of all projective geometries. CGA probably provides the transformations of all the other geometries (their groups of transformation are, as far as I know, contained in the conformal group of the same dimension), but it does not provide the projective transformations of that dimension, only the incidence properties. We should be clear about what is said (and avoid wording that can be interpreted with false implications). Li is being horribly imprecise here, and maybe we should be very careful of how we reflect what he says in general. Leaving the reader to draw inferences from phrases repeated from a text is not what we should be doing. So we still need to fix this one, despite my toning down of what was claimed from the arguably false "manipulation" to the undefined "provides a framework". —Quondum 12:07, 6 September 2017 (UTC)
 * Current remaining "hole" if you want to think of it that way is indeed homography (projective transformations), Dorst/Li's (3,3) looks like it might well fix that problem (creates another, just got efficient 32 dim and now 64, will need some tricks there), we'll see. I was thinking of including this aspect somewhere, not sure where tho.Selfstudier (talk) 12:22, 6 September 2017 (UTC)
 * Personally, I draw inferences from texts all the time, I think most people do that; you are at liberty to provide contradictory or alternative cites, you cannot just say a cite is wrong because you think it is wrong. If you want to pick up on the "hole", just cite Dorst and Li for it.Selfstudier (talk) 12:28, 6 September 2017 (UTC)
 * To produce an example in a low number of dimension does not justify a general claim. I don't really have the time to research everything that you put into the article or even to correct what is clearly incorrect; I was just trying to alert you that are putting things in that seem to amount to speculation, either by a GA source or by you.  —Quondum 13:34, 6 September 2017 (UTC)
 * Well, the nice thing about Wikipedia is it doesn't just depend on you (or me), if we don't have the time or the inclination, someone else will come along who does and they will edit and/or cite, hopefully resulting in a better article as time goes by.Selfstudier (talk) 14:34, 6 September 2017 (UTC)
 * Li was not precisely wrong in his statement, the "hole" comes about because you have to step outside the GA formalism (into a non-metrical version of Cayley algebra) and then step back in after doing whatever (resulting in need to translate between formalisms and additional coding etc). What GAers want is the whole lot in one formalism (it used to be half a dozen or more and many still do it that way).Selfstudier (talk) 17:12, 6 September 2017 (UTC)
 * CGA, and its extension "Extended CGA (k-CGA)" does provide a general framework for planar orthographic projections, planar perspective projections, and spherical (or hyperpseudospherical) projections (projections into the surfaces of spheres) of general degree k polynomial plane curves, such as quadratics, cubics etc, as entities. The curves are represented by general degree k surface entities intersected (by wedge) with a plane entity. The details about these projections, and of k-CGA, are not published well yet, so it is all in the original research (OR) phase right now. Some details are to be published later this year. The formulas of these projections are in a proceedings paper of SSI 2016 in Japan and later in an extended paper in their SICE journal JCMSI. A preprint that gives the formulas (which really generalize down to CGA for general degree 1 lines, up to general degree k curves in k-CGA) is found in the paper on vixra: vixra.org/abs/1511.0182 (this is not cracked pottery, even though it is on viXra!). So, what I am concluding is that, Hongbo Li may not be entirely wrong when he supposedly says that CGA provides a general framework for projections as well. For general polynomial plane curves, there are formulas (products) that give a variety in projections. These formulas are based on intuitive geometric concepts or products, involving versors, contraction, inversion, and standard projection formulas. However, these formulas are not simply versor operations, which some other research has sought. Most literature, if not all, focuses on projections of points, so the CGA methods of projections are of entire curves, which may be a significant advancement, in theory. Something that has not been worked out fully are shearing transformations, but even these are probably possible on at least general plane curves (probably whole surfaces of general polynomial degree, in theory) by using combinations of the hyperbolic rotor (boost), rotor, and some projections (pseudoscalar subspace projections), so again, these are not pure versor operations but may involve versors, projections, and perhaps contractions (part of projection). Too much OR, but just wanted to say, there are theoretical possibilities and some working formulas. Twy2008 (talk) 13:54, 7 September 2017 (UTC)
 * More work for me, haha. Hitzer has a long association with GA, I know that; as you say, it's a research area, good to know that it is active, I don't want to wait until next AGACSE for news, lol. Wonder what else is hiding in these 200 year old math archives..? Selfstudier (talk) 17:57, 7 September 2017 (UTC)
 * Twy, I saw that you have created an article entitled Quadric Geometric Algebra, maybe there could be a small section for Research Geometry Models or something of that sort and it, the above and maybe one or two others could be mentioned in it. Selfstudier (talk) 19:13, 27 September 2017 (UTC)

Definition
The definition section used to give the clear, succinct definition that a geometric algebra is the Clifford algebra with $$q(v) = \|v\|^2$$ (i.e. it is the quotient of the tensor algebra by the ideal $$(x \otimes x - \|x\|^2)$$). I really like the following: "the geometric algebra for this quadratic space is the geometric algebra $\operatorname{G}(V, g)$, together with the exterior algebra ${\textstyle \bigwedge}V$." This reads like, and to me is, a circular definition. You cannot define a geometric algebra in terms of itself, when the reader has not been introduced to a definition yet! Also, "together with" is vague. Overall, this is really not a definition at all, and is extremely confusing.

For this reason, I highly suggest reverting back to the earlier definition. Clifford algebras are more general than geometric algebras (the exterior algebra is one example, and clearly not the same as the geometric algebra) and the two should not be treated as synonymous. It is useful to emphasize how the geometric algebra includes the exterior algebra, but that is not part of the definition, but rather a corollary of the definition obtained by noting that the product $$\mathbf{a} \wedge \mathbf{b} = \frac{1}{2}(\mathbf{ab} - \mathbf{ba})$$ has the same properties as an exterior product.

I think the axiomatic definition is more intuitive for readers (it certainly was for me before I learned abstract algebra), but has other problems, namely that it is not clear what the geometric product operates on or returns. Implicitly, it appears that we mean that the geometric product is the "freest" product subject to those axioms, but the notion of being free is not elementary, and thus I was left scratching over my head about the result of the geometric product of two vectors. For this reason, the formal definition should be the succinct, abstract one at the start of my post, with the axiomization noted as an informal definition unless we have an elementary way to explain the nuance of being "free". Otherwise, it's a chicken-and-egg problem when we define multivectors using the geometric product since a mathematical operation is a function, needing a domain and (importantly) a codomain.--Jasper Deng (talk) 07:07, 5 May 2019 (UTC)


 * Well, Wikipedia is a game of secondary sources (ie personal opinions are irrelevant, all that matters are the sources). Perhaps the simplest way to proceed would be to examine suitable secondary sources (about geometric algebra) and see what they say as regards definition. Like the Foundations section of this one, for example https://people.kth.se/~dogge/clifford/files/clifford.pdf. You will likely find that there is some discussion as to definition, in which case that discussion should as well be reflected and sourced.Selfstudier (talk) 09:13, 5 May 2019 (UTC)
 * Sourcing is irrelevant if the definition given is blatantly circular like it is here. In any case, the paper you linked uses the definition I advocate, not the nonsense that the current article has (sorry if that appears WP:UNCIVIL but the tautology "A geometric algebra is a geometric algebra" does nothing whatsoever to define that term).--Jasper Deng (talk) 10:24, 5 May 2019 (UTC)
 * This is blatantly a circular definition. It has been introduced on October 2017 by  without providing any source nor any explanation. I have restored the previous definition, without checking whether the numerous other Selfstudier's edits at this time were correct or not. D.Lazard (talk) 11:07, 5 May 2019 (UTC)
 * Feel free to add sources, no-one is preventing it. It's a good idea and it is after all, the way Wikipedia operates.Selfstudier (talk) 11:14, 5 May 2019 (UTC)
 * To be clear, the burden of proof (or rather, verification) is on you since you were the one who originally inserted that change. It seems like you were careless (though not in a bad-faith manner) in replacing the Cl(...) notation, doing so blindly without reading the surrounding text. As long as you agree that you did that, and promise to be much more careful in the future, I don't really have a problem with the text anymore.--Jasper Deng (talk) 07:41, 6 May 2019 (UTC)

Non sequitur claimed
These are two different isomorphisms (I have not verified your claimed isomorphism, but its validity is irrelevant anyway). Please carefully review what you are claiming. —Quondum 00:18, 14 July 2019 (UTC)
 * I have put what is intended in plain English, no notation. Better now? If so, then I will add the wikilink.Selfstudier (talk) 09:09, 14 July 2019 (UTC)
 * At least it does not make incorrect claims now – previously it confused two different isomorphisms. Now you have simply added a claim to how I had corrected it.  The relevance of the isomorphism of 3-d Euclidean space with the even subalgebra of the Clifford algebra of Minkowski space is not clear in the historic context from the sectrion, and it is not clear what the "three approaches" refers to.   —Quondum 19:49, 14 July 2019 (UTC)
 * If you don't like it, edit it out, I will not engage in pointless discussions.Selfstudier (talk) 21:38, 14 July 2019 (UTC)

New Versor Section
This is reasonably crucial concept in GA not really dealt with properly at the moment. The ideas around sandwiching and relationship with rotation, reflection, rotor (not to mention spinor) should get wrapped up in here somehow. (I will be adding to this section in pieces rather than writing it all up and doing it one go so a little patience).Selfstudier (talk) 18:03, 21 September 2017 (UTC)


 * An edit I made under this heading has been twice wrongly reverted by an editor here with false reasons on both occasions, despite being informed that it is standard terminology well sourced in the literature. I have rereverted.Selfstudier (talk) 19:09, 22 September 2017 (UTC)


 * Well, forgive me for failing to understand that you meant something other than what you said with your edit comment "Check Dorst, hestenes, multiple sopurces for "k-vector" sign is another issue.."; I only reverted because I took it as evident that you had mistaken something. To describe it as "wrongly reverted" is a bit strong.  I'm finding the strong emphasis on Hestenes's words to the exclusion of a clear understanding of the subject somewhat antithetical to the style of article that I thought WP was about: articles that make sense to those who have an interest in more than a narrow presentation of the topic.  So perhaps this is an article in which my interest will wane.  —Quondum 23:06, 22 September 2017 (UTC)


 * If, by "Hestenes words" you are referring to the cite I provided for first use and definition of "versor" this now replaces the following footnote that was in the article (misplaced in the section about reflections):
 * I am sure you will agree that having the actual event is an improvement over a random reference to a claim about the event. I quoted Hestenes/Sobczyk in order to show that the prior historical usage of the term "versor" by Hamilton was acknowledged as well as to show that the usage is in the way of a generalization (which I will come to in due course). As for the matter of the k-versor itself, I provided the cite since that is the original source for it and a further cite from Dorst for the avoidance of doubt. I can supply other sources if required, the terminology is commonly used and afaics per se unobjectionable. As usual you may provide alternative and/or contradictory citations.
 * Lastly, adding additional/relevant/properly cited material to an article is not a "narrow presentation", it is the contrary. Perhaps you might see how the section develops before any leap to judgement.
 * Selfstudier (talk) 09:49, 23 September 2017 (UTC)

The §Versor is promotional for and does not deserve to be here. Note that the section dares not make the link to versor! — Rgdboer (talk) 01:56, 16 February 2020 (UTC)


 * The term is well used in GA and multiple sourced. Alternative usage is acknowledged, not contested. (It has nothing to do with APS either, your 2009 comments on the versor talk page refer). Find a source supporting your statement that versor is promotional otherwise its just your personal opinion.Selfstudier (talk) 10:12, 16 February 2020 (UTC)

Inconsistency in Electromagnetism section
The definition of the electromagnetic field tensor is inconsistent with the formulation of Maxwell's laws. The source that gives $$F=(E+icB)\gamma_0$$ also gives Maxwell's laws as $$DF=J/c\varepsilon_0$$, not $$DF=\mu_0 J$$ as stated in the article. We should either change the definition to $$F=(E/c+iB)\gamma_0$$ or change our statement of Maxwell's laws. I recommend changing $$F$$ to be consistent with other pages. Ghartshaw (talk) 03:25, 25 September 2020 (UTC)

Add geographic applications
No mention to "geo" in the article. Easy to cite, see for example https://doi.org/10.1080/19475683.2019.1612945

See also the old (first issue) Map algebra and "modern map algebras" in GIS applications.

See also geo-objects and geo-fields conceptualized by GIS theory in https://doi.org/10.1080/13658810600965271  (also transformations from objects to fields and from fields to objects).

— Preceding unsigned comment added by 2804:431:C7C0:76DC:CFAF:85AE:A3B2:C1A6 (talk) 11:35, 21 February 2021 (UTC)
 * This is an article of mathematics (I have clarified this by editing the first line of the article). The concept referred to in your links is not clear, but it is certainly not related with the one of this article. D.Lazard (talk) 12:35, 21 February 2021 (UTC)