# Talk:Gyrovector space

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## A comment

Gyrovectors are equivalence classes that add according to the gyroparallelogram law, just as vectors are equivalence classes that add according to the parallelogram law (Abraham A. Ungar (2009), "A Gyrovector Spacce Approach to Hyperbolic Geometry", Morgan & Claypool Pub., ISBN 1598298224, 9781598298222). A gyroparallelogram is a hyperbolic quadrilateral the two gyrodiagonals of which intersect at their gyromidpoints, just as a parallelogram is a Euclidean quadrilateral the two diagonals of which intersect at their midpoints.Aungarab (talk) 19:29, 11 February 2009 (UTC)

## Inline comments from article

"may be not a true vector, but an Euclidean point: mixed states lie on interval, what hyperbolic geometry?"

Until the wikipedia article gets expanded, you'd have to read Ungar's papers.

"Algebra is a branch of mathematics, is it really what did you mean?"

The word algebra can also be used to mean a particular algebraic structure. For example see Universal algebra.

"Hungar is an alternate spelling? or, probably, incorrect translation? have such note a Russian title?"

Probably incorrect translation.

"what of Euclidean do you see here?"

Euclidean geometry is a special case of a gyrovector space. Gyrovectors can be used to unify the study of Euclidean and Hyperbolic geometry. Charvest (talk) 18:31, 31 August 2009 (UTC)

## Clarifying the axioms

1.) Is there any reason not to use the less ambiguous term 'magma' instead of 'groupoid' when describing the axioms? Based on usage, I'm assuming the term 'magma' is technically accurate, if someone can confirm.

2.) I had to read axiom #4 several times before I understood what it was saying. Maybe instead of "The map gyr[a, b]:G → G given by c → gyr[a, b]c", would it be clearer and still accurate to write "The function gyr[a, b]:G → G given by gyr[a, b](x) = gyr[a, b]x for all x in G..."? I do not want to make this change however if the notation as currently used is as or more common. I assume the goal was to avoid parentheses alongside brackets, but I'd rather err on the verbose side... Bmord (talk) 16:35, 8 September 2009 (UTC)

1) Both terms magma and groupoid are regularly used in the literature, however in the context of gyrogroups the term groupoid is the one I've seen used.
2) I agree that the notation can be confusing. I'm just following the notation used in the sources. Charvest (talk) 19:20, 8 September 2009 (UTC)

## Crackpot maths?

Somebody please convince me that this is not crackpot maths.

Seconded. The article sounds like it was written by Ungar himself. --97.115.195.95 (talk) 02:27, 12 October 2009 (UTC)

I have had no reply to my above request so I suggest that the article is proposed for deletion. Martin Hogbin (talk) 17:21, 12 October 2009 (UTC)

Martin, do you agree with these points:

1) for non-colinear velocities the velocity composition is not commutative i.e. u+v ≠ v+u
2) it isn't associative either i.e. (u+v)+w ≠ u+(v+w)
3) that one can ask how u+v and v+u are related i.e. find f(x) in u+v = f(v+u)
4) and ask how (u+v)+w and u+(v+w) are related
5) the magnitude of u+v is the same as the magnitude of v+u, so u+v and v+u are related by a rotation rather than a scaling, so f(x) is a rotation.
6) if |w| = |u+v| and gu,v(w) is the rotation that transforms w into u+v then finding an expression for gu,v(w) answers the question of what f(x) is in 3) above
7) Ungar found gu,v(w) was given by gyr[u,v]w = -(u+v)+(u+(v+w)) where Ungar uses the notation gyr[u,v]w rather than gu,v(w).
8) Ungar found that the same operator gyr that "repairs" commutativity also "repairs" associativity: i.e. left associativity is u+(v+w) = (u+v)+gyr[u,v]w and right associativity is (u+v)+w=u+(v+gyr[v,u]w)
9) The gyr operator is just Thomas rotation and if the velocities are colinear then gyr is the identity gyru,v(w)=w and u+v=gu,v(v+u) reduces to u+v=v+u so there is no Thomas precession for colinear velocity composition.
10) The notation gyr[u,v]w although potentially confusing at first can be clearer than gu,v(w) because the subscripts u,v can be the other way around as in the right associativity law above, and subscripts are not as easy to notice.
11) lack of associativity means velocity composition isn't a group (mathematics) but it is a quasigroup, and Ungar calls a quasigroup with gyroassociativity a gyrogroup.
12) vector spaces are based on commutative groups so velocities with relativistic addition which doesn't form a group, doesn't form a vector space, but Ungar calls the resulting structure a gyrovector space.

Charvest (talk) 23:33, 12 October 2009 (UTC)

This looks like OR to me, actually taking place on this page. This is an encyclopedia for established mathematics not a forum for establishing new ideas. If the concept of a gyrovector is generally considered useful there will be references to it in mathematical literature and text books. Can you provide references showing that this concept is in widespread use? Martin Hogbin (talk) 12:27, 13 October 2009 (UTC)
The only point in the above that is OR is 10). I felt like using a different notation, and point 10) is just a discussion about notation. As for being widespread, that depends on your definition of widespread - there are a fair number of books and papers about this. Charvest (talk) 16:39, 13 October 2009 (UTC)
The problem seems to me that all references seem to point back to one man who has come up with an idea that the rest of the mathematics community seems to find startlingly uninteresting. Can you prove this wrong? Martin Hogbin (talk) 16:59, 13 October 2009 (UTC)
Well, the study of quasigroups of which a gyrogroup is an example, is not exactly a hugely popular area of research. Charvest (talk) 17:10, 13 October 2009 (UTC)
Can we not get an expert at this stuff to have a look at the page? I still have my doubts but admit to not having the expertise to come to a real decision. Martin Hogbin (talk) 22:19, 13 October 2009 (UTC)
I have myself worked in this field but my approach is quite different from that of Ungar and in my opinion he is in error. over non-associativity. However I dont feel free to say why or refer to my own work because that would constitute research. Ungar's theory is not established mathematics; it only seems so because he has managed to publicize it so well - I feel somewhat smothered by all the talk of gyrovectors. Although it is interesting and important to discuss these matters I think we should obey the rules and stick to established mathematics. JFB80 (talk) 21:07, 11 October 2010 (UTC)
See #Nonassociativity and noncommutativity section below. 92.29.71.161 (talk) 14:18, 12 October 2010 (UTC)
Well all the discussion of relativity is completely sui-generis and of absolutely no current or historical interest, and I strongly suggest this mess be deleted. This is an author posting about his own interests and methods which, whatever small merit these ideas may possess, is completely against the spirit of the WP. Antimatter33 (talk) 06:20, 7 November 2010 (UTC)
There is no doubt that Ungar published his ideas but I can see no evidence that his work has been embraced by anyone in the wider maths community. As a minimum, the article should be reduced to a paragraph or two stating that Gyrovectors were a concept proposed by Ungar. If the concept is considered by experts to have no value at all, the article should be deleted. It is currently nothing more than a vehicle for private OR. Martin Hogbin (talk) 10:17, 7 November 2010 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── 89.241.230.184, it is not that I do not like this article it just that I do not want it to give the impression that gyrovectors are in widespread use in relativity or any other field. From what I can see it was an idea by Ungar that he consider might be useful but has not yet proven to be so. As far as I can see no one else has actually used this formalism in their own work. It is essentially a one-man subject. On the other hand, the quote below from a cited book review convinces me that it is not complete crackpot maths.

"As with most new formal methods, it has little to recommend its study in the way of new physical results or insights, but may be compared with the spacetime approach in terms of the elegance of respective proofs of selected theorems. One might also compare Ungar’s method to those of his immediate forerunners. On both counts, the gyroformalism proves to be worthy of physicists’ attention."

I suggest that we add to the article something to make clear to the reader who stumbles across that this is not in the same class as Minkowski spacetime for example. I also suggest that we remove links to this page from relativity articles. Ther is no evidence that gyrovectors have ever been used in that field. Martin Hogbin (talk) 13:49, 7 November 2010 (UTC)

I wouldn't say no evidence:
The book "Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics" 2001 edition, has a couple of pages about gyrogroups: pages 141–142.
The book "Special Relativity: Will it Survive the Next 100 Years?" has several pages about it.
The book "Special Relativity: An Introduction with 200 Problems and Solutions" refers to Ungar's work, as do various papers which can be found in Google scholar.
We could add your book review quote to the article to address the issue of not being widespread. Gvsip (talk) 17:13, 7 November 2010 (UTC)

## Useful search engine links

The following links were restored because they were "useful", but their usefulness to prospective editors should be limited to the page discussion, their usefulness to readers is clearly mandated by WP:ELNO #8. I have included them here for reference purposes (and maybe debate?):

Sincerely, Le Docteur (talk) 01:52, 16 October 2009 (UTC)

There exists a theory of hyperbolic relativity independent of Ungar's Gyrovector spaces and the two are confused in the above contributions. The basic fact on which the hyperbolic theory is based is that Einstein's addition law for vectors can be interpreted as an identity in hyperbolic trigonometry giving the 3rd side in terms of the other two (analogous to the cosine rule in spherical geometry) From this it can be deduced that relativistic velocities form a hyperbolic space (Lobachevsky space)with negative radius of curvature c. This idea was due to Robb, Varicak and Borel early 20th century It was not mentioned in the article. 62.103.225.96 (talk) 19:12, 13 July 2010 (UTC) JFB July 2010

The article does say: "Soon after special relativity was developed in 1905 it was realized that Einstein's velocity addition law could be interpreted in terms of hyperbolic geometry". Bethnim (talk) 13:38, 17 July 2010 (UTC)
If you know something about this subject, perhaps you could comment on my suggestion above that the article is deleted. Martin Hogbin (talk) 21:34, 13 July 2010 (UTC)

Since the later part of the entry 'gyrovector spaces' is so sketchy I think the entry should be reduced in size to the bare statement of axioms plus references - that is until you can find someone to write a decent account. I note there is a second entry 'gyrovector' which is almost vacuous. That can go. The first 5 entries coming up on searching wikipedia+gyrovector seem to be due to Ungar. My opinion is that the gyrovector idea is not sound - the basic assertion that this is a method of discussing vectors in hyperbolic space is never proved. 62.103.225.136 (talk) 07:43, 17 July 2010 (UTC) JFB 17 July 2010

In what way do you mean it is not sound? You've already admitted above that relativistic velocities form a hyperbolic space. And surely you don't dispute that the velocity addition is neither commutative nor associative. Bethnim (talk) 13:49, 17 July 2010 (UTC)

Certainly they form a hyperbolic space but do they form a gyrovector space? Are the two the same? Relativistic addition of two velocities is not commutative (as already observed by Sommerfeld 1909) but when we extend to 3 or more vectors the situation becomes unclear. Ungar extends the maths formally to 3 vectors and finds a contradiction from which he tries to escape by non-associativity theories. 62.103.225.195 (talk) 14:14, 18 July 2010 (UTC) JFB 18 July 2010.

I might add that there is virtually no published material on this subject except that by Ungar. Martin Hogbin (talk) 14:39, 18 July 2010 (UTC)
Yes you are right. Ungar himself has a very great output but other writers are involved only in a few joint papers of his. I would like to add a further comment in answer to Mr Bethnim. It is that there are two ways of adding vectors in hyperbolic space called by Varicak 'inner' and 'outer' addition. The inner one is the usual one in special relativity - effectively Einstein's addition law. It is this which Ungar modifies to his gyro addition. It is non-commutative (Sommerfeld: Phys Z. 1909) The outer one however seems to be commutative. So it is too soon to say addition is non-commutative (and non-associative) JFB80 (talk) 20:48, 6 October 2010 (UTC)
See #Nonassociativity and noncommutativity section below. 92.29.71.161 (talk) 14:18, 12 October 2010 (UTC)

## Nonassociativity and noncommutativity

The nonassociativity and noncommutativity of velocity addition in special relativity is not some invention of Ungar's. Here are some references which mention nonassociativity and noncommutativity:

• "Instead s possesses a non-commutative relationship (29) that should inform the unwary that things are very different, even for the rudimentary enunciations of the principle of relativity, from what he had imagined. A knowledgable reader might now consider that “addition” is, in generality, actually a non-commutative group operation. But this too is incorrect. As we shall see, (30), “addition” is, in fact, generally also non-associative." from Page 3 of The theory of relativity-Galileo's child by MJ Feigenbaum

92.29.71.161 (talk) 14:18, 12 October 2010 (UTC)

• Your references are from 1990 and 2008. The earliest paper of Ungar is I believe 1988 as follows

Ungar A.A: The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities. App. Math Lett 1(4) 1988 403-405. But the question apparently was discussed before that e.g. Mocanu C.I: Some difficulties within the framework of relativistic dynamics. Archiv für Elektrotechnik, 69 1986 97-110. Noncommutativity dates from the paper of Sommerfeld 1909 in Phys.Z and Silberstein's 'Theory of Relativity' 1914.JFB80 (talk) 19:05, 14 October 2010 (UTC)

In Einstein's original 1905 paper Zur Elektrodynamik bewegter Körper (On The Electrodynamics Of Moving Bodies) in Part 1, Section 5: "Additionstheorem der Geschwindigkeiten" ("The Composition of Velocities"), Einstein says: "Das Gesetz vom Parallelogramm der Geschwindigkeiten gilt also nach unserer Theorie nur in erster Annäherung." ("Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation."). The parallelogram law says you can add vectors by the head to tail method in either order A+B or B+A i.e. vector addition is commutative, and Einstein says in his first paper introducing relativity that addition of velocities does not follow the parallelogram law i.e. addition of velocities is not commutative in relativity. This quote of Einstein appears in Ungar's books. 89.241.239.0 (talk) 20:44, 14 October 2010 (UTC)
Yes I forgot that. But Einstein did not go into detail and did not mention rotation angle. Sommerfeld did also explaining it using his spherical representation.JFB80 (talk) 06:14, 16 October 2010 (UTC)

See Sommerfeld (1909): s:On the Composition of Velocities in the Theory of Relativity. Gr. --D.H (talk) 11:34, 16 October 2010 (UTC)

I've just had a look through Wikisource:WikiProject_Relativity/Wikisource:Relativity. Wow! D.H, that's great work you're doing. 2.97.28.188 (talk) 17:59, 16 October 2010 (UTC)
Further remark on non-commutativity. It is important to bear in mind that normally velocity addition means addition of relative velocities in a 'head to tail' way. So the basic rule is: velocity A to B + velocity B to C = velocity A to C. It it is put in this way it is immediately obvious that you need to know what you are talking about when you reverse the order of the addition. Silberstein (1914) is clear on this point. JFB80 (talk) 17:28, 23 December 2010 (UTC)

## is you is or is you ain't a operator?

(gyr isn't a single operator but depends on v and u, i.e. gyr = gyr[vu] )

This could be read as "gyr is whatever you need at the moment". But what operator doesn't depend on its operands? Should it be called a function? —Tamfang (talk) 17:58, 11 February 2011 (UTC)

Here u and v are not viewed as operands but as part of the definition of the operator. The gyr operator doesn't act on u and v but on a third vector w. gyr is shorthand for gyr[u,v]. gyr[u,v]w is a spatial rotation of the vector w with the rotation angles depending on u and v. Although it might look at first to be "whatever you need", it can be proved from the axioms of a gyrogroup that gyr[u,v] is specifically given by the identity $\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))$.
In the identity $\mathbf{u} \oplus \mathbf{v} = gyr(\mathbf{v} \oplus \mathbf{u})$, which is more properly written $\mathbf{u} \oplus \mathbf{v} = \mathrm{gyr}[\mathbf{u},\mathbf{v}](\mathbf{v} \oplus \mathbf{u})$, the operator $\mathrm{gyr}[\mathbf{u},\mathbf{v}]$ is acting on the expression $\mathbf{v} \oplus \mathbf{u}$.
This is similar to how a partial differential operator depends on which variables are being differentiated but the operator is acting on an expression e.g. $\frac{\partial^2 }{\partial u \, \partial v}$ acts on a function, say f=u^2+v^2. 2.97.17.122 (talk) 10:36, 12 February 2011 (UTC)
Thank you. I reckon I didn't read as closely as I might. —Tamfang (talk) 00:46, 13 February 2011 (UTC)

## Proposed deletion.

I am going to propose this article for deletion unless anyone can show that the subjects meets our notability criteria. Martin Hogbin (talk) 23:40, 20 October 2013 (UTC)

I support you in this. I was going to suggest the same, but was less sure and ended up making the recent edit to at least make the reader aware. It really looks like a novice's attempt to formalize a very complicated and incomplete path (read: lacking insight) to a simple and well-understood topic, namely Lorentz transforms. As a generalization (i.e. a mathematical structure not used in relativity), it would not have achieved notability unless it was shown that it can achieve something that groups cannot achieve. While the velocity-addition formula may appear in non-tensor approaches to special relativity (and contains much of this article already), to elevate this to a generalized formal axiomatized mathematical structure seems just silly. — Quondum 00:11, 21 October 2013 (UTC)
You seem to understand more of the maths than I do but I came to much the same conclusion as you. It seems to be the work of one individual, A. Ungar. Most of the cited references lead back to this one person.
One alternative to deletion might be to reword the article so as to reflect the true status of the subject as an idea of A Ungar which has never moved into general mathematical usage. At the moment the article is misleading in that it reads as though mathematicians and physicist regularly use this concept and find it useful or insightful in some way The article currently reads as a promotional article for somebody's bright idea.
If the supporters of this article, and there are a few, would agree to accept this approach I would accept not proposing it for deletion. Martin Hogbin (talk) 08:42, 21 October 2013 (UTC)
(From WT:MATH). I concur. I see no evidence that the axiom system is studied, but the concept of velocity addition in special relativity seems to deserve an article. — Arthur Rubin (talk) 08:38, 26 October 2013 (UTC)
We do already have Velocity-addition formula. Martin Hogbin (talk) 12:15, 26 October 2013 (UTC)

Having proposed deletion I am going to propose a possible reprieve as suggested above. My real objection to this article is not that it exists at all (Wikipedia is a big place and has space for subjects that would not be found in a written encyclopedias) but that it presents a gyrovectors as concept that has become part of mainstream mathematics and physics. In fact it would seem that Ungar was an academic who came up with a an idea that he thought would be useful to mathematicians and physicists but the concept was not taken up by any significant body of mathematicians or physicists and that it remains one of many bright ideas that never came to anything.

To delete the article, and thus to pretend that the gyrovector we never proposed and that Ungar never existed, does not do our readers a service. It may be an extreme backwater of mathematics and of only marginal notability but I think it probably does just meet the required standard for inclusion in WP.

The problem is that some editors may try to edit the article back to its current form, which suggests that gyrovectors are an important tool in maths or physics. Providing this is strongly resisted I see no reason why the article should not remain.

I propose heavily editing the article along the above lines to see how it looks. We can then decide if it should be deleted. Martin Hogbin (talk) 12:15, 26 October 2013 (UTC)

Most people disagree, but I find this topic interesting. Nevertheless, notability is more important...
Thanks, Martin Hogbin, for taking the time to rewrite it. I'll review it in more detail later. M∧Ŝc2ħεИτlk 14:21, 26 October 2013 (UTC)
I'm happy with Martin's argument that keeping an article as marginally notable doesn't hurt as long as it does not give undue weight to the topic, though the notability guideline lends little support for existence of the article in the longer term from what I can tell, but rather suggests that it may be discussed in Velocity-addition formula, or a similar merge. It is unclear to me that this is used in any other context (i.e. for something other than velocity "addition"); until we find that it is, I would eventually lean on the side of deletion/merging. Some references other than by Ungar to gyrovectors do seem to exist. (PS: Maschen – your first link is by Ungar.) — Quondum 16:27, 26 October 2013 (UTC)
Sorry, I forgot to add this to my watchlist, until now...
I know the first link is by A. Ungar but he collaborated with another person: Jing-Ling Chen. M∧Ŝc2ħεИτlk 15:43, 27 October 2013 (UTC)

## Started rewriting

I have rewritten the lead to give a better overview of the subject.

I think the body should start with an introduction rather that a set of axioms that seem to use undefined and non-standard terms. Has anyone got any ideas? Martin Hogbin (talk) 13:46, 26 October 2013 (UTC)

I tend to agree. This, probably being unfamiliar to laymen, physicists and mathematicians alike, would probably have to take an explanatory route, carefully introducing background motivation and interpretation along the way. Though I can't say that I'd have the energy; I have the strong feeling that this is merely a very convoluted way of representing a restriction on a Lie group. I'm sorely tempted to critique its use in relativity, but... — Quondum 17:30, 26 October 2013 (UTC)
I am not a mathematician but a physicist and do not claim to understand this subject although I can say that I gave never seen gyrovectors used in SR. They seem to offer no advantages of brevity, simplicity, or insight to me.
As a mathematical layman, the article does not explain to me what gyrovectors are. It seems to follow the pattern of bad teaching which asserts that if the reader does not understand the subject they should not be reading the article. Without some introduction explaining simply what gyrovectors are I think the current 'explanatory' text is unhelpful and until someone can add a simple introduction to the subject which makes the rest more comprehensible my inclination is to delete it all. Martin Hogbin (talk) 09:36, 27 October 2013 (UTC)
Something along the lines of Four-vector would be useful, if anyone is so inclined. Martin Hogbin (talk) 10:02, 27 October 2013 (UTC)
Do you mean to write an intro to gyrovectors similar to the lead in four-vector? Maybe I could try at some point, if I understood more about the topic... M∧Ŝc2ħεИτlk 15:43, 27 October 2013 (UTC)
This (capturing the gist in the lead) is actually a severe challenge, even thought of only in the context of SR, unless one naĭvely glosses over how one translates velocities between frames so that they can be gyroadded. Having explored the same math in my youth, I'm guessing that the scalar velocity components are transferred between frames (without any transform) using aligned respective orthogonal bases in the only way that works: aligned spatial axes, and '"forgetting" the time axis. To say "it is a generalization for 'adding' velocities in special relativity using a 'translation' of the second velocity to the reference frame of the first" is sidestepping the core and whole motivation of the exercise: to be able to use a particular velocity translation that takes quite some defining, before gyroaddition. The velocity addition article glosses over this too. — Quondum 17:32, 27 October 2013 (UTC)
Maschen, I was referring more to the body of the Four-vector, which actually explains, reasonably simply, what four-vectors are. Rather than the detail we have here I think a simple explanation of what gyrovectors are is all that is needed here, bearing in mind that there are very few (or maybe no) sources showing them actually being used. Martin Hogbin (talk) 18:14, 27 October 2013 (UTC)
Your feedback is interesting Martin Hogbin, since I rewrote most of the mathematics of the 4-vector article earlier this year (and added the thermodynamics examples last year). It still needs work BTW.
It would take me time to rewrite this article on a par with the 4-vector article, maybe next week or so, sorry... M∧Ŝc2ħεИτlk 20:22, 27 October 2013 (UTC)
The 4-vector article was only an example. My main point is that the article seems to consist of cut-and-paste sections from various papers with no simple explanation of what gyrovectors are. Bearing in mind that they do not seem to be used by anyone, I think all we need here is a brief and simple explanation of the subject. Martin Hogbin (talk) 23:36, 27 October 2013 (UTC)

### Heading level changes

As I do not feel qualified to make major changes to the mathematics and do not want to delete it all, I have lumped it all under one main heading and reduced the heading levels of the sub-sections to give it less prominence. I still think it should be simplified and reduced somewhat but I do not know how to do this. Martin Hogbin (talk) 12:52, 29 October 2013 (UTC)