Talk:Möbius transformation

Fixed point formulae are ambiguous

1) Since the argument of the sqrt operator is in general a complex number, the operator must be a complex number sqrt not a real sqrt.

2) Therefore, some mention of normalization needs to be made. That is, where in the complex plane is the branch cut being made. A usual place is along the negative real axis. In that case, the polar angle of the sqrt is normalized to +-90 degrees (+- pi/2 radians).

3) Once you have normalized the square root, the two values are the +-sqrt values. These are points that mirror each other across the origin. The quadratic equation has only two roots. With a precisely defined sqrt operator, the standard quadratic formula unambiguously specifies both roots.

4) A geometrically clearer, and therefore preferable form of the quadratic formula is formed when, given f(z)=a*z*z+b*z+c, let p=-b/(2*c) and q=c/a. Then the roots are given by p +- sqrt(p*p-q). The value p is an extremum of the function. The roots are symmetrical about p. In the case where p*p==q you have a double root. The other form is rubbish from the educational system and muddies the water. — Preceding unsigned comment added by 96.55.28.32 (talk) 23:05, 24 February 2017 (UTC)[reply]

What's written in the article seems clear enough to me. Concern over the doublevaluedness of the complex square root seems confused, since it is both roots we are considering simultaneously. There is no need whatsoever to introduce a branch cut. Sławomir Biały (talk) 23:32, 24 February 2017 (UTC)[reply]

Applet

Say, does that java applet work for everyone?

It works for me. Evil saltine 09:02, 17 Dec 2003 (UTC)

This comes in handy when doing computing, as terms with possible zeroes in the denominator can be multiplied out.

I cut that out.

One can get rid of the infinities by multiplying out by $z_{2}-z_{1}$ and $Z_{2}-Z_{1}$ as previously noted.

I left this in - but there's a problem with this and the previous comment, as stated. The implication that one can happily divide by zero is not good. In fact it is OK here, as can be seen by talking more systematically about homogeneous coordinates. This is implicit in the second comment, which is why I left it for the moment.

The article still needs work, to adapt the imported material.

Charles Matthews 10:27, 23 Dec 2003 (UTC)

I have updated the applet to demonstrate that the cross product is invariant under a transformation. Sweet. Should update the page too, to mention it.

Also: I understyand the inverse pole a little better now. Will update the text re the poles.

Higher dimensions

The Möbius transfomation is not just a two-dimesnional thing. In fact in higher dimensional Euclidean space the Möbius transformations, which are defined by stereographic projection rather than using complex numbers, are the only conformal mappings. Is this worth putting in? I have actually used these in an applied context, and have a paper with a man made of spheres and cylinders Möbius transformed to make the point [1].

What do folk think? Worth mentioning in the article or is it two long already? Or should be somewhere else (conformal transformations in differential geometry or something) Billlion 14:32, 6 Sep 2004 (UTC)

I'm not convinced that linear fractional transformations in more variables are normally called Möbius transfomations. They are certainly of interest. The point about the conformal group in higher dimensions is already mentioned (at conformal geometry?). It would be well worth expanding on that, somewhere.

Charles Matthews 14:41, 6 Sep 2004 (UTC)

There should be a discussion somewhere of the real Mobius transformation on the n-sphere, or equivalently, on the one-point compactification

R^{n}\cup {\infty }

. This is indeed called a Mobius transformation, see for instance Beardon, The Geometry of discrete groups. Gene Ward Smith 06:21, 21 April 2006 (UTC)[reply]

So, you could try amplifying conformal transformation. I've made a conformal geometry category.

Charles Matthews 14:49, 6 Sep 2004 (UTC)

I think this point is worth considering again. The Beardon text describes Mobius transformations on R^n and goes on to show that many of the results in C generalize. To start off the article saying Mobius transformations are transformations of C is misleading. 65.183.252.58 20:21, 18 March 2007 (UTC)[reply]

I think there needs to be a disambiguation notice at the top, explaining where to find the material on higher dimensions (there's some stuff at conformal geometry). I think Charles' point is that "Mobius transformation" will in many cases refer to linear fractional transformations of the Riemann sphere. I agree with that although I also think that a good many mathematicians think of a Mobius transformation as something more general. I was certainly surprised to see the general case is not even discussed here. Seems odd. --C S (talk) 13:30, 1 June 2008 (UTC)[reply]

In the meantime, two additions concerning higher dimensions have been made: [2] in the section on Lorentz transformations and [3] in the introduction. The introduction is now in contradiction with most of the article, which uses "Möbius transformation" to refer only to what the introduction now calls "Möbius transformation of the plane". I think this needs to be more systematically integrated into the article, but I don't know enough about how the term is usually used. If the comment above by C S is correct, the article should perhaps also mention that the term is sometimes used to refer only to the two-dimensional form and sometimes more generally. Joriki (talk) 08:55, 10 April 2009 (UTC)[reply]

more

I'm fiddling with this User:Pmurray_bigpond.com/Geometry_of_Complex_Numbers but it's not anywhere near ready yet. If it ever is, I might promote it to a topic "Geometry of Complex Numbers (book)".

Lorentz group

Hi all, very nice article, but you've been forgetting something important! Namely the connection with the Lorentz group To be precise, the proper isochronous Lorentz group SO⁺(1,3) is isomorphic as a Lie group to the Möbius group PSL(2,C). This is terribly important in physics, and adds interest to the mathematics of Möbius transformations, which are admittedly already sufficiently interesting to deserve a long article devoted to them.

I just added a bunch of really good citations and some discussion of the physical interpretation of three of the conjugacy classes. (Wanted: pictures of parabolics.)

Congragulations to whoever added the illustrations of individual Lorentz transformations--- beautiful! But I think animations of continuous flows would be even more vivid. Or at least figures showing the flow lines for typical elliptic, hyperbolic, loxodromic, parabolic transformations.

I like the fact that the article tries to explain the derivation of the classification of conjugacy classes, but right now the organization seems a bit try. I think this can literally come alive if someone takes the trouble to add animations.

Curiously, except in the comments I just added, the one-parameter subgroups are referred to as "continuous iteration". Since the rest of the article is written at a fairly high level of precision, I feel it is probably worth rewriting this language in terms of one-parameter subgroups.

I use the term "continuous iteration" because I am a computer programmer, not a mathematician. My original interest was in animating trees laid out in hyperbolic space in response to movements of the mouse. See the inxight website. 203.10.231.231 05:03, 10 August 2005 (UTC)[reply]

I'd like to bring out more clearly the fact that in this article we are classifying elements of the M group up to conjugacy, but this is essentially the same as classifying the one-parameter Lie subalgebras up to conjugacy. More generally, I'd like to add some discussion of Lie subalgebras to this article as well as to the Lorentz group article, including a nifty graph of the lattice of subalgebras. I'd like to say a bit about the interpretation of the coset spaces (homogeneous spaces) in terms of Kleinian geometry:

the coset space of the four dimensional subgroup of similitudes on the euclidean plane, which is the stabilizer of the origin in the Moebius action, and the stabilizer of a null line in the Lorentz action, corresponds to conformal geometry of the sphere,

the coset space of the three dimensional subgroup E(2), the isometry group of the euclidean plane, which is the stabilizer of a null vector in the Lorentz action, corresponds to the momentum space of massless particles (see Wigner's second little group), and is none other than the degenerate geometry of the light cone. (The stabilizer of a null vector is a subgroup of the stabilizer of a null line, and conformal geometry on the sphere can be regarded as a simplified model of the geometry of the light cone.)

the coset space of the three dimensional subgroup SO(3), which is the stabilizer of a timelike vector in the Lorentz action, corresponds to the momentum space of a massive particle, which is none other than hyperbolic space H³ (see Wigner's first little group).

In general, since these two articles (which were unlinked until today!) are discussing the same thing, it seems fair to put the more mathematical discussion in this article, and the physical discussion in the other article. But Penrose's interpretation of Möbius transformations is so vidid that I think it is justifiable to have a bit of overlap. I might add pictures of the flow lines of the one-parameter subgroups to the Lorentz group article, which I would draw using Maple. Hmmm... actually, I know how to produce the kind of animated pictures I was asking for above, but I'm not sure if I can massage them into a form which will run on the Wiki server. Any suggestions?

This article is already getting rather long, so I'll see if I can explain clearly but concisely the connection between the action on Minkowski spacetime and the action on the Riemann sphere in the other article. ---CH (talk) 2 July 2005 03:57 (UTC)

P. S. Why do I want to see pictures of parabolic transformations? These are very important in both math and physics. In physics, they are called "null rotations" and are needed for new articles related to Petrov classification of exact solutions of Einstein's field equations.---CH (talk) 2 July 2005 04:07 (UTC)CH

P. P. S. Another important and interesting theme is the way that the Moebius group arises as the point symmetry group of an ordinary differential equation. This also has close connections with Kleinian geometry. ---CH (talk) 2 July 2005 19:53 (UTC)

Yes well, seeing that the article is getting rather long, it might be more appropriate to think about how a general intro might be written, directing the user to other articles that review assorted Lorentz gp representation bits and pieces. Having been the last to make major edits here, my apologies on the lack of mention of SO(3,1); oddly enough, I have an entire book devoted to SL(2,C) (although not in the title): Moshe Carmeli Group Theory and General Relativity; rephrases most of GR using SL(2,C). Never liked it much, I admit, but maybe its time to re-read/skim it again. For what its worth, I've yet to see a book on modular forms or Riemann surfaces that even hints at "Lorentz" or physics anywhere in the book. There's oodles to be said on this topic; note also that the interplay of representations is the launching point for supersymmetry. What we really need to do is to think about how such an obviously broad topic can be split into bite-sized chunks. -- linas 3 July 2005 04:31 (UTC)

On reviewing your edits, it seems that one possible new article might be physical interpretation of Mobius transforms which reviews the the various physical, intuitive correspondances between the various forms and the physical realizations of corresponding transforms. Maybe the graphics should be moved there as well. I'm already finding the current article to be rather unbearably long. linas 3 July 2005 05:09 (UTC)

Naming Conventions

I've encountered some different terminology. Sadly, mathematical definitions aren't as standard as we'd all like, so I thought I'd put it out there to figure out if this is at all standard or if I'm crazy.

I've heard the term "fractional linear transformation" or "linear fractional transformation" for functions of the form $z\mapsto {\frac {az+b}{cz+d}}$ , and the term "Möbius transformation" for those of the form $z\mapsto {\frac {z-a}{1-{\overline {a}}z}}$ ; that is, those that, combined with rotations, constitute the fractional linear transformations mapping the unit disc to itself. This is the terminology used by Mathworld.

Also, is there another term which specifically refers to those of the form $z\mapsto {\frac {z-a}{1-{\overline {a}}z}}$ ? If so, an article should be written.

Usually, Möbius transformations means the same thing as a fractional linear transformation. Either refers to any automorphism of the Riemann sphere. I think the Mathworld article is wrong (or at least unconventional). Someone please correct me if this is not the case.

To the best of my knowledge we have no article on the automorphism group of the unit disc, which is the group PSU(1,1) given by transformations of the form

z\mapsto {\frac {az+b}{{\bar {b}}z+{\bar {a}}}},\qquad |a|^{2}-|b|^{2}=1

nor on the automorphism group of the upper half plane, which is the group PSL(2,R). Of course, these two groups are isomorphic and we should probably devote an article to them. I'm not sure what the best name would be. -- Fropuff 7 July 2005 05:12 (UTC)

Stellar movement

There is stuff in this article on how transformations with fixed points directly "ahead" and "behind" correspond to the affect of accellerations in space on where the stars appear to be. What about if the fixed points are not ahead and behind? Presumably that is the situation when you rotate about an axis that is not the same as the axis of "boost".

In general, what path will a point in space, with an initial velocity, appear to follow when an observer accelerates and spins around an axis? Stars are a special case, when the "points" are infinitely far away. What shapes will geometrical objects (planes, lines, spheres) be distorted into?

Given that straight lines remain straight lines under lorentz contraction, and that the klein and poincare models do the same thing at the border of the unit circle, I suppose that things would would be distored in a manner similar to the way the klein-model isometries distort things (only in 3d, not 2).

projective transformations

I found the projective transformations section to be a bit odd. Then I worked out what was going on: the original was written in simpler terms, and then someone had repeated the same thing in more formal terms. After puzzling out that π was referring to the mapping introduxed in the first sentence and that much of the formalism was simply saying the same thing again, I have reorganised the paragraph to integrate the two streams of thought a bit more closeley.

It needs reviewing by someone who understands math (g). It looks liek π is being defined twice, because we use "π:". In fact, π is defined by the words "let us call this mapping π", and the two expressions are not defining it but merely saying something about it. I'm not sure how you "say" this in <math> language.

edited class representative for parabolic

I edited the "class reporesentative" for parabolic transforms to be z+a. This is because the only "pure" parabolic transform (ie, k=0 without anything else) is the identity transform, but putting that in would not really help anyone.

Pretty pictures

Whew! I have bean meaning to do this for some time, and now its done. I have added some 3d pictures of transformations being stereographically projected. The "loxodromic/arbitrary" ones are inaccurate .... but I'm leaving it as-is for the time being because the params are consistent with those in the other pictures.

Parabolic transformations

This article is great! It answered some questions I needed answers to. One problem: it refers to "the subgroup of parabolic transformations". Contrary to what this suggests, the set of all parabolic transformations is not a subgroup! Parabolic transformations of the form

{\begin{pmatrix}1&a\\0&1\end{pmatrix}}

form a subgroup, which is indeed a Borel subgroup. Heck, I guess I'll go and fix this. John Baez 17:19, 7 March 2006 (UTC)[reply]

I am not sure I agree it is a Borel subgroup. You need to allow diagonal entries too (same problem in main article).85.1.13.169 (talk) 20:24, 5 February 2008 (UTC)[reply]

I agree that this isn't a Borel ... I changed this in the text ... hope you don't mind. —Preceding unsigned comment added by 86.25.180.89 (talk) 15:14, 25 May 2009 (UTC)[reply]

Fraktur

Is the use of fraktur for matrices standard in the literature? Since it is not elsewhere... Dysprosia 08:11, 15 May 2006 (UTC)[reply]

it's used in "The Geometry of Complex Numbers", Hans Schwerdtfeger. He uses them for modius transformations and for 2*2 Hermitian matrices that he uses to represent circles.

TeX is a Pain in the Ass

how to format this in TeX? (is wikipedia using TeX or LaTeX?

f1[Z]:= Z+D/C                  (translation)
f2[Z]:= 1/Z                    (inversion and reflection)
f3[Z]:= - (A D-B C)/C^2 * Z    (dilation and rotation)
f4[Z]:= Z+A/C                  (translation)

in parlticular, what i want is a block with 2 vertical alignments, and, the second column should not be in math format.

thx.

Xah Lee 22:44, 25 July 2006 (UTC)[reply]

I'd probably use wiki markup mixed with TeX. Something like this:

$f_{1}(z):=z+{\frac {d}{c}}$	(translation)
$f_{2}(z):={\frac {1}{z}}$	(inversion and reflection)
$f_{3}(z):=-{\frac {ad-bc}{c^{2}}}z$	(dilation and rotation)
$f_{4}(z):=z+{\frac {a}{c}}$	(translation)

Actually, MediaWiki software uses neither TeX nor LaTeX, but a strange approximation called texvc. There is an effort underway to switch to new software called blahtex, which supports more of LaTeX and can output MathML. For better or worse, TeX markup is the familiar standard within the mathematics and science community. Try to appreciate it; it could be worse! Consider the MathML markup for a quadratic formula:

   <math xmlns="http://www.w3.org/1999/xhtml">
     <mfrac>
       <mrow>
         <mrow> <mo>-</mo><mi>b</mi> </mrow>
         <mo>±</mo>
         <sqrt>
           <msup><mi>b</mi><mn>2</mn></msup>
           <mo>-</mo>
           <mrow> <mn>4</mn><mo>&it;</mo><mi>a</mi><mo>&it;</mo><mi>c</mi> </mrow>
         </sqrt>
       </mrow>
       <mrow> <mn>2</mn><mo>&it;</mo><mi>a</mi> </mrow>
     </mfrac>
   </math>

All this just to produce

{\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.

The TeX version is orders of magnitude more user-friendly (and brief!). --KSmrq^T 23:12, 31 July 2006 (UTC)[reply]

hi KSmrg, actually i have some personal beef with the entirety of TeX. I think, it has done huge damage to the math community. I did some rant that touched the gist of my views http://xahlee.org/Periodic_dosage_dir/t2/TeX_pestilence.html , which i hope to clean up and add proper format and references in the near future... In fact, despite the verbosity of MathML, i think it is far, far better alternative than TeX simply because that it started with the right basis. Xah Lee 02:29, 1 August 2006 (UTC)[reply]

Inverse funciton wrong!

the inversion function on the page is wrong!

the inverse of mobius transformation is actually: (d z - b)/(-c z + a).

The expression for the inverse function as given on the page is actually correct only if a d - b c = {1,0}

i've moved the wrong inverse expr from main page to below:

« with the following two special cases:

the point $z=-d/c$ is mapped to $\infty$
the point $z=\infty$ is mapped to $a/c$

We can have Möbius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity.

The condition ad − bc ≠ 0 ensures that the transformation is invertible. The inverse transformation is given by

z\mapsto {\frac {\left({\frac {d}{ad-bc}}\right)z-\left({\frac {b}{ad-bc}}\right)}{-cz+a}}

with the usual special cases understood. »

also, the statement about where the points -d/c is mapped to should be addressed somewhere else, as a detail about the mobius transformation. It is not “with these special cases”. Rather, if we want to state that there, it should be something like “MF has 2 points that are notworthy”. Also, we don't “need” to “augment the domain with a point at infinity”... any, my point is that this block of phrasing are terrible.

Xah Lee 08:35, 27 July 2006 (UTC)[reply]

Here is formula^{[dead link]}. --Adam majewski (talk) 09:41, 30 October 2021 (UTC)[reply]

Silly to try answer a 15 years old post that is blatantly wrong. D.Lazard (talk) 11:12, 30 October 2021 (UTC)[reply]

https://www.mathwords.com/i/inverse_of_a_matrix.htm --Adam majewski (talk) 18:16, 4 November 2021 (UTC)[reply]

critical problems in the article

i like to point out a few things in this article that i think needs fix.

• the angle-preservation property, which is a fundamental property of the MT, is never discussed in the article (other than saying it is a conformal map). It needs at least some discussion on how or why. (the proof of it can be deferred to the circle inversion page)

• the decomposition of MT into simpler affine transformations + circle inversion is not mentioned in the article. This is critically important, as it gives insight of what MT really is from geometric transformation point of view, and the circle inversion is most fundamental key to the whole MT business. This also needs to be mentioned.

• that the inverse function of MT given in the page is incorrect, as i've talked about before. I have given the correct formula in my last edit.

These issues, i tried to correct in my last sequence of edits (see here: http://en.wikipedia.org/w/index.php?title=M%C3%B6bius_transformation&diff=66701420&oldid=66351090 ) I'm not sure a full revert is necessarily a proper course of action, even if my formating and presentation style can be improved.

Some of the style and presentation as they currently are, in my opinion, suffers from jargonization symptom as in most texts, as well as from wikip's collective editing nature. But regardless, it is important to get the above critical math contents present and correct, however or whoever does it.

I hope people will correct these problems. Thanks.

Xah Lee 02:15, 1 August 2006 (UTC)[reply]

Unlike some people I don't think reverting major edits is a good way of accomplishing anything positive. As such I've reverted Oleg's revert. --MarSch 10:22, 19 September 2006 (UTC)[reply]

quaternions?

Just as the (proper) conformal transformations of S^2 are (P)SL(2, C) also known as Moebius transformations, the (proper) conformal transformations of S^4 are (P)SL(2, H), where H stands for the quaternions and not for the complex upper half-plane. The name "Moebius transformations" probably applies, but fractional linear transformations certainly also applies to the quaternionic variant. The article is very biased towards the complex case. --MarSch 10:46, 19 September 2006 (UTC)[reply]

Organisation

Who the heck ersaed the formula for computing the inverse? Luckilly, it's on this talk page. And I had to totally ferret around in order to find the formula for deriving the characteristic constant from the matrix terms - the whole "characteristic constant" section has been erased altogether, and all that's left is a mere mention in section 6.1. It's like someone with a different idea of what these transformations are all about has been erasing stuff.

Could we please lever things of interest to people like me, who need the equations so we can write computer programs? I appreciate that some people feel that the focus should be the beauty and inner meaning of the equations, but some of us want to calculate things. Maybe there's a way we can have both? I would, however, like to thank the eraser for not wiping section 10, despite its being of practical use.

Oh - and wikipedia for some reason has decided that some of my images didn't have copyright, even though I recall filling out that "yes, I generated these bits" pages for each of them. In any case, we need a set that also includes the parabolic cases. —The preceding unsigned comment was added by 124.176.37.159 (talk) 09:15, 17 December 2006 (UTC).[reply]

Further problems

There are two further problems I've noticed. Firstly, the division by zero/infinite issue isn't properly handled ( c=0 => f(infinity)=infinity, else f(infinity)=a/c, f(-d/c)=infinity). This doesn't contribute to a proper understanding of the nature of a function (some people may derive from this that, for example, 1/0=infinity). Also, the general decomposition does not work for c=0. c=0 => d=/=0 by def'n. Then let f1=(a/d)z, f2=z+(b/d), and composed f2of1. I'd change it myself but...I suck with TeX, and I might be missing the point Wrayal 19:26, 29 May 2007 (UTC)[reply]

Arnold, Rogness video link

isn't this a better link for the Douglas N. Arnold and Jonathan Rogness video in external links? The creators page about the video with both low res youtube version and high res download, in addition to some images. Images which by the way are currently CC'd, but perhaps could be dual licensed for the article, if someone wanted to contact them. but that's probably not necessary. ._-zro ^t_c 03:04, 8 August 2007 (UTC)[reply]

awkward phrasing

Hi,

"may be performed by performing" seems a bit awkward.Randomblue 18:47, 28 September 2007 (UTC)[reply]

when decomposing the Möbius transformation in four simple funtions, wouldn't it be nice to specify, for funtion 3, the scaling ratio and the rotation angle?Randomblue 18:51, 28 September 2007 (UTC)[reply]

automorphism of the Riemann Sphere

Is it clear to everyone else what is meant by automorphism of the Reimann sphere? Does it mean invertible complex-analytic? Perhaps that should be said, since the Reimann sphere is not a group and there are many real analytic (as well as only smooth, as well as only continuous) bijections on the sphere. MotherFunctor 21:18, 2 December 2007 (UTC)[reply]

Reference for Opening Statement

Does anybody know a reference for the second sentence in the article?

"Equivalently, a Möbius transformation may be performed by performing a stereographic projection from a plane to a sphere, rotating and moving that sphere to a new arbitrary location and orientation, and performing a stereographic projection back to the plane."

I can write down a proof, but it seems to be a folk theorem. The (American) complex analysts I've asked haven't been able to find a written reference. Is this characterization better known outside of the United States, perhaps? —Preceding unsigned comment added by 128.101.152.151 (talk) 17:59, 4 December 2007 (UTC)[reply]

I removed the sentence on the grounds that it appeared to be false based on my initial reading of it. However, if one correctly understands "moving that sphere to a new location", then it is true. This could be restored if it were made more precise. I'm not too worried about finding a reference for it, though. As soon as it is properly stated, I think it can be filed under "obvious". siℓℓy rabbit (talk) 14:49, 13 August 2008 (UTC)[reply]

So the article as it stands only defines Möbius transformations in the plane. For n>2 Möbius transformations are defined just as it says. Of course to include dilations we must allow a choice of stereographic projection. For a reference see Kulkarni R S and Pinkall U 1988 Conformal Geometry (Braunschweig: Vieweg). p 95. I used these maps in my paper, jou may like to look at picture of a man who has been Möbius transformed! http://www.iop.org/EJ/abstract/0266-5611/14/1/012/. It is a long time since I wrote that paper though so I am not about to fix the articel.Billlion (talk) 16:21, 19 March 2009 (UTC)[reply]

Your mistake was to ask complex analysts. You should ask geometers.Billlion (talk)

There is now a reference for this statement. See the article "Mobius transformations revealed", by Arnold and Rogness, published in the Notices of the AMS, volume 55, number 10. best, Sam nead (talk) 14:28, 2 March 2012 (UTC)[reply]

Parabolic fixed point

Why does it say in "Fixed Points" that "every Moebius transformation" has two fixed points? The transformation z->z+1, the archetypal parabolic transformation surely has only one fixed point. I think this is on the whole a great page, but I don't think it's very helpful to say this. Alunmw (talk) 17:58, 21 March 2008 (UTC)[reply]

It's explained in the next sentence. Fixed points are always solutions of quadratic equations, so there are always two of them provided one takes into account the appropriate notion of multiplicity. siℓℓy rabbit (talk) 14:40, 13 August 2008 (UTC)[reply]

Loxodromic/Parabolic

I changed the definition of Loxodromic transformation, according to the definition in G.Jones & D. Singerman "Complex Functions". I believe it is preferred to have four distinct and non-intersecting types of maps. Paxinum (talk) 09:35, 3 October 2008 (UTC)[reply]

Missing images

There are a lot of images that were deleted because they weren't tagged as being in the public domain. Oddly, it seems that only half of the images in each group were deleted, instead of all of them. Is it possible for someone to re-generate the deleted images? If they can't be re-generated directly, for consistency per group, could complete batches be generated? --ΨΦorg (talk) 05:45, 1 December 2008 (UTC)[reply]

Elliptic if and only if |λ| = 1? (Minor remark)

Under the header elliptic transforms, it is stated that "A transform is elliptic if and only if c"

Shouldn't that be "A transform is elliptic if and only if |λ| = 1, λ not equal to 1."?

If λ = 1, then the trace would be equal to 4, in contradiction to the classification. Nielius (talk) 09:25, 4 March 2012 (UTC)[reply]

Symmetry or Conjugation

Recently, a IP user has added a new subsection called "Symmetry Princple" (sic). The subject is relevant to the section in which it has been inserted. However, he use his own terminology, calling "symmetry" an involution that is usually called "conjugation" and is never called "symmetry" (in geometry, the symmetries are usually isometries or, at least linear, which is not the case here, except when the generalized circle is a line). Thus I have edited this section to use the correct terminology and make the link with harmonic division, which is fundamental here.

The IP user has reverted my edit for the reason that he prefer his own version. I have reverted again for the above reasons. D.Lazard (talk) 10:10, 18 February 2013 (UTC)[reply]

John B. Conway uses the term "Symmetry Principle" in Functions of One Complex Variable I (p. 51, 3.19). He also defines "symmetric" (p. 50, 3.17) --50.53.43.172 (talk) 04:48, 26 September 2014 (UTC)[reply]

Picture of Smith chart?

I made this image a while ago -->

The Smith chart is a famous depiction of a Möbius transformation -- it's a complex unit circle for a parameter called Γ, but the circle is filled with markings and labels so that for any Γ you can immediately read off the value of z=(1+Γ)/(1-Γ). It's used ubiquitously in high-frequency electrical engineering. Maybe this image has a place in this article somewhere? (I'm not sure where it would go, or whether it's too off-topic.)

Just a thought. :-D --Steve (talk) 14:59, 12 July 2013 (UTC)[reply]

This looks very interesting, and surely Smith charts should be described somewhere in Wikipedia.

But the subject is not really Möbius transformations — rather it is electricity. I think this topic belongs in some closely related article that is mainly about electricity, or possibly it deserves its own article (if it is adequately explained, not just illustrated).

I say this for the same reason that if some arithmetical calculation shows something about, say, farming, that would not belong in the article on arithmetic.Daqu (talk) 23:10, 15 August 2015 (UTC)[reply]

I actually added it to the article a while ago (after no one had objected for a long time). It's there now.

There is a certain Mobius transformation with an important practical application (electrical engineering); and people in this application area have a graphical method that they use all the time for calculating this Mobius transformation. That seems quite on-topic to me. (I was half-hearted earlier but I'm feeling more confident now that it belongs.)

What would be the farming analogy? I think the right analogy would be: Suppose there is a slightly-obscure arithmetical calculation (say, calculating quadratic residues), and let's say for some reason farmers needed to do this calculation all the time, and that the farmers have a special kind of table that they used exclusively for doing this calculation, and it was so important that every tractor has this table inscribed on the dashboard. If that were the case, I think it would be entirely appropriate for the quadratic residue article to say that it's used in farming and to show the table that the farmers use for doing the calculation. --Steve (talk) 01:17, 16 August 2015 (UTC)[reply]

Fair enough. My only reason for mentioning farming is that the application of arithmetic to another thing is something that is principally about the other thing, not about arithmetic.

To elaborate on my main point: The use of Möbius transformations in a Smith chart is not about Möbius transformations; it is about electricity. The article is for information about Möbius transformations. In this article, it would be appropriate to mention that Möbius transformations are used in Smith charts, but with a link to the article that contains the details about Smith charts.

Wouldn't it be better for Smith charts to have its own article? Then it would be entirely appropriate to add something about who Smith was, when they invented the charts, what problem this was devised to solve, what kinds of situations they are used in today (in the age of software), etc.Daqu (talk) 04:53, 16 August 2015 (UTC)[reply]

The article about Smith charts is Smith chart. "Mentioning" it here is exactly what I proposed and then did. You can check the article - it's one figure plus a two-sentence caption with a link. Nothing else. --Steve (talk) 14:06, 16 August 2015 (UTC)[reply]

Comment on Möbius transformation#Lorentz transformations

Isn't it more conventional to write for $(x_{0},x_{1},x_{2},x_{3})$ , associate the Hermitian matrix

X={\begin{bmatrix}x_{0}+x_{3}&x_{1}+ix_{2}\\x_{1}-ix_{2}&x_{0}-x_{3}\end{bmatrix}}

to be in better agreement with $\sigma ^{\mu }$ etc.? Freeboson {talk | contribs} 22:05, 30 July 2013 (UTC)[reply]

"Next, the Möbius group is connected, so any map is homotopic to the identity."

I just changed the phrasing of this claim. It's not true that any map is homotopic to the identity on a connected space; see the antipodal map on S^(2). Someone more knowledgeable than me should explain *why* the connectedness of this group shows that any map is homotopic to the identity (I don't think it follows from connectedness). DeathOfBalance (talk) 22:20, 18 November 2013 (UTC)[reply]

I think that because the sentence you quote as the title of this section is expressed poorly, you may have mistaken what that sentence is attempting to say. My guess is that they were only trying to say that each element of the Möbius group (which of course, as a linear fractional transformation, is a holomorphic map S² → S²) is in the same path-component as the identity, since there is only one path-component.Daqu (talk) 05:02, 16 August 2015 (UTC)[reply]

The quote and title of this section says that the group is connected (and says nothing about the space it acts on).

Since a conncted Lie group is pathwise connected, this does imply that all Möbius transformations are homotopic to the identity Möbius transformation. — Preceding unsigned comment added by 2601:200:C082:2EA0:189E:F594:5A2E:F75E (talk) 16:09, 20 September 2023 (UTC)[reply]

several redirects now point to a separate article called linear fractional transformation

There is a separate article called linear fractional transformation. The redirects similar to "linear fractional transformation" that had pointed to this article have all been changed to point to it. Here is a complete list:

--50.53.43.172 (talk) 02:13, 26 September 2014 (UTC)[reply]

Higher dimensions

I moved a lengthy digression on Möbius transformations in higher dimensions here, because the term Möbius transformation is also very widely used to refer to conformal maps of the n-sphere. It is not just in the context of Liouville's theorem. See, for example, the second volume of Marcel Berger's "Geometry", or R.W. Sharpe's "Differential geometry: Klein's generalization of Cartan's Erlangen program". Papers of Shigeo Sasaki and Kentaro Yano refer to the Möbius group in this higher dimensional setting. The term is firmly established in geometry, and so should at least be mentioned in this main article. The case has been made that attributing these to Möbius is not historical, and the article can also mention that if we have reliable sources on the matter. 12:01, 18 February 2015 (UTC)

I think you might have forgotten to sign completely and thoroughly here Sławomir

. YohanN7 (talk) 19:54, 6 March 2015 (UTC)[reply]

Why miscount the fixed points?

Since the parabolic transformations fix only one point, is it really necessary to almost completely ignore this (I did say "almost"), and repeatedly claim that every Möbius transformation fixes two points?

Sure, go ahead and mention that if the fixed points are counted with multiplicity, then the sum of the multiplicities is always 2. But that is not the number of fixed points for a parabolic Möbius transformation.

So please don't pretend that 1 = 2. Even three-year-old children know this is false.Daqu (talk) 23:03, 15 August 2015 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Möbius transformation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Could do with some orginisation of the page, currently seems to be rather bitty with no overal page low. As ever more history of the subject. Salix alba (talk) 10:33, 2 November 2006 (UTC)[reply]

Last edited at 17:59, 14 April 2007 (UTC). Substituted at 02:22, 5 May 2016 (UTC)

"also (...) named (...) linear fractional transformations"

The article says : "Möbius transformations are (...) also variously named homographies, homographic transformations, linear fractional transformations (...)"

It seems that a Möbius transformation is a particular type of linear fractional transformation. Indeed, there is another article with title Linear fractional transformation. Marvoir (talk) 10:26, 14 December 2016 (UTC)[reply]

False statement needs to be replaced

In mathematics, the number 2 means two. It does not mean "two when counted with multiplicity."

If you want to add the multiplicities of the fixed points together and claim that the sum is always equal to two — that would be true. But as stated, this is simply false:

"Topologically, the fact that (non-identity) Möbius transformations fix 2 points corresponds to the Euler characteristic of the sphere being 2:

\chi ({\hat {\mathbf {C} }})=2.

"

And No, the fact that the previous section redefines "fixed point" as "fixed point with multiplicity" does not justify this. It is a very bad idea to redefine words in an encyclopedia. But then to use the redefined word in another section??? That is a very, very bad idea.2600:1700:E1C0:F340:295B:3FA:6619:54BE (talk) 22:28, 9 November 2018 (UTC)[reply]

Continued fractions?

This article is listed in the category Category:Continued fractions; how are Möbius transformations related to continued fractions? I couldn't find any mention in the article about it. —Kri (talk) 21:15, 9 March 2021 (UTC)[reply]

I have no particular opinion on the categorization, but, to answer the question:

A continued fraction

a_{0}+{\cfrac {b_{0}}{a_{1}+{\cfrac {b_{1}}{a_{2}+{_{\ddots }}}}}}

can be viewed as a composition of Möbius transformations

\textstyle z\mapsto a_{k}+{\frac {b_{k}}{z}}

. There's some literature taking this view, especially in the context of the (somewhat tricky) topic of defining notions of convergence for infinite continued fractions. Among articles on Wikipedia, Generalized continued fraction is probably the one that comes the closest. Adumbrativus (talk) 05:38, 9 March 2021 (UTC)[reply]

That is true; I hadn't seen that connection before. Thanks for the explanation. —Kri (talk) 16:06, 5 October 2021 (UTC)[reply]

Clarity about the scope of the name "Möbius transformation"

The current state of this Wikipedia page does not (in my non-expert experience) accurately reflect the usage of the term Möbius transformation in academic literature. I have seen this term regularly used for linear fractional transformation over hyperbolic numbers and dual numbers (e.g. by Kisil (2012) Geometry of Möbius Transformations) or multivectors in Clifford algebras or conformal transformations of 3+ dimensional Euclidean or pseudo-Euclidean spaces (e.g. by Ahlfors (1986) "Clifford Numbers and Möbius Transformations in Rⁿ"), and so on.

Emphatically stating that "Möbius transformations" only apply to the complex plane seems at least somewhat misleading to readers. It's fine to have an article focused on that narrower topic, but someone should try to do a bit of literature survey and make sure that the text accurately reflects prevailing usage. (The text might say e.g. "the term Möbius transformation is also regularly applied to more general settings, but this article will focus on transformations of the complex plane.") Ping @Quondum, who just made a change here about the relationship between Möbius transformations and linear fractional transformations. –jacobolus (t) 19:48, 20 May 2023 (UTC)[reply]

I tend to agree with you. My change was made because the emphatic definition followed by the indication that the terms were equivalent implied a restriction of all the terms to the narrow definition was clearly nonsense. I expect that terminology is not consistent in the literature, i.e. that various authors use the term differently. Such terminology variation is pervasive in mathematics. It makes sense to settle on a primary terminology in WP (which should be researched; some WP conventions exist). I suggest that we "sidestep" the issue here until someone adequately surveys the literature for prevailing usage by introducing the article by saying that only the Möbius transformations over ~~R and~~ C are considered here, but that various generalizations exit with names such as [... here we can list the various names]. —Quondum 15:24, 21 May 2023 (UTC)[reply]

Looking at this a bit more, a linear fractional transformation is a transformation of the form x ↦ (ax + b)(cx + d)⁻¹, where the variables are all elements of any chosen ring. That some authors have extended chosen to extend the term "Möbius transformation" to the rings other than the complex numbers is understandable, but examples are not evidence of a norm. The text at Homography § Homographies of a projective line suggests that the primary use of "Möbius transformation" applies only to the complex case, that any use of the term as generalized to any other ring may be sporadic, and that "linear fractional transformation" is the preferred term in the more general case. Linear fractional transformation confirms this: "In the most basic setting, a, b, c, d, and z are complex numbers (in which case the transformation is also called a Möbius transformation)".

The extension to a higher number of dimensions (in the sense of a projective transformation) seems somewhat dubious, and I'm not sure Ahlfors even meant it as transformations of a higher-dimensional space: from the introduction (which is all that I can see), his reference to 2×2 matrices of Clifford numbers makes me assume he is referring to fractional linear transformations of Clifford numbers, not of the vector space over which the Clifford algebra is defined. Note also that a Möbius transformation is a map onto the input space, whereas a homography is an isomorphism rather than an automorphism of projective spaces, which is strictly more general (thus, no-one would claim that "Möbius transformation" is synonymous with "homography" unless they meant it in the very general sense of a "conformal mapping", and the restriction of "Möbius transformation" to complex numbers is affirmed there too).

In all, I am now of the view that WP should present a Möbius transformation strictly as a linear fractional transformation of the complex numbers. That is, we should delete the qualification "of the complex plane" in the first sentence of the lead. —Quondum 01:23, 22 May 2023 (UTC)[reply]

No it is not (that I have seen) ever the same as a general "conformal transformation" if you include the 1–2 dimensional cases, where conformal transformations have significantly more flexibility. Möbius transformations in this extended use would always refer only to conformal projections (automorphisms) of the full space which map all circles to circles (where "circle" includes straight lines) and preserve cross ratios, of the type that can be obtained by sphere inversions (including reflections). Cf. inversive geometry. In 3+ dimensions this is the only kind of conformal transformation you can have, but in the plane there are infinitely many other kinds of conformal transformations.

Another (perhaps even more common) use of Möbius transformations is to refer to what we might call "circular transformations" of the Euclidean plane, without necessarily considering the transformed points to be complex numbers per se.

Many sources also include transformations which are orientation reversing among "Möbius transformations".

You can't treat Wikipedia text (of any of these articles) as reliable here. It was written by some random person who never bothered to do any kind of survey, and was most likely also not an authority on these subjects. It's also tricky to take any other individual sources as authorities, as they very often are focused on one particular thing and will give it a name, even while other authors use the same name for something else.

What do you think Ahlfors might mean by a "Möbius transformation of

\mathbb {R} ^{n}

"? I can tell you: he means conformal circle-preserving transformations in n-dimensional Euclidean space. All of which can be represented by transformations of the form

x\mapsto (ax+b)(cx+d)^{-1}

if you take the domain to be vector-valued and the coefficients to be multivector-valued ("Clifford numbers"). Ahlfors himself wrote a few papers about this topic, one or more of which you might be able to look at: https://scholar.google.com/scholar?q=ahlfors+Möbius+Transformations Ahlfors credits the idea to Vahlen (1902) https://eudml.org/doc/158052 (who calls it a "Vector-Transformation", in German which I do not read). But we can also extend the idea to transformations of some other kinds of multivectors if you like, and this has been done in other people's papers, e.g. you can find papers about "Möbius transformations" of Quaternions. –jacobolus (t) 04:59, 22 May 2023 (UTC)[reply]

Looking at the papers citing Ahlfors might also give you somewhere to start. But here's another example https://akjournals.com/view/journals/10998/14/1/article-p93.xml –jacobolus (t) 05:28, 22 May 2023 (UTC)[reply]

You don't need to convince me that the term Möbius transformation gets used by some with various (not necessarily compatible) broader meanings. This is likely true for most mathematical terms, but this is not a reason to include every meaning that has ever been used for each. The point is to determine the dominant modern interpretation of the term as what we should adopt for WP, and examples of individual scholars' use is of no value for this. I understand the risks of use of other articles to draw conclusions; however, if we were to make a change in one article without updating all the other articles accordingly, such a change would produce a confusing mess of broken cross-referencing in WP. It is far less problematic to consistently settle on a particular meaning of a term in WP than to use it differently in different articles (unless there is detailed disambiguation in each article, which is itself not worth the price). Accordingly, we need to be very sure about the dominance of a term before changing it everywhere or anywhere. In this article's case, the change would not be to change the article content, but to change the title to "Möbius transformation of the complex numbers", "Complex Möbius transformation", or similar. An example: the term "ring" originated to mean what is rng in WP, despite extensive historical and even modern literature that calls the latter "ring". However, WP has settled on "ring" having an identity element. Since this was settled, working on articles using these terms has become far less frustrating and time-consuming, and reading them is far less confusing. Similar approaches have been taken with respect to adoption of a convention for defining divisor, zero divisor, zero ring. I suggest that at this point, we should solicit the opinion of a fairly broad group of editors who might have familiarity with the field (at WT:MATH?).

To repeat my point: primarily we need to determine a definition for use in WP, and secondarily this should be the predominant prevailing meaning. Ahlfors is merely one (possibly dated) data point of the statistics that are to be gauged. For example, if the predominant term for a broader class is "linear fractional transformation" or "projective transformation", then a simple comment in the related articles to the effect that "the term Möbius transformation is used by some authors to refer to this concept". —Quondum 12:44, 22 May 2023 (UTC)[reply]

I don't think the title needs to change. I just think the text should make sure to be a bit less emphatic that this is "the" meaning of the term, since readers commonly take such claims in Wikipedia as gospel. To take your ring example, see Ring (mathematics) § Notes on the definition. –jacobolus (t) 13:10, 22 May 2023 (UTC)[reply]

I disagree: often it is problematic to attempt to force consistent use of terminology across Wikipedia articles. This is especially true in mathematics, where often the terminologies used by mathematicians, physicists and engineers all disagree, yet we need to write articles that work for everyone, or sometimes we need to write separate articles for different audiences.

In the case of the present article, I think we should have an indication of the existence of the "Higher dimensions" article in the lead section. It is reasonably common to hear people say "the only conformal transformations in dimension 3 or higher are Möbius transformations" and these are not sufficiently distinct objects to merit being discussed in a separate article. —Kusma (talk) 13:13, 22 May 2023 (UTC)[reply]

The picture I'm getting is that there are two uses that need to be highlighted, and this article should point out that it is about the transformations of the complex numbers; a hatnote might also point out where other uses of the term might be found.

We don't have to "force" consistent terminology, but we should strive to make links coherent with concepts, not blind to differences of the intended meaning. Anything that links to this article should mean to reach this content. Anything that means the "higher dimensional Möbius transformations" of a space should link to a different but suitable destination.

jacobolus, I'm not sure what you would like to change to make it less emphatic (it already qualifies it as "a Möbius transformation of the complex plane"), aside from perhaps adding another hatnote. I don't have objections to making it seem less emphatic; do you have suggestions? —Quondum 15:15, 22 May 2023 (UTC)[reply]

Möbius transformations in 3+ dimensions aren't different enough from those in 2D to need their own article, unlike the Möbius transform. —Kusma (talk) 16:18, 22 May 2023 (UTC)[reply]

I think we should add another sentence or two at the end of the lead about how the same kinds of transformations of the Euclidean plane (not complex numbers per se) are also called Möbius transformations – I believe these are indeed what Möbius originally studied, though I am not an expert – and that the same idea generalizes naturally to higher dimensions. We already have a section "Higher dimensions" discussing this, though it could probably be expanded. It also may be worth discussing the orientation-reversing transformations at greater length in the article. –jacobolus (t) 16:34, 22 May 2023 (UTC)[reply]

I've added a mention of generalization in the lead; if you want to make it clearer that the transformations of any Möbius geometry can also be called "Möbius transformations", feel free.

Presumably Möbius studied the Möbius plane; I have no idea whether he extended it. As a geometry it has naturally been extended to other fields and higher dimensions, though Möbius geometry seems complex. The 'higher dimension' section assumes a positive-definite metric, which probably does not seem to do justice to the topic, even as a mention. —Quondum 18:15, 22 May 2023 (UTC)[reply]

Sort of... but article Möbius plane is about incidence geometry rather than anything Möbius did directly. –jacobolus (t) 20:39, 22 May 2023 (UTC)[reply]

That just goes to show what I (don't) know ... it is ages since I looked at this stuff, and then even then it was fleeting. Please tweak the article to your satisfaction. —Quondum 21:35, 22 May 2023 (UTC)[reply]

I think I would actually change the basic definition to say that Möbius transformations are the such-and-such transformations of the Euclidean plane [leaving unstated here because I'm not sure what the best phrasing for a definition is], and that if we identify points in the plane with complex numbers, these can be represented as fractional linear transformations of the form

{\textstyle z\mapsto {\frac {az+b}{cz+d}},}

etc.

That is, I think it's better to initially think of "Möbius transformation" as meaning a transformation of the plane rather than thinking of it as a complex function. In just the same way we'd define "translation" as or "reflection" as a geometrical concept rather than starting out with how they can be represented as complex functions. –jacobolus (t) 19:36, 23 May 2023 (UTC)[reply]

While I have no in-principle objection to the article with this title describing the said transformations of the one-point compactification of the Euclidean plane, and agree that this is likely more correct. Yet, changing to this geometric perspective would seem to imply rewriting most of this article, which is written from the perspective of transformations of the complex projective line, which is simpler that a pure geometric approach in many ways, so I see its attraction. One can do as you say, in the the sense of defining it geometrically, but then switching to the complex projective line for the bulk of the article. This is less of a rewrite than the pure geometric approach, but does suggest keeping the mindset that this represents a real geometry, and relating back to the geometric interpretation everywhere. Editing the article to match this mindset would take some work, and I'm not signing up for this. —Quondum 20:28, 23 May 2023 (UTC)[reply]

I think a more significant rework of this page is worth doing for the benefit of (especially non-expert) readers, but I'm not going to tackle that right away either. I'd like to get to it eventually after a big pile of other related projects on other mathematical pages.

I also think we should split Riemann sphere from complex projective line (with the former focused on interpretations that are directly spherical, and the latter also including other interpretations). –jacobolus (t) 23:10, 23 May 2023 (UTC)[reply]

That makes sense. As I understand it, the Riemann sphere is a graphical representation of the complex projective line, but this does not imply that they are synonymous, whereas the article sort of mixes them. —Quondum 23:49, 23 May 2023 (UTC)[reply]

Ads/CFT connection

The bottom of this article has a mention of the connection between MTs and hyperbolic space. This is agreed-upon mathematics that you can find with some googling. A citation should probably be added but I don't have time.

It also had an enigmatic sentence after this description, which was "This is the first observation leading to the AdS/CFT correspondence in physics". It had no citation, so I started googling.

I could find nothing. I note that de sitter and AdS are usually associated with a metric like (2,3) (see Dirac) or (2,4) (see Lasenby), whereas 3D hyperbolic space has the metric (1,3). Plausibly there is an interesting connection, but obviously this would need to be demonstrated explicitly in a peer-reviewed paper before being cited on wikipedia. And for sure if studies hyperbolic space "leads to" AdS/CFT in any sense, there should be a citation of an introductory textbook.

All this points to it appears to be someone's original research, so I have removed it. Hamishtodd1 (talk) 11:17, 8 March 2024 (UTC)[reply]

Thanks. A claim like this should definitely have a source, especially since it's not precisely clear what the claim is saying.

My understanding is that de Sitter spacetime is a Lorentzian manifold of uniform sectional curvature with the same metric signature (1, 3) or (3, 1) as the flat Minkowski spacetime, and likewise for anti de Sitter spacetime, but either can be embedded as a hypersurface in some flat pseudo-Euclidean space of higher dimension in the same way a curved 2-sphere with intrinsic signature (2, 0) can be embedded in 3-dimensional Euclidean space of signature (3, 0).

I don't know anything about AdS/CFT correspondence, but it is true that the conformal transformations of hyperbolic 3-space can be related to conformal transformations of the 2-sphere, which can be thought of as the boundary of 'ideal points'. I can't tell you whether AdS/CFT correspondence is a direct higher-dimensional analog of this. –jacobolus (t) 18:43, 8 March 2024 (UTC)[reply]

Our article De Sitter space is not great, and Anti-de Sitter space could also be better. I'd also like to see articles about the 2-dimensional analogs, which could e.g. be titled De Sitter plane and Anti-de Sitter plane. There has been enough written about these that we could definitely put together an article about it, and I think it would be quite helpful to readers trying to understand the 4-dimensional examples. –jacobolus (t) 18:53, 8 March 2024 (UTC)[reply]