What if we define 1/0 = ∞? | Möbius transformations visualized
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- Опубліковано 1 лип 2024
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Defining 1/0 = ∞ isn't actually that bad, and actually the natural definition if you are on the Riemann sphere - ∞ is just an ordinary point on the sphere! Here is the exposition on Möbius maps, which will explain why 1/0 = ∞ isn't actually something crazy. And this video will also briefly mention the applications of the Möbius map. As is the case for all videos in the series, this is from Tristan Needham's book "Visual Complex Analysis".
There will also be things like circular and spherical inversion, which are really neat tools in Euclidean geometry to help us establish lots of interesting results, this one included.
This video was sponsored by Brilliant.
Video chapters:
00:00 Intro
02:38 Chapter 1: The 2D perspective
08:43 Chapter 2: More about inversion
14:33 Chapter 3: The 3D perspective (1/z)
19:38 Chapter 4: The 3D perspective (general)
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SOURCES:
[That 2012 paper] Rigid motion 1-1 Möbius map: scholar.rose-hulman.edu/cgi/v...
Möbius transformations revealed: • Möbius Transformations...
Accompanying paper: www-users.cse.umn.edu/~arnold...
Unitary iff rotation: users.math.msu.edu/users/shap...
Möbius iff sphere: home.iitm.ac.in/jaikrishnan/M...
Rotation of Riemann sphere: people.reed.edu/~jerry/311/ro...
Circle-preserving implies Möbius: onlinelibrary.wiley.com/doi/e...
Problem of Apollonius video: • Problem of Apollonius ...
Power of a point: www.nagwa.com/en/explainers/7...
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MORE CONNECTIONS OF MÖBIUS MAPS:
Sir Roger Penrose lecture on the book with Rindler (Spinors and space-time): • Sir Roger Penrose on c...
The book: www.cambridge.org/core/books/...
Hyperbolic geometry: assets.cambridge.org/97811071...
Conformal mapping (fluid mechanics): math.berkeley.edu/~iliopoum/T...
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Music used:
Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River
Asher Fulero - The Closing of Summer
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I know that this title is quite clickbait, but hopefully this is a good enough exposition on Möbius maps. There are lots of links in the description to help you learn more as well, because I have missed something there, like group properties, or its relationship with matrices (which give rise to the PSL(2,C) I mentioned in the end) [Actually I thought about “generalising" the idea of 1/0 = ∞ to rational functions and more general still, meromorphic functions, but this video is long enough.] Again, this is basically an animation of the book Visual complex analysis by Tristan Needham, so that the ideas can come alive. Hopefully you will like this video!
1/0 can be -infinity
or -infinity =+infinity
like -0=+0
1/-0=1/+0
@@user-bt1hf9cr5n In some sense, yes! The infinity here corresponds to infinity in all directions, so it can be approached via the positive real axis, in which case, it would intuitively be called +infinity, or the negative real axis, or -infinity; or the imaginary axis with positive imaginary part, so i*infinity and so on...
thanks for solving my problem.
For any electrical engineer wondering: the Smith chart is indeed a particular case of Möbius transformation, of course with the physical context (impedance, reflection coefficient, etc.) that makes it actually useful in the design of RF circuits.
I do not know that!
@@mathemaniac The Smith Chart maps the half-plane Re(z) ≥ 0 to a circle.
I've used them for years, but just recently learned that the Smith Chart is one example of a Möbius Transformation.
Whats more useful is that we can take reciprocals just as easily on the chart. Admittances are much easier to work with when u wanna do multiple stub matching
@@douglasstrother6584 Strictly speaking it also maps the negative resistance, or Re(z)1. MWO has this implemented for example.
Also, this means that you can think about the impedance matching as spinning and translating the Riemann sphere such that the load impedance is on the lowest point :D
@@Danilo_Djokic True. Where would we be without oscillators & amplifiers? It would be pretty quiet!
It's unfortunate that Mathematicians don't mention the Smith Chart as an Electrical Engineering example of a Möbius Transformation, and that Electrical Engineer Professors don't mention that the Smith Chart is a useful application of a broader class of transformations in the complex plane.
Completely agree with the math education rant! I wish that balance of big-picture ideas and small details could be more common in lectures and textbooks
Yes exactly!
I wish....
But my professional expectation is that its going exactly in the opposite direction...
I think so too, that's what makes math fun! I mean... it's not wonder that most people absolutely hate math when all it's ever been to you is rote memorization and calculation, that IS pretty boring! But math can be super exciting and fascinating, and perhaps even FUN!
I think part of the problem is that on some level, the beginning kind of _has_ to be boring and simple. You need a solid foundation to be able to get to the interesting bits (or to even understand why they're interesting at all!). I don't know the best way to handle that, but I know that what we're doing definitely isn't it.
I genuinely _love_ math. I think it's one of the most fascinating things to learn about, and I bet you do too. I've have always loved math, and I owe that to a string of truly wonderful passionate math educators, but most people aren't that lucky. Which is exactly why this (and many others on YT) are vital in helping people to see what mathematics can be with a good teacher.
I really appreciate what they're doing here because I die a little bit inside every time somebody says they're scared of math. No! You shouldn't be afraid of math! Math should be afraid of YOU because you're going to solve the shit out of it!
where does he say that I can't find
sorry but this visual intuition is completely useless. It doesn't help a bit
That moment when Mathemaniac möbbed every complex function in the room with infinities was truly an achievement of all times. Mörbios is definitely one of the transformations ever invented.
I laughed when Möbius said "It's Möbin time" and möbed all over the real number line.
Great video as always!
7:46 If c = 0, then neither a or d can be 0 since ad-bc is not 0. Then f(z) = (a/d) z + (b/d) is some combination of Enlargement/Shrinking, Rotation and Translation.
8:10 Yes we need all 5. But I don't know how to prove it.
11:24 At the center, the tangent lines to the image circles are parallel to the original lines respectively. Thus the angle formed by image circles remains the same.
13:30 Let a point P be d away from a circle C of radius r(relatively much larger than d) with center O, i.e. |OP| = r + d (the case of r - d is similar).
The image of P after the inversion is P'. Then |OP'| = r^2 / (r + d) = r - d + d^2 / (r + d) and PP' is orthogonal to the circle.
As r approaches infinity, the circle becomes a line L. |OP'| approaches d. Thus P and P' are both d away from L while on different sides of L. Also PP' is orthogonal to L. Thus P' is the reflection of P about L.
Thanks for attempting the exercises! For 8:10, we actually don't need all 5, and there is freedom to choose which to give up!
At the last part, when you say |OP'| = r - d + d^2/(r + d), that's already enough to conclude reflection as a limit: as r goes up, the fraction vanishes!
also, for the first question, there is actually no rotation necessarily. as f(z) = (a/d) z + (b/d), this is a scaling (what he calls Enlargement/Shrinking) by a factor of a/d followed by a translation by b/d, no rotation
@@mathemaniac I'd say that the 5 transformations are a bit loosely defined though. can a rotate about any point? what about a point at infinity? lots of stuff like that. if all these transformations can only be done at a fixed point, then you need all 5 of them.
I thought this said Morbius transformations :(
This video was pretty dense, but in a good way (so will def need to watch it more than once to unpack everything). But WOW. This video was amazing, and my mind was particularly blown when you explained a stereographic projection in terms of a 3D inversion map. Thank you for this wonderful content!!
Thanks!
I started learning complex analysis when you first announced this series in the 1st video and I have completed learning a month ago.
was it good
Sorry that each video just takes a very long time to make haha
Congratulations. Are you currently working toward completion of a degree?
You completed... learning? I'm sorry, but my processor does not comprehend that sequence of words... would you mind explaining?
Beep boop.
@@idontwantahandlethough I think he means copleted a subject or a course. Suspect a loss of detail in translation. To quote Tigger: "You have to read betwixt the lines."
This hardly relates to the video, but as a layman with limited math experience, and a programmer, I personally have found tons upon tons of cases where defining division by zero as being equal to zero has been extremely useful to me, even if it is ultimately invalid by some definitions.
It has been incredibly useful in game dev to treat division by zero as equal to zero, for lots and lots of vector math, trig, etc. It's been so useful, I've often even gone so far as to create code that'll handle division by zero for vectors and numbers this way, as well as certain trig outputs, and it often produces consistent and accurate results.
When you don't define division by zero you admit that a literally infinite number of operations don't map to *any* value in the set of real numbers. I don't like that at all, it doesn't make much sense to me. Contrary to imaginary numbers where you get a 1:1 mapping for your inputs to outputs, in this case you get a mapping of your inputs to a single output. However, I would argue that for 0 this is actually valid, because if you multiply anything by zero.
This isn't really proof or even evidence but a fun piece of intuition, you can make the statement that the average of the sum of the values of 1/x and 1/-x will always be zero where it is defined so we can extrapolate that it should also be zero where the divisor is zero. 1/0 and 1/-0 are the same (-0 = 0) and the only single number a which sums to 0 like this is 0. This is nice because it means scale or proportion is irrelevant. This is as far as I know pretty consistent and works correctly in lots of formulas and equations, and in my opinion, it's a better definition than undefined because of that.
where else is it found useful?
Can you make a video about conformal mapping? I'm struggling with it and I find your videos really enlightening!
Thanks for the compliment! There might be a mention of it for the next video about complex differentiation.
Absolutely blew my mind when you mentioned stereographic projection
Glad to hear that!
From what I can tell, of the 5 types of transformation in a Möbius map, if we allow arbitrary axes or circles, then translations and rotations are just compositions of reflections (translation if the reflections are parallel and rotation otherwise). Meanwhile, scaling can be achieved by inverting across two concentric circles. _But wait,_ the video covered that reflections are really just inversions across circles passing through infinity. After all that, Möbius maps should all be capable of being boiled down to a series of inversions across multiple different circles.
This is awesome! Thanks for making it! The only suggestion I have us at 10:40 or so. The green and blue lines are a little hard to distinguish on some screens. It might be wise to use two colors that are further apart next time.
Yes, noted. I usually do colour code things in my videos, like when I draw objects, I usually colour the first one yellow, then the second one cyan, and then green and so on. I might consider changing those colors because they do look similar!
@@mathemaniac perhaps just a deeper blue than cyan
Oh. oh my. _Oh my god._ @8:34, when you say "however, the transformation looks a bit too complicated to understand - is there an easier way?" and at that exact moment the background projection hits the point where the 2d image distorts to become a 3d object (or rather, a shape on a 2d plane that our brain immediately interprets as 3 dimensional), and then totally unfolds into your standard 2-dimensional plane... holy crap my dude. If that was intentional it was truly a genius choice. Amazing work :) If that was your video editor, please give them a raise. If it was you, please give yourself a raise. Or at least a cookie. Chocolate chip maybe?
(seriously though, did you purposefully make your words and the background animation sync up like that? I _need_ to know!)
No, that was not intentional, because the rendering of the animation was simply to smoothly transition from one Möbius map to the identity map. It just turns out that in the middle it might look like it is a 3D object.
Brilliant is the definition of a well targeted sponsored video. Than you two =)
Love your videos! I would love if you could make a video on pythagorean spinors and quantum mechanics. There are a couple of papers relating them to apollonian gaskets. You can find them if you google them. I find the connection fascinating and I think that given your expertise, I bet you would be able to flesh out some really unique.
I will check them out!
Thanks for the brilliant video!
(That was 25 minutes?! Didn't feel that long)
2:05, You almost got me there.. A certified Vsause reference
Yooooo morbius transformation
No way, it's morbin' time
Thank you so much for this amazing video!
Glad you liked it!
Love your rant on math education, I agree!
New here, just wanted to say, your use of squares to silently show your work is intuitive.
Thanks!
Arguably Great video !!! Ali!!!
Thanks!
This is awesome! Thank you :)
Thanks!
The statement at 9:00, I wish all math authors followed this!
Beautiful video and explanation. If I only may suggest a tiny clarification, the Riemann sphere is a model of the EXTENDED complex plane
Which is the complex projective line P1(C), not to be mistaken for the real projective plane P2(R)…
Really high quality
Thanks!
Your videos are great at making hard mathematical concept look easy. Thank you and keep up the good work!
Thanks so much!
I toyed with this idea many years ago, I defined 1/0=inf, but I also defined 1/-0=-inf, I haven't really comprehended what that really implicates to this day but I drew some nice circles and parabolas and philosophised on axies so that was fun
8:15 off the top of my head I remember that every isometry is a composition of at least 3 reflections, meaning that if you can chose any lines to reflect upon, you don't need rotations or translations
What about stretching and squishing? Must it be there, or can it be a combination of the inversion and reflection?
8:10 :We only need inversion
Inversion with circle passing infinity makes reflection
2 reflections make rotation and translation
2 inversions on circle with same center make things larger/smaller
Congratulations on getting a sponsor!
Thank you!!
Great video! I was wondering about this for a long time and it's a great addition to the math videos on UA-cam. What happens if we set the degrees in a circle from 360 to a larger highly composite number with more divisors?
There is no way this is true. The Sumerians definitely knew of numbers much larger than 360 that were highly composite. They wrote about such large numbers frequently. 360 is just the most highly composite number that is also closest to the number of days in a year, according to a solar calendar, so that 1 degree of orbit approximately matches one day of the year.
Also, your question is misconceived. There is no reason why we should think of uniformly partitioning the circle into any number of segments to devise a unit of rotation. Mathematically, this is unnatural. This is not an issue with 360 being arbitrary choice for the number of degrees in a circle, but an issue with trying to define a unit of measurement for angles in terms of fractions of a rotation to start with.
What is most natural is to think of angles that are congruent, and place them in the same equivalence class. We want them to be congruent under direct isometries, not indirect ones. The class of angles in Euclidean space can be partitioned into these equivalence classes, which can be totally ordered, such that they are order isomorphic to the interval (-p, p] for any real p > 0. Since each real number corresponds uniquely to a class of directly congruent angles, this real number can then be thought of as the measure of the angle. p is the measure of a straight angle, and so represents a half rotation. p is arbitrary, and a choice of p amounts to a choice of scale factor. p = 180 is equivalent to using degrees to measure angles, but this is by no means a natural choice, in light of what we are working with. There is one obvious choice that is natural, p = 1, but there is a most natural choice that is not obvious, and this choice comes when thinking of a circle whose center is the vertex of an angle, and thinking about the relationship between angles and arclengths. The choice is given by p = π. The reason arclengths are more natural, and more fundamental, is because of how they generalize the notion of angle to the other conic sections, and even to other metric spaces, where arclengths can be defined, but not angles.
Congrats!
In most cases we want to keep "1/0" as undefined. But the point of the video is to show you that, under a suitable context, we may assign a symbol "infty" to the expression "1/0". At times, "infty" is just a symbol, but sometimes (like in this video), "infty" is in fact a "number".
Perhaps the phrase "infty is in fact a number" unsettles people. Indeed it is a slight abuse of words, but it can be made rigorous by the following two steps: 1. extend your number system to include a new point denoted by "infty". eg in this video we have complex_union_{infty} 2. the (extended) number system is a set, and by a "number" we simply mean that it is an element of this set.
The caveat is that step 1 requires a suitable context. Therefore the take-away is: If people tell you that "1/0 = ...", ask them "Under what context?". This video answers: let me give you one suitable context through two ingredients, the Riemann sphere and the stereographic projection.
That's a nice way to put it.
@@mathemaniac Keep up the good work =)
@@strikeemblem2886 Well, I disagree that this is sufficient to say that ♾ "is a number." Despite the analytic applications, you cannot do arithmetic with ♾ consistently, which is a defining feature of what we almost always mean when we talk about a number system. But I agree with the rest of your comment.
@@angelmendez-rivera351 valid and great point, you insist that a number system has to be a field so as to do arithmetic. Then you would also have a problem with things like quarternions and ordinal numbers. Here, it is simply meant that we extend the binary operations + x, possibly dropping any algebraic structure in the process.
.
There is a nice discussion on stackex titled "Is infinity a number?", and I pretty much agree with the top answer.
@@strikeemblem2886 *valid and great point, you insist that a number system has to be a field so as to do arithmetic.*
I did not say it has to be a field. Notice that the natural numbers form a semiring, yet they are still what most people would consider a number system, and you can do arithmetic with them. Addition and multiplication are well-defined. The integers form a ring, but not a field. These are still algebraic structures, and you can do arithmetic with them. The projective complex plane, or Riemann sphere, while interesting in terms of its applications in analysis and its topological structure, it lacks sufficient algebraic structure. Sure, technically, - and / are well-defined, but + and · are not.
*Then you would also have a problem with things like quarternions and ordinal numbers. Here, it is simply meant that we extend the binary operations + x, possibly dropping any algebraic structure in the process.*
In what sense are we extending the operations? I know that, by notational convention, a·♾ = a for nonzero a, for example. But since things such as ♾ + ♾ and 0·♾ are not defined, I am not sure in what sense this forms an algebraic structure. I suppose that, if one considers functions from a proper subset of (C+{♾})^2 instead, then one gets a unital semigrouid for the addition and multiplication. But at that point, I am fairly certain most people would not feel comfortable calling this a number system, and I think this goes for many mathematicians as well. Now, I understand this is largely a matter of semantics. There really is no formal definition of the word "number" that most mathematicians have agreed to, and the boundary between just an algebraic structure and a number system can get a bit vague and fuzzy. And for the record, I am not dismissing the utility or the interest or partial structures, or what have you. But to the extent that these are drastically different from the algebraic structures that we typically associate with number systems, to the extent that I think most people would not recognize them as such, I am not convinced that grouping them with number systems is the most conceptually healthy intuition to instill on people, hence my objection.
I mean, I know none of this actually matters when you have a degree in mathematics and you actually know what you are doing. Whether you call ♾ a number or not does not matter to a person that understands the underlying theory: calling it a number will not bring about any silly mistakes by someone educated on said theory and actively researching it. But what I am concerned about here is educating laypeople who, at most, have a bit of undergraduate level education in mathematics, and are just looking at Internet resources to strengthen their understanding or are just doing it for fun. For these people, whether you classify something as a number or not does matter, and tends to evoke a pretty consistent image in their heads. When they ask "is ♾ a number?," what they really mean is, "can I add, subtract, multiply, or possibly even divide ♾ with all other numbers in a given number system in a way that is consistent?" And since that is the image people have, I think that is the operational definition of "number" with which we should approach the question, especially in light of the fact that many people who make videos on UA-cam always say "♾ is a useful concept, but not a number; you cannot do arithmetic with it or compute with it." Since it is apparent that, at least intuitively, number systems have to be a total algebraic structure of some sort, I find it extremely unhelpful to take the liberal approach used in higher mathematics of treating "number" as synonymous with "literally any mathematical object" for this specific context of discussion. It does not answer any questions, and it does not give people the deeper understanding they need about how they should think of ♾. Among higher level mathematical discussions, that usage if the word number is perfectly fine, but I find it unfriendly and obtuse to do so in the context of answering the question "is ♾ a number?".
i'm not sure I understood anything but as an engineer, 1/0=infinity works just fine anyway
Are you okay with 0*infinity=1 ?
I could barely keep up with this video. I'm gonna have to rewatch this a bunch of times to get all of it🤯
Edit: I loved it
This video is amazing. Can you do a more in depth one about Penrose's Twistor identity at the end? His hand-drawn projector slides are nice but I bet you could make something so much more striking with this modern animation.
bro Ive been saying this for years now
Please help me with this part 21:44
Why is not just the composition of the two rigid motions?
excellent content really mind blowing
Thanks!
for the problem at 7:49, here is my solution:
if ad-bc=0, then ad=bc, therefore, bc-ad=0.
since bc-ad=0, the problem shown below turns into 0/(z + the initial translation by d/c and the enlargement/shrinking by c^2), which reduces down to 0.
now, the equation simply is f(z) = a/c + 0/xz, which still eventually reduces to a/c.
therefore, if ad-bc=0, then all complex inputs would be reduced to a single output, i.e. a circle with radius 0.
at least i think i got this right. correct me if i'm wrong!
You don't need complex numbers for this argument. You could see that inversion on the real line is the same as rotation on the stereographically projected circle. Then 1/0=∞ since 0 and ∞ swap places under this rotation. Also I would like to learn what the relationship between rigid motions of spheres and PSL(2) is (because we know that they both correspond to mobius maps). Also I love the video : )
Yes, that's right - since this is a series on complex analysis, this has to be an argument on complex numbers...
I think the relationship between rigid motion of spheres and PSL(2, C) is the most direct when we link them to Möbius maps, but there might be some more direct ones that I don't know of.
أذكر عندما كنتُ صغيراً وأفكر في حل مشكلة القسمة على الصفر اعتقدتُ أننا بحاجة لاستبدال خط الأعداد بدائرة الأعداد، حيث يكون الصفر في أسفل الدائرة والنقاط على اليمين تمثل القيم الموجبة وعلى اليسار تمثل القيم السالبة، وتكون النقطة في أعلى الدائرة هي النقطة التي تمثل ±∞ في وقت واحد وهذه النقطة تمثل إحداثية ناتج القسمة على الصفر، وهكذا تم التغلب على فكرة الاقتراب من +∞ من اليمين ومن -∞ من اليسار، لكن بهذا المفهوم فإنّ جميع الأعداد التي ستقسم على الصفر ستعطي النتيجة نفسها وتمثل بهذه النقطة
Inversion can be interpreted as reflection on a circle. Compositions of such reflections generate the whole conformal group.
Compared to the affine group emerging from usual line reflection. The conformal version adds dilatation and contractions. They come from the composition of inversions on circle sharing the the same center.
Yes! (I think you haven't mentioned rotation, but this should be easy: successive reflections in two different lines) So only inversion would work!
@@mathemaniac Exactly.
Since lines are flattened circles passing through ∞, the reflections, translations (reflections with 2 parallel lines) and rotations (reflections with 2 concurrent lines) can be recovered from them.
hope you got a mil one-day.
That's way too ambitious, but thanks for your appreciation!
thank you very muh for this video.
Glad you like it!
This makes me totally want to do relativistics in terms of the Riemann sphere and then have a look how black holes look from that perspective! That should shine a light on the idea that our universe is in fact inside the black hole of another universe! Unfortunately I have no time, so I hope someone else will do this...
You know what got me is studying point set topology. It's why I recognize the symbols and it teaches how to create proofs ultimately leading to proofs for algebra. And I did 1/0 on my chromebook. It correctly displays the infinity symbol
can you point me to any good topology resources you've found online? The wikipedia pages on the topic are.... _dense,_ to put it mildly, and I'm having a hard time knowing where to look. Any help would be greatly appreciated (regardless of the density of the reading material! Normally that doesn't bother me, I'd say I actually prefer it, but the topology pages on wikipedia are truly something else lol)
edit: or not online! I'll take anything I can get my hands on at this point
@@idontwantahandlethough Zach Star's channel might have some videos on books on the topic.
An in depth video. Back in 2014 @Numberphile had a demonstration of inversion by hand. The video was titled _Epic Circles_ . Keep publishing good mathematics videos!
Thanks!
I'm not sure that video exists anymore, but here is a newer video from a year ago from his channel ua-cam.com/video/bJOuzqu3MUQ/v-deo.html
@@floydnelson92 Thanks! Love this video.
yes, intuitively, division-by-very-small numbers, ALMOST overflows into multiplication-by-very-large-numbers, but for this to make sense, you need a few constrains, that are not always useful.
No it does not make generally sense to equate the 2. Whenever you divide (by a very small value (that approaches 0)), this is a clear tell-tale, that your model/axioms approach a much simpler special-case == near-asymptotic-case, that often does not exist in a special case. The division-by-0 is an emergent property of your generalization, and it points out a constraining-range of a generalization.
Where ever you multiply in a function, your inverse of that function introduces a division (unless the function is its own inverse or something like FFT) , and your possible-parameters may or may not enable a division by 0, and you then have to filter out a boundary, to evade the precision loss, that results from dividing by too small numbers, by using something like SmoothMinimum(a,b)
within that low-precision-area (where you divide by almost-0) you then use a simpler special-case function, and outside of it, you must use the generalization, that added the division-by-0 case.
The circle of inversion part brought back flashbacks to Numberphile's Epic Circles video.
The infinity itself is an undefined concept. There is axiom of infinity, induction and lim, but no infinity itself. Especially not as a scalar.Usually the symbol is shorthand for "any positive number" or "any negative number". If you consider infinity to be a number larger than any other numbers, then dividing 1/infinity, 2/infinity etc would give distinct infinitesimals. It also preserves sign (0 is unsigned, and loses such information). Same way, 2*infinity would be a different kind of infinity from 3*infinity They did that in earlier calculus, before introducing lim. In the end you an collapse them to normal 0 and infinity.
I've always thought that you could define translation in terms of rotation. By using a circle with infinite radius (a line), a rotation along that circle is effectively a translation along that line
This is something I haven't thought of, although you can't really define the angle of rotation that way... But among those 5 transformations, both translation and rotation can be reduced to some combination of the others.
The "angle" of rotation is in fact simply the distance being translated. You can also go the other way and argue that standard angles of rotation are instead distances in spherical geometry. My reasoning for defining the angle as such is from the relation between complex numbers, hyperbolic numbers, and dual numbers. With complex numbers, e^iφ moves φ units along the unit sphere, while hyperbolic numbers (aka split-complex numbers) e^jφ moves φ units along the unit hyperbola, which is sometimes called the "hyperbolic angle." Dual numbers behave similarly, but in flat geometry, so e^𝛆φ moves along the straight line at x=1 by φ units.
Translations in certain systems behave somewhat like dual numbers, which is why the comparison is relevant.
im pretty sure all transformations can be described by 3 reflections. i got super into projective geometry while studying perspective projection in computer graphics, and i remember that fact (along with many other concepts like point/line duality) sticking out as being astounding
I think that projective geometry is the best way to motivate mobius transformations instead of just rational linear functions. the mobius transformations are the automorphism of the projective space over C.
congratulations for the 69k subscribers!
Nice
Do asymptotic representations of functions survive the introduction of such operation?
No, not quite.
Bosons with spin 2 are represented by a complete revolution of the Mobius strip, which is central to understanding how mass, energy, space, time emerge from a singularity, the two poles of the Reimann Sphere.
That's just a way of representing the fact that the spin group is a double cover of the special orthogonal group though. That's not directly related to mobius transformations (although they also form a semisimple Lie group)
22:41 I didn't find a paper that demonstrated it, but the video "Möbius Transformations Revealed [HD]" (which you also have linked to) demonstrates it, is created by people different that the one who wrote the paper, and was copyrighted in 2008.
I don't think the video demonstrated it - the point here is Mobius map can be *uniquely* represented by a rigid motion of the Riemann sphere, but the video only shows that a Mobius map *can* be (but not necessarily uniquely) represented by a rigid motion. The paper I mentioned explicitly wants to show the uniqueness, as shown by their introduction: "The main result shows that for any given Mobius Transformation and so-called admissible sphere there is exactly one rigid motion of the sphere with which the transformation can be constructed."
@@mathemaniacAh, okay, that may be right. So maybe the connection had already been known for a long time, only that the one-to-one correspondence was first shown in that paper?
Yeah - but that is quite a "basic" result in the sense that it doesn't involve advanced mathematical tools, which is why I was surprised it was only shown so recently.
A very, very nice video :) Thank you very much. A question - is this right, that the inversion reverses orientation? 1/z is a holomorphic map and holomorphic maps are true to orientation and conformal. You must reflected your curves on the real axis, then it is true to orientation, isn't it? Best wishes from germany
So for the **geometric inversion** (or inversion with respect to a circle), it does reverse orientations (hence anticonformal), and of course, a simple reflection across the real line also reverses orientation. Both operations preserve the magnitude of the angles.
Since 1/z is a combination of both operations, it preserves both orientation and the magnitude of the angles, and hence yes, it matches the prediction that holomorphic functions are conformal (where the derivative is not 0, but 1/z does not have derivative 0 anywhere on the normal complex plane anyway).
Don't bother with Brilliant if you are on the level you watch this for fun - 2x year subscriber
19:03 very nice wok.
thank you
Wow the three R's ! Ty
Very entertaining video.
Thanks!
Vote up, nice video, thanks for sharing :)
Thanks for the compliment!
How long does it take you to edit a video like this one? Thank you
No exact figure, but easily 70+ hours.
It's not ridiculous and makes sense even in 2D. The inversion works with non-unit circles (of non-zero radius) too, producing the same infinite 'circle' regardless of the radius of the original circle. Proving once again anything divided by zero is not just undefined, but more precisely, infinity.
¡Gracias!
It's Möbin' time.
Are the moebius maps another name for affine transformations from linear algebra or is there some subtle difference?
You need all the transformations, I think, and you can prove that with matrices.
No - affine transformations and Möbius transformations are really different: for a start, affine transformations maps lines to lines, but Möbius transformations and map lines to circles. Affine transformations might not be conformal, but Möbius transformations must be.
You don't need all transformations, and you can do everything with just one of them!
7:30 note that differentiating with respect to z we get |M|/(cz+d)²
is there any relationship between mobius maps and matrices?
YES! I have said in the pinned comments that this is something I have missed - just that the video is too long so I didn't include this part on matrices...
I don't see any pinned comment?
Yes, every Möbius transformation az+b/cz +e is uniquely described by the matrix (a,b)(c,d). And if you concatenate two Möbius transformations you multiply their matrices.
Möbius transformations (meromorphic bijections of the Riemann sphere) form a group under composition which is isomorphic to the projective general linear group PGL(2,C). Each transformation corresponds to an invertible 2x2 matrix (see Caspar's comment) uniquely up to multiplication by a nonzero complex constant.
@@Caspar__ Uniquely up to multiplication by a nonzero complex constant. Also e vs d is a typo :)
When Robert Siliciano wrote "Constructing Mobius Transformations with Spheres" he was still an undergraduate.
Can you do a video on homeomorphism and diffeomorphism?.
It might not be a full video, but this is something that I can consider.
Alright 👍. Looking forward to watch your videos on them.
"By abuse of notation...". Wow. This is candid, even for a mathematician.
Isn't translation just a combination of rotations?
And inversion is a combination of scaling & reflection?
If I'm right, the 5 tranformations can be simplified to just scaling (grow/shrink), rotating, & reflecting
Can you elaborate more on why inversion is a combination of scaling and reflection? (I am referring to reflection only across a line here)
@@mathemaniac I guess I was picturing your animation at 3:57 where tracing the circle counter clockwise traces the inverted circle clockwise. But now that I think about it, we don't need points to be unique (map to specific new points) so reflection does nothing & inversion is scaling + translation (which is just a combination of reflections)
@@AlexanderQ689 In particular, since we know all the actions preserve circle, we can just translate C1's center to C2's center, and scale the diameter till they match
Although circles do map to circles, inversion doesn't do the scaling and translation *uniformly*, i.e. a circle with radius r does not always map to a circle with radius kr, where k is a constant. It depends on the position of the circle.
In relation to the discussion division by zero, you can watch this video ua-cam.com/video/nFk9GNSLAyo/v-deo.html
This gives another perspective to the subject.
Haha 😂 loved that Vsauce reference at 2:05!
I believe that there has been a historical mix of things going into the education of math in schools. Some of the jargon is complex because it builds on other complex ideas and has to be very precise so that people can communicate exactness correctly. But also much of it was made overly difficult so as to bar it from being taught to the commoners. Sometimes very simple ideas are given very large names that intimidate people who do not know what the word means. I believe that modern society should rename some of the jargon to be more indicative to the lay people what is meant by the idea of the jargon. I think the jargon should be about as "simple" as the idea that it means to express. In many cases, this is the case, but in many cases it is not.
I don't think math jargon is created with the intent to intimidate, just so that is can be reffered to later, for the sake of compactness
Then you need to learn. There are places where women aren't allowed an education. The same goes for caste society in India, some people are barred.
This isn't exactly what was stated, but why the hell were bibles written in latin for such a long time when only the all-male clergy spoke latin?
division of anything by zero is left undefined in general because of the lie everyone's accepted that fractions are numbers. of course it makes more sense to define it some specific way in specific cases, because the reason it's left undefined is a result of there being multiple such cases which contradict each other. this results in division by zero not behaving like a function (surprise! probably because it's the slope of a vertical line, which by definition is not a function... what a shocker!). but in a world where people expect every division problem with defined numerator and denominator to yield a number... this is a hell of a paradox.
the solution to the paradox is to realize that fractions are always vectors, and actually never numbers (by which I mean scalars), and as such it's usually extremely important to understand the relationship between the units of the components even more than computing the ratio between the components. for instance, atan(2/0) is obviously pi/2, but if you attempt to compute 2/0 before looking at the function that's being applied over it you end up stuck. and even taking limits won't necessarily help, as the result can yield either positive or negative infinity, and atan(-2/0), which is what it will appear to be if you try to take the limit from the left, is 3*pi/2, which is obviously not correct.
moral of the story, math is not about computing values, so rushing to compute things before looking at what you're doing is a foolish move. as such, 1/0 being best handled as infinity in some cases while remaining undefined in general makes sense, and honestly this kind of thing should be shown to students far, far sooner than it is. because the mentality that we need to teach the basics first, and computation is basic... is completely wrong.
Can you tell me, why is infinity so important for the complex analysis so that it is meaningful infinity representing as a point on the sphere? I mean why is the extended complex plane so important? The only reason I can introduce is, that you can whole functions (functions that are holomorphic in the complete plane) classify after the kind of isolated singularity in infinity. Do you know more reasons?
I do not know too much, but I think it is way easier to represent Möbius maps if we just include infinity, as demonstrated in the video. (Excluding infinity will make Möbius maps lose the group properties)
But I suspect the main reason has something to do with geometry and topology. It makes the complex plane compact, and the description of meromorphic functions(i.e. holomorphic functions from C to Riemann sphere) more convenient.
Its importance is less about complex analysis, and more about projective geometry and algebraic topology.
A question. The inversion is a one to one mapping - one point maps onto one other point.. But all the points at infinity map onto the one single point of the origin. So that's for me actually a strong argument for the indefinitness of 1/0 in the context of mapping operations.
In the projective complex line, which is a way of making this rigorous, there is only one point at infinity.
2:06 LOL at the Vsauce reference
Wow , it's live
In GLSL, 1.0/0.0 is Inf but 0.0/0.0 is NaN
In fact iirc any number divided by zero (except for zero) is Infinity
I looked glsl up and i was almost offended at how obvious your statement is! Thanks for the insight!
There is a much graceful definition for an algebra that allows division by zero. Actually the periodic table of elements presents such an algebra.
I don't know what you mean. How does that algebra work and what does it have to do with the periodic table?
I have to agree with Alexander here. What are you talking about?
totally agree!
Thank you..With all my respect I wish you have completed the series yourself.
If I walk far enough on a mobius map, will I get to the other side?
Someone please answer 11:22 why
Ima need to rewatch this when it’s not 4:30am 🙃
its like there is 1 of something that will be given to 0 persons, it can't be given to 0 persons because that just doesn't work like that, so that 1 of something just stays here and never gets given, so technically its infinity as long its still will be given to 0 persons
Nice
Is this video related Geometric Algebra ? I saw lots of similarities.
Conformal Geometric Algebra is precisely the algebra that describes these angle-preserving transformations. It rarely gets much attention since people tend to focus more on pure Euclidean Geometric Algebra, Projective Geometric Algebra, or the Spacetime Algebra depending on their field.
I am really hoping that Mathemaniac makes a video about Geometric Algebra at some point.
It confused me at the beginning but when it went 3D, I was like "Ohhh"... I finally get what's going on.
That's exactly what I was hoping!
This needs to be made clear: you *cannot divide by 0.* This video is not in contradiction with this.
Why can you not divide by 0? It is very simple: if addition and multiplication are well-defined, and 0 is the identity element of 0, and multiplication distributes over addition, and addition is cancellable, then it can be proven that 0 is not ·-cancellable. This is true, regardless of what other mathematical objects you include in the system. Otherwise, you have to give up distributivity, or you have to give up cancellability for +.
So why is this video claiming you can divide by 0? Well, the video is *not* claiming that. When people write 1/x = y, *typically,* they are using a notation where 1 = x·y is implied, and this is what it means to talk about division. But when a mathematician says 1/0 = ♾, in the context of projective geometry, they are *not* saying 1 = 0·♾. The notation is misleading. In fact, one does not need to use this kind of notation to talk about the Riemann sphere, and some mathematicians do not do so. So when you watch this video, you should not think of 1/0 as division by 0. You should think of Möbius transformations, which is what the video is about. In the Riemann sphere, there is the compactification point, which we commonly denote as ♾, for the sake of convenience, and there is a continuous function which satisfies f(0) = ♾ and f(♾) = 0, and this function f just so happens to coincide with the reciprocal function for nonzero complex numbers. So notating f(z) as 1/z, while technically an abuse of notation, is not all that uncalled for, and is helpful for the sake of analogy.
Ultimately, this is all about doing geometry, not arithmetic. We are not dividing by 0, and at no point have we allowed ourselves to add, subtract, multiply, or divide by ♾. So the mathematical theory behind the Riemann sphere is not in contradiction with the theorem I presented earlier in the video that prevents us from dividing by 0 in the context of algebra and arithmetic.
So, no, division by 0 does not result in "infinity," and yes, dividing by 0 is still "forbidden." But when you get the privilege of playing around with notation in mathematical theories that do not involve arithmetic, suddenly, you can allow yourself to say things like 1/0 = ♾, and for the purposes of that theory, it can be helpful to do so. And that is ultimately what this video is doing: using intuitive notation to present a complicated concept in projective geometry.
A long time ago I noticed if you take basic math and you try to divide by zero you get every real number in existence you could literally arbitrarily decide the answer, which in a way is infinity.
I don't understand imaginary numbers enough to say if you could get those as well or not
I can't believe you didn't mention Smith Charts!
I didn't know Smith chart, only now that one of the top comment mentions it that I know...
@@mathemaniac yes, the Smith Chart is related to Möbius transforms. It was used more in electrical engineering before computers were prevalent, but it's still taught. Great video, btw.