Moebius Transformations Revealed
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- Опубліковано 27 вер 2024
- A short film depicting the beauty of Moebius Transformations in mathematics. The movie shows how moving to a higher dimension can make the transformations easier to understand.
The full version is available at www.ima.umn.edu...
The background music (from Schumann's Kinderszenen, Op. 15, I) is performed by Donald Betts and available at www.musopen.com.
Not only is this beautiful, but it's incredible how skilfully it's presented
I've come back to this vid over and over across the years. There's something about it that gives me great peace and satisfaction.
I think this video demonstrates quite nicely how cool and interesting (visual) complex analysis is, one of the most magical and beautiful fields of mathematics.
Fantastic! I've been using these transformations for years in tools like Photoshop and never saw the underlying mathematical principle until now. Kudos
Alright gang let's see who the Moebius Menace really is!
*takes off mask*
GASP! Old man Sphere Projection!
This has been one of my favorite videos for so long that it is like an old friend. Great choice of music. Schumann Kinderszenen opus 15 number 1 beautifully played by the late Donald Betts.
i *heart* mathematics
wow i cant believe i saw you here lol
I am usually averse to math, but I feel like watching more of this. A hundred rounds of applause for the soothing music.
Then, put a hypersphere above THAT sphere!
You are a madman and I will have no part of this insanity!
I intend to fund your research into this! :O
Thank you. Finally, a way I can visually explain it to my friends without spending forever trying to get them to understand. You simplified it so well. Sweet video. 5 stars.
That is beautiful :]
This is such a good example of why I love mathematics.
They are mentally stimulating on such a greater level than most people understand.
And when coupled with their potential (in their own field and others), the possibilities are endless.
Very nice visualization, it really gives you this cool effect of realization ("Aha!") the kind of which good maths is full of (and also shows the inherent beauty). And I think it's even cooler in that it touches many people that don't care much what an inversion or Moebius transformation really is ... very well done!
fantastic...
What a wonderful teaching tool. If I had been taught using a similar approach, I probably would have grasped mathematical and geometrical concepts to a much greater extent. It explains WHY theory is what it is, not just set up a series of solution rules to remember by rote.
I want to pay you money for taking time to build this animation
Seriously, I felt spell bound...Thanks so much
Wow, amazing video dude. Somehow brings out the emotions and feelings within you out into the open. All the colours seem to entrance you and to be honest, you really can't stop watching the sphere and when the end comes, you let out a small sigh of contentment. Well done dude!
Hey! I saw an artical about this in the news paper, and had to check it out.
I had no idea math was as interesting as this! I wish they would teach stuff more like this in highschool.
Also, great music.
Anyone here from MathPath?
Yep!
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This is an extremely thorough explanation that stays very concise. Great job!
606 thumbs down, really? What the heck are reggaeton-lovers doing here?
+Martin Cigorraga why reggaeton lovers though?....
How math concepts are taught largely determines how widely - and profoundly - they may be comprehended. Arnold and Rogness prove this quite elegantly; the popularity of their brief video should speak volumes to math teachers worldwide.
Thanks. Worth a thousand words. There is nothing like the beauty of mathematics, and the introduction of the sphere was worth a big smile. I'm off to see the full version...
-whispers very softly-
wow...
At first I thought this video will go beyond my understanding, but the thing with the sphere that the inversions can be described as a surface on a sphere?!? This is genius! Very nice video!
I think that is the first time math ever gave me goosebumps. Beautifully clear.
I think it's really cool. I had trouble visualizing the inside out transformation until the relationship with the sphere was shown. Nice job.
A+! If mathematics was made this interesting when I was in highschool, I may have actually paid better attention. Thanks for this. Opens my mind up to mathematics once again. Very cool.
Excellent. This video has become my main example of what to aim for with some I'm trying to make myself. A great book for this sort of thing is "Visual complex analysis" by Tristan Needham.
Moebius transforms are precisely the analytic bijections of the extended complex plane, and can also be characterised by their vanishing Schwarzian derivatives. THE way to think about rotations, naturally leading to quaternions.
Rogness, you crazy mathematician/filmmaker. Looks like you've got a hit on your hands here. Very nice.
When I was in school, we called this The General Projective Transformation and it consisted of a 3 x 3 matrix by which the original figure was transformed by matrix multiplication. Since then, TGP has taken over in the implementation of computer graphics. Using the sphere is a nice visual of it.
I'm not a math scholar but this definitely helps me look at all of life from different perspectives. And after viewing I somehow feel like the explanation was very logical. Thank you!
This is a demonstration of a mathematical transformation which basically means using a pre-determined formula to "map" (not like map in geography) every point to a new location but not in the sense of motion. An example is that in some engineering applications, your stresses act in directions that don't correspond to the axis so you apply a transformation to rotate/translate/skew/etc. your coordinate system, not your piece.
Beautiful, simple, profound, calming.
whenever i lose track of something, i watch this video. it's amazing how i suddenly regain overview. sometimes it's just the different perspective.
Absolutely, "Visual Complex Analysis" is a GREAT book, shows how higher mathematics can be made exciting and understandable without any dumbing down.
Awesome. Got my attention with the fancy-looking inversion, THEN showed how elegant and intuitive Riemann's geometry made it. So I thought the order of presentation made it really good.
I wish I had been shown things like this more often in school...
BEST Video i've ever seen on Mobius transform.
I actually used to visualize this but plane was Circle and Mobius transformed circle was used to be entire complex plane except circle.
This video makes a good simulation than my imagination!
this is seriously one of the coolest things ive ever seen.
This is one of the best Mathematical discoveries in modern history. We should marvel at it, and use it in future academic endeavors!
I love having my mind blown so gently.
Very nice man. I remember lost days & nights studying Moebius Transformations
@SingerAvril It's a projection from the light source at the top of the sphere, to each point of colour then to the plane. There's no refraction or reflection, just straight line mapping of each colour point to 2 space about the light source. Pretty awesome new way of visualising transformations.
On a deep level this illustrates much of the patterns that my imagation invisioned. It may show much of the dynamics of how the Universe behaves; for example, if gravity bends space perhaps the ends really are the beginings; nothing if extended enough can be truly flat. The paradox is that it may seem that we exist in a very large pad of paper so dense that it makes marble look like a marshmellow and a diamond as lofty on the inside as Ider down.
Magical explanation of the wonders of sacred geometry and it's universal applications like healing for example. Thanks JR.
OMG this is one of the best videos I've ever seen (and I've seen way too many). Get's the concept of a Moebius strip across in a really easy to understand visual. Thank you!
Amazing video. This really makes Moebius transforms a lot easier to visualise.
I liked this- it really simplifies the visualization of a transformation... kind of what happens when you use apply transformations in multi-variable calculus or linear algebra- except here you can actually see what's going on!
I think this was the video Terrance Tao was talking about in that talk.
Wow! Thanks! I'm here from researching the Z-Transform. I needed a concrete visual and THIS is it. THanks!
The colors are just a way to keep track of things. If you wanted to see "what the map does outside of the square" you would just move that square over and apply the map, or start with a larger square with similar coloring. You might try to see a pattern the map follows by looking at what appears in the video.
Whoa... it's kinda scary that I actually understand all that, lol. This is a pretty good explanation and the visuals would help people who haven't been exposed to this topic yet to grasp it. Very nicely done.
I'll be honest with you.
This video's topic meant nothing to me. But I understood what was going on through the visuals. It's a nice little informative vid, the music was lovely and fitted with the images. While I may not be a student of this math/science(?) and not have a clue what was going on I can appreciate it all the same. Good work, mate!
This is soo beautiful! I gather that this is probably very complicated usually for lay people to understand but it is sooo simply explained with these graphics that I can appreciate the beauty of it! It's so elegant in its simplicity and perfection!!! O my gosh! Maybe I should become a math major!!! AAAAAAAAAh!!
Great explanation of a mobius in a short clip for people who like such things, like myself... thank you. It's all mathematics and it's all relative.
because it is a very interesting and fantastic way of explaining an otherwise complicated form of maths and also quite artistic.
Wow, why were we never told that the Moebius transformations are just elementary movements of the Riemann sphere?! Although I never found them hard, this makes them much easier to understand and more beautiful. Fantastic video.
most stunning video ever seen on youtube
great job mate(prof).. I always had difficulties visualizing these transformations in riemann sphere... gives a crystal clear idea... Ur video of 3 mins gives a hell lot better insight than 3 weeks of lectures in my complex analysis class...
could u post more such videos on complex analysis... (i would request complex integration over lines and loops... pleeese...)
a real testament to the greatness and ingenuity of the human mind.
wow i dont really understand it but the music was really beautiful and added to alot of the transformations beauty. That was cool that actually made me smile to watch that.
Amazing. I have never seen that demonstrated so clearly before. I have struggled with that concept since I was 16 y/o. I finally understand it. Thank you.
Beautiful animation....leaves all quiet clear!
I admit, I am a Math-head myself, and proud of it. I've heard a lot about Mobius, but I just had to see it in action. It seems complicated, but it's not hard to understand if you follow the sphere concept in relation to how the plane moves. I'm sure when applied to more complex problems, it can understood on a full scale.
This is an excellent example of how our human brain tries so hard to simplify/over complicate things.
A wonderful new way of thinking about plane geometry. Great music too.
probably the most useful thing ive ever seen on youtube
And now I get how certain graphic filters work!
You never let this video die, Mr. Rogness.
It is the description of the video.
It is lovely
It's the flower of life minus the center. There are different ways to get to the same answer using different formulas. If you don't know what I'm talking about, look it up. It's very interesting and the pattern is seen in nature, including the division of zygotes in human development.
wow this is great we're learning this in analysis but my entire class has no clue what's going on thanks for the edge!
OHHHHHHH...... now I understand why the 2 dimension illustration is revealed by adding the 3rd dimension.
That's awesome.
Fascinating in its simplicity
Yes, zero and infinity are antipodal to each other on the Riemann sphere (depicted in the video).
Very nice, I always like having these little revelations in mathematics.
more!!! this presentation captures the essential beauty of mathematics!!!
@essenceofzagnut Vom fremden Ländern und Menschen by Robert Schumann. It's from his set of piano pieces called Kinderszenen or Kinderscenen. Usually in English titled Of Foreign Lands and Peoples from the set Scenes from Childhood. This is the second most well-known of the set, the most well-known being Träumerei, Dreaming.
This video is nicely done. Next time you might want to include the coefficients for the transform as well.
cool vid. I'm not a mathematician but I can recognise the thought process, and the logic behind it. This would be a great learning tool I think. Visually entertaining, concise and informative, great job.
Sumanraj you seem to be critical. I actually learned from this. Excellent video. Please do tell Sumanraj10, when do you plan your even more enlightened video?
Wow. Impressive Professor Rogness. I'm going to miss UMTYMP next year.
Really Nice. Everyone* knows the good old moebius strip, but i'd never thought of that principle being extrapolated like this (but then, I'm not a mathematician either...)
Beautiful. Can all mathematical constructs be illustrated so neatly?
Very cool - math and visualization is a very good combination
What it reveals is that, if you shine a light through one of these squares, you get Tweety-Pie's head. (1'03") Remind me, what is the mathematical value of Tweety Pie? 3.14....
I hated algebra and was just getting into calculus (which I fell in love with) when I was forced to quit school. This is a fabulous example! I wish I'd gotten that far. :D
Yep, a Schumann piece in a simple AABA form...I always struggled on the B part, but you play A enough times to have it easily memorized.
man good job a simple complex geometry equation can be understood with your animation well done.
simplifying i.e. to learn the alphabet will encourage others to grasp math and science and possibly apply them to their thought process. i remember in high school with the dam priests they tried to use the equations on the blackboard with chalk and a projector. i was pissed no one understood them the entire class failed.
Puppy, I'm not sure what you mean, but what we see here is exactly what we would see in an ACTUAL 3d representation.
First, notice that the light is on the very top of the sphere. So, every point in the sphere not on the top, would be below that light source, and thus be projected on the plain(If theres a ray coming from above, you will see shadow). The only problem is the point on the very top of the sphere, which is not projected. So we map it to 'infinity'.
I hope I understood you:-)
And they say there is no beauty in science.
I shall sit here and be in awe at the simplcity and rightness of pure maths.
It would be really impressive if you could do a similar visualisation but with a fourth spatial dimension (e.g. hyperspheres, hypercubes).
Excellent excellent excellent. As a mathematician, I have to say that was as good as "smoking a bowl."
Wow!! That's pretty neat. I'm impressed with the graphics...it's very hypnotic.
Well, I love this thing because the colored grid is analogous to a 2d randomized block statistical design, where the colors indicate the effect of a uniform extraneous variable (or resulting vector), where the vector is perpendicular to the blocks. I just happened to be thinking, would that conform also to the existence of a 3d gradient? Well, to be sure...
Simply amazing with nice piano music dubbed in also.
Complexity emerging from simplicity. And with wonderful music! Good vid, great math.
i wish there were more maths based videos like this on youtube
this gives me the same thrill as a good twist in a good mystery novel...
I clicked this vid by accident and it thought me something that i couldn´t understand in class. Thanx for uploading it.
That was beautifully done, a perfect introduction for those of us with no experience in this section of math.
Also, *gorgeous*.