Moebius Transformations Revealed

[willbirful]

love the video man

[jessyjessy4]

some really good stuff here

[sprattysy]

i enjoyed this vid

[paologr8]

great screensaver!

[joevideowatcher]

Beautiful.

[happyman]

this gives me the same thrill as a good twist in a good mystery novel...

[julianforestphoenix]

Yeah, this a really cool. Neat to watch.

[SnowmansDream]

This is one of the best videos I have ever seen.

[TheQedqubit]

only 2million views sofar...

visual explanations like this one can do so much for idiots like me, who have a deslyxic allergy for greekalgebra , and just want to 'get the(big) picture'

this movie helps understand how iMaginary value(sqr-1) Really matters Reality Explane X=10s, existense (and for religion... iMaginary is not Real,and 4 science,spirituality is imaginary value, thus conciderable as source of Reality)

complex paradox hey ? seek&find next : numbertypes & the unit circle

[intrptr]

That was too cool. I liked the transition of 2nd to third dimension: In the second,"dilations" and the third, "bouncing". I wonder what it translates to in the fourth?

[kenjokenjo1]

BEAUTIFUL!

[Photon98]

In similar ways can you explain other transforms like Laplace, Fourier, Z.??

[6199applelover]

i'm so confused

[SofondaDildos]

This is the best video of Moebius transformations I've seen in my entire lifetime. Everything became much, much easier to understand now. Thank you for creating it.

[johng904]

That is by far the best movie about Geometry in YouTube!

By the way, have you heard of the new type of (discrete) derivativative from 2010? You should try and search it on YouTube: "Eureka! a new approach to calculus". This video depicts a simple kind of derivative.

[ReddishBocarnity]

OOOOH! PRETTY COLORS!

[lizzabet4]

@ReddishBocarnity woooooooooooooooooooow…………………

[Photon98]

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!

[fritspas]

Ah, it's just projection, now i get it :) bit like mercator :P

[Vertsk8er419]

the transformation to the next dimension, above the planar one, now involving the sphere, makes me think of the opening of the third eye. what if the pineal gland is the lightsource and your eyeball is the sphere..?

Or the sun and the moon...

[Photon98]

@Vertsk8er419

Good!!

[1morphdude]

OK .... I want the software

[nmacgre]

God, I stumbled on this peace of crap looking for the new TRANSFORMERS movie!, Thanks, now I'm stupider.

[iickle]

@nmacgre

Fuck you.

[lilverson1333]

Outstaning

[auntjemima52]

Math ≥ Interesting

[PairInstabilityType]

@jchino723 But science and math can be proved, making them rules worth beleiving in. Science has helped us make sense of our world - we know that walking off a cliff will result in gravity pulling us to the ground. We know that heat can be used to purify water and cook our food. Religion is based on stories older civilisations used to try and make sense of their world.

Great video btw, i'll watch it again after i've smoked some ;)

[whatever2964]

MORE MORE MORE!!!!

[ratulloch1]

very well done :-)

[ddrmaxman]

Wow. Never looked at it like that.

[SASNIGHTCRAWLER]

@jonathanrogness

Hi, dimensions are clearly non physical, these represent the information encoded within energy and matter. Your working with the dimensional appearance or holographic representation of combined information. Nice vid btw.

[LegenderyMan]

OK, what would happen if i turned the sphere inside out?

[BeetleAngry]

@LegenderyMan You just blew my mind man

[SingerAvril]

does the sphere have a specific material like glas or mirror?? I do not understand that "the points on the plane follow...

[JaySmith91]

@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.

[SingerAvril]

maths can be sooo harmonic...

[dominiccss]

wow

[maleiva]

Beautiful colors

[pokemonkiller20]

i didn't know inversion is a transformation. i just learned something new.

[brainwormemc2]

this is superiest video ever about geometry ^_^

[TimJSwan89]

Oh, I see now, they are not claiming that the image is the same as the first one, so therefore, yes the final transformation is in fact a mobius transformation, but one where the apparent image is originally different from the previous original images because it is shifted to begin with. It looks misleading because it is not explicit as the sphere moves off screen before it is changed and moves quickly back on screen making one assume that it is the same sphere and pattern as before.

[TimJSwan89]

I originally loved this video until I realized that there was a mistake and the image on the sphere is different after moving off screen in 2:04. The image on the sphere is stretched on the side so that it only appears square on the plane when the sphere has the image rotated to the side. Before messing with the sphere like this, stick to your claim that the mobius transformations can be formed by sphere movements alone and show, without messing with the pattern, a sphere tilted on 2 axis'.

[TimJSwan89]

Wait wait wait wait wait... 2:02 watch before it translates off screen it is centered then it returns rotated, but the image is still square to the plane... something isn't right because the first time it is rotated upside down the image distorts. (1:52)

[roman78x]

Wonderfull

[AgentMidnight]

My mind is full of rainbows, spheres, and fuck

[Xenogamer777]

i was searching for the meaning of the word moebius online because of the upgrade moebius reactor in the game StarCraft 2 o _o

interesting

[bernardoxbm]

Schumann music is sublime!

[npisvaj]

HEY!! this is an AWESOME video! haha!! i just saw the professor that made this give his presentation on how this works (3/1/11 @ University of Minnesota Twin Cities)!! It was fascinating!! amazingly i didnt fall asleep :P

[bestmathematicstutor]

When I was in elementary school our teacher had the class make Mobius strips. I think this is when I really started to get interested in mathematics. Thanks for this video.

[Odelrobg]

Beautiful.

[DivineNucleus]

= (1(CollinCreatedWormHole&Black­HolePhysicsAllPercentLoyaltoCo­llin--)1) =

[97kevinhuanle]

@DivineNucleus

You're like EVERYWHERE

[DivineNucleus]

@97kevinhuanle YES, like in the way that is blissful --

[GeorgeAlexanderOz]

Excellent. Thank you so much. Who are the 590 dudes that disliked this piece of art? Wake up guys, this is incredible.

[jobriq5]

I love this animation

[pbesi]

Great! A lot of clarifications! Thank you

[wavery56]

wonderful !!

[agostinoperna]

Eccezionale, per veri intenditori

[tatunkha]

wonderfully depicted now let's go to a higher dimension

[michaelw2711]

well done riemann my boy!!

[hassmann2000]

Maths. Bitch.

[happyman]

more!!! this presentation captures the essential beauty of mathematics!!!

[RandomAXISfilms]

MOEBIUS wohoo

[WonderWatcher1]

If this is projecting 3d onto a 2d plane, what would projecting a 4d onto a 3d space look like?

[WonderWatcher1]

You just helped my mind evolve more than I can measure.

[Nani101]

That's beautiful! I would totally want a sphere like that for decoration! :D

[yazeed87]

FUCK MELVIN VID

[shivanshub]

Great .. I got every thing in 2:35 min which i did not get in my engineering study .. amazing ...:)

[HugRunner]

Very nice.

[formerlycheebo]

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...

[BenZeneProject]

Beautiful and elegant.

[legendofzelda101]

I'd do acid to this.

[shikamarusgirl17]

Coolz. =D

[tyniehawk]

Not sure what the significance is of this... Must do some research?

[nursty100]

this is some confusing edumacation lol no i get it......like 75%

[MarkArandjus]

And now I get how certain graphic filters work!

[Teekles]

its eversion, not inversion in the text of the video

[intothetimevortex]

The beauty of applied mathematics :)

[essenceofzagnut]

Loved the animation! That was spectacularly elegant. What was the music playing in the background?

[essenceofzagnut]

nvm just read the description

[Ibuiltatower]

@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.

[Ihatebrendon]

is this analytic geometry or topology

[goodgoodgoodful]

beautifully explained.

[manucyan]

Oh thanks for this beautiful post !

[mitchman52]

this video would be epic if you where high.

[JoakimfromAnka]

What the hell happens when it turns inverted? Does it get infinite size or what??

[krenthabohl]

@JoakimfromAnka yes something like that, the origin gets moved to infinity if i'm correct,1/0=infinity (maybe not the best way to describe it). I think most math books covering complex analysis treat this.

[JoakimfromAnka]

@krenthabohl

So the limited surface and infintie empty space turns into limited empty space and infinite surface?

[krenthabohl]

@JoakimfromAnka yes that is the idea :). pretty interesting stuff, I myself am now following a course on complex analysis. It is pretty handy to figure out how circles/lines tranform by a complex function if you can verify that it is a moebius transformation. Because they will then also be transformed to lines/circles.

[joke312]

Truly a wonderful animation.

[elrevelde04]

that esphere simplifies things very much

[nahaymath]

(strict) translation: a=d !=0, c=0 translate by b/d

(strict) inversion: b=c !=0, a=0, d=0

(strict) dilation: b=0, c=0, d !=0, a/d real, dilate by a/d

(strict) rotation: b=0, c=0, d !=0, a/d absolute value = 1, rotate counterclockwise by arg(a/d)

[sjsawyer]

Shit.. I wish I found this video before my complex analysis exam...

[DanielEndy2]

Attribution! That's my quote below.

[wildersara]

"We are a single curved mirror bent so it can look at it's own infinite reflections of itself."

[Himax9]

Wow... beautiful.

[fonsidream]

Imaginary Numbers - Breakthrough in Mathematics

/watch?v=MO5LgzTsI58

[Quinzio]

Thank for this video.

Amazing

[Carthsting]

Creator of this must have lots of IQ

[ladyluvbuds]

how can this be applied to alchemy>?

[Illustratos]

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.

[cheezmunky]

I invented this.

[cusanusnicolas]

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...

[Upquark2]

Excellent!

[Distortedbackdrop]

Simply amazing, just watching this .. a picture holds a thousand words; well this video may just hold many more than I have ever witnessed in one spot! =P creativity plays a big role in this, but it just makes it easier to grasp a logical explanation of reality its self.

-Peace.

[zlakerbeetch]

beautiful :)

[Samuraionthewall]

Genius.

[danzigvssartre]

wonderful

[a2gregjockca]

holy fuck. i'm a hater troll but this shit is trippy ass cool. i have to admit. now fuck you, pricks, and die. thanks! LOL!

[AClarke2007]

@a2gregjockca You must be white, not coloured.

[MrBauchnabbel]

Nice visualization.

Music from Schubert:

Kinderszenen Nr. 1 (Von fremden Ländern und Menschen)

[ArchRascal]

Great visual demonstration, thanks for the video!

[RocketSpecialist]

wow how can 580 people dislike this video??! Probably thy failed maths in high school... Anyways, great video :)

[YanKizon]

fan-tas-tic!!!! XDDD

i'm not that good in maths (i'm a medical student °_°) but the explanation is so good that i easily understood!!

tks a lot!

[xandeudii2]

Very beautiful! I'd like to know what song was being played during the video.

[MrBauchnabbel]

@xandeudii2

Music from Schubert:

Kinderszenen Nr. 1 (Von fremden Ländern und Menschen)

[Umoyk]

@MrBauchnabbel

Its Schumann, not Schubert.

[GixxxerKim]

What I see/feel/experience during a salvia trip!

[jurisnake8]

That is a beautiful demonstration I wanted to make, but like I see someone else did it already:)

[rikonor]

bravo

[antideborah]

Beautiful!

[CharlesPLopezt]

huhutag und nacht träume ich davon dass sich jemnd findet der mich vor meiner langweile erlöst^^

[pixelcomet]

well done animation

[astrochicken]

Great video.

Assuming the sun is the lightsource and the moon is the sphere that redefines ones perception of reality.

[jamesblackburnlynch]

If we are only interested in the projection of the moon's reflection on the Earth, I see that.

[Hal2222222]

Relative lights shining in the dark. B, A, ware of Ix's and why's still in the lerch. Eve-n sheep can be counted with z's.

[Ajm4756]

Perfectly explained!! Well done!.

[Cityj0hn]

very informative

[jnar7]

wow... actually holds truths with fractals n how lsd works with the brain. craazy stuff

[latex123456]

i like it

[ferdem01]

Amazing, thanks!

[kikaro11]

this is like an acid trip for the math geeks

[alekk]

! illuminating !

[ThorkilKowalski]

Wicked

[jyandow]

Beautiful

[skidmark68]

Well done.  Thank you!

[SergeLktrRckr]

Beautiful animation....leaves all quiet clear!

[UOYxSSIMxI]

woahhhhhh.....

[TimJSwan89]

what music is that

[gerardpc]

@TimJSwan89 Schumann Kinderszenen 1

[TimJSwan89]

Thanks

[johnAshpool]

Complex analysis is too good to be real.

[signorellil]

LOL - that's a good one!

[Apollys]

Wow! Thats so sick.

[AussieVu92]

riht... another thing im sure to think about before going to bed... in other words ill be up for a couple of extra hours lol

[tybug31]

1:03.. tweety bird?? anyone? anyone?

[dys4ikH2]

@tybug31 thats awesome

[royalsteven]

This is a perfect way to torture you're brain O_O

[ortega24024]

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.

[signorellil]

Absolutely, "Visual Complex Analysis" is a GREAT book, shows how higher mathematics can be made exciting and understandable without any dumbing down.

[therockhopper]

Good job Jonathan.

[lupocephalic]

I find my ego is boosted by the fact that this has been in my favourites for at least a year now, and now that Magma has it as one of thier top 100 my foresight is vindicated.

Yay me.

[lloplop]

youre pathetic man

[lupocephalic]

Am I?

Goodness me. I'm ever so concerned about what you think is pathetic.

Please don't berate me any longer, I have a fragile constitution.

Fuck yerself fagboy.

[gallupintk]

is this one of walt disney,s i like the colours they make me dizzy. is there any dope involved?

[gbxgbxgbx]

@gallupintk lol

[rparl]

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.

[revscat]

@rparl Upvote for being the first intelligent comment I have ever encountered on YouTube.

[mikeriveraprod]

Although I have no knowledge about this field, I find the animation splendid! :)

[coolintruddle]

Neato

[ryandenki]

I wish teachers always explained math this way!

[TheRealmsOfGold]

@ryandenki Especially with Schumann in the background. :-)

[Meejoe27]

Incredible.

[lunchboxboy]

lol this is exactly what i imagined

[kaosgrunt]

wow.

[ArkBlitz]

The formula uses 4 variables, does this mean it's on R4?

[malevolentfreak]

No, actually the coefficients are complex numbers. Möbius transformations map the extended complex plane onto itself.

[typosensei]

No, it's 2 dimensional. f(z) and z are the two independent variables. The others are just coefficients and constants.

Furthermore you know it's two dimensional because it is shown on a 2-dimensional cartesian coordinate plane.

[ArkBlitz]

Ah, OK. The transformation maps an R4 plane in an R2 plane, then?

[Fensterplaetzchen]

z is a complex (2-dimensional) number z = x+iy.

the formula is

x'+iy' = z' = f(z) = (az+b) / (cz+d) = (ax+iay+b) / (cx+icy+d)

that is {a,b,c,d}+{x,y} --> {x',y'}

so you could intepret it as a mapping from R6 to R2.

But you usually won't because a,b,c,d are just parameters of the transformation. Like in the real world R3 you can rotate around 3 axes and translate in 3 dimensions, but you don't consider that a mapping from R9 to R3.

[Fensterplaetzchen]

oh, I just realized a mistake on my side in my first coment:

{a,b,c,d} are all complex, so in your way of thinking this would even be a mapping from R10 to R2.

I also wanted to add, that since for example you can't choose c=d=0, bacause that would mean to devide by zero, there are obviously points excluded so it wouldn't even be all of R10. (without giving a proove here, all points with ad-bc=0 are excluded, which includes c=d=0)

[Fensterplaetzchen]

to give you a bit background information, the transformations presented here do indeed include translation and rotation in 2 dimensions, as you can see in the video. the formula is f(z) = (az+b) / (cz+d) for {a=c=1, d=0} you get the translation f(z) = z+b in the same way, for |a|=1 (that is a is a phase) f(z) = a z is a rotation for a=real f(z) = a z is a scale transformation and the actually significent part about the möbius transformation, the inversion for {a=b=0,c=1,d=0} is f(z) = 1/z