The Beautiful Function

June 26, 2007 2:13 AM

Möbius Transformations Revealed [yöutube alert] See also: Stereographic Projection Demo.
posted by chuckdarwin (17 comments total)

Your Geometry teacher probably already told you all about it one morning when you weren't paying attention, but a basically Möbius Transformation is "a map of the complex plane to itself in which a point "z" is mapped to a point "w" by w=(az+b)/(cz+d), where a,b,c, and d are complex numbers and ad-bc does not equal zero."

Got that? There's going to be a pop quiz on this tomorrow, so pay attention.

Propers to tdj and his beautiful blog for the lesson.
posted by chuckdarwin at 2:16 AM on June 26, 2007

For a brief moment while watching that I completely understood curvature of space, relativity, and quantum theory. Then I lost it, poof! What a post. Blowed my mind. thanks.
posted by Blingo at 2:56 AM on June 26, 2007

Science!
posted by flippant at 4:49 AM on June 26, 2007

That's nicely done.
posted by gleuschk at 4:51 AM on June 26, 2007

Yes.
posted by DU at 4:56 AM on June 26, 2007

Very nice indeed.
posted by languagehat at 5:24 AM on June 26, 2007

So can he show us what a Möbius transform looks like when applied to 3D objects? I'd like to see how a complex model looks as it's turned inside out.
posted by Tzarius at 5:34 AM on June 26, 2007

Fantastic. For number theory fans, Mobius transformations are crucial to the theory of modular forms -- sort of a higher-dimensinal version of trigonometry in which the zoo of transformations of the plane depicted in this video replaces simple translations of the line.
posted by escabeche at 6:00 AM on June 26, 2007

I am currently studying for a PhD qualifying examination in analysis, the field where most students first encounter these types of Moebius transformations.

The animation was very pretty, and a good depiction of what's going on.
posted by King Bee at 6:16 AM on June 26, 2007

Only loosely related, but you can find a couple of classic math videos on Google now: Outside In (sphere eversions) and Not Knot (hyperbolic knot complements).

Then there's the Indra's Pearls book and this batch of Java visualizations (check out the punctured torus / Kleinian group limit set app).
posted by Wolfdog at 6:40 AM on June 26, 2007 [3 favorites]

Thanks, Wolfdog. Cheers, everybody; this is mainly down to me having such a shower of smart-arses on my f-list.
posted by chuckdarwin at 6:45 AM on June 26, 2007

Ever since that Björk thread, every time I see an "o" with an umlaut over it, I see a little surprised face.

ö

See?

It's probably good I'm not German.
posted by LordSludge at 6:48 AM on June 26, 2007 [1 favorite]

Very nice. Thanks.
posted by OmieWise at 7:02 AM on June 26, 2007

Beautiful - thank you!
Would that this had existed when I was trying (and mostly failing) to learn such things.
posted by speug at 7:37 AM on June 26, 2007

that was really lovely. i think the part of my brain that likes ayn rand just had an orgasm.
posted by es_de_bah at 8:20 AM on June 26, 2007

i think the part of my brain that likes ayn rand just had an orgasm.

That's so disgusting in so many ways that I feel like I need a shower.
posted by OmieWise at 8:35 AM on June 26, 2007

Famous Surfaces !
posted by hortense at 9:34 AM on June 26, 2007