# Conformal Models of Hyperbolic Geometry

March 12, 2012 8:22 AM Subscribe

Whoa! No idea of the maths, but the animations are wonderful.

posted by carter at 8:53 AM on March 12, 2012

posted by carter at 8:53 AM on March 12, 2012

I took a course in some of this stuff. We never got to the

posted by BungaDunga at 9:19 AM on March 12, 2012

*multiple armed banded whatsit*which is just incredibly cool.*A Julia model of the hyperbolic plane*?!?!posted by BungaDunga at 9:19 AM on March 12, 2012

*A Julia model of the hyperbolic plane?!?!*

possibly Gaston Julia, who, afair, taught Pierre Mandlebrot.

posted by marienbad at 9:38 AM on March 12, 2012

Also, (sorry for double post) I also seem to recall that Poincare taught Julia. (Wiki says otherwise)

Poincare was truly brilliant; from wiki:

"Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity."

posted by marienbad at 9:42 AM on March 12, 2012

Poincare was truly brilliant; from wiki:

"Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity."

posted by marienbad at 9:42 AM on March 12, 2012

marienbad: Pierre Mandelbrot? Surely you mean Benoît?

posted by crazy_yeti at 9:48 AM on March 12, 2012

posted by crazy_yeti at 9:48 AM on March 12, 2012

*possibly Gaston Julia*

Well yes, it's a mapping that looks like the Julia set, but I have no idea what that actually means except that it looks

*awesome*

posted by BungaDunga at 9:57 AM on March 12, 2012

Oh, I figured this would be about airliners.net's declarations of the Airbus 380 being the most amazing bestest thing EVAR or the most absolute colossal waste of money in the history of the universe.

I'll get my coat.

posted by ricochet biscuit at 10:45 AM on March 12, 2012

I'll get my coat.

posted by ricochet biscuit at 10:45 AM on March 12, 2012

BungaDunga: The Riemann mapping theorem states that it is possible to construct a conformal map from the disc to any other simply-connected open proper subset of the complex plane. (This map is essentially unique: if you specify the value and derivative at a single point in the domain, there is only one such map). Since Julia sets are simply connected, one can construct a conformal map from the disc to the Julia set - this is basically what Bulatov is doing here. However he's mapping the disk to the OUTSIDE of the Julia set instead of the inside ... even though the outside is not simply connected, this is still possible - one just uses an inversion z -> 1/z to exchange the inside and outside of the unit disk.

posted by crazy_yeti at 11:04 AM on March 12, 2012 [2 favorites]

posted by crazy_yeti at 11:04 AM on March 12, 2012 [2 favorites]

*"marienbad: Pierre Mandelbrot? Surely you mean Benoît?"*

posted by crazy_yeti

oops! Yes. sorry for the error.

posted by marienbad at 12:27 PM on March 12, 2012

Bathsheba Grossman has been printing 3D mathematical models in metal for quite some time now as well.

posted by TheCoug at 11:09 PM on March 12, 2012

posted by TheCoug at 11:09 PM on March 12, 2012

It's sad in a way that hyperbolic geometry isn't better known. It's a consequence of Thurston's conjecture (which was proved by Perelman) that most 3-dimensional worlds (aka compact 3-manifolds) have a hyperbolic geometry, so maybe you're in a hyperbolic world now.

Maybe this is easier to understand one dimension down. Most 2-dimensional worlds are hyperbolic. Let me back up. By "world", I mean a compact manifold. By "compact manifold", I roughly mean a space that can be mapped with an atlas of charts using a book of maps with only finitely many pages. The idea is the same as mapping the surface of the Earth using an atlas that has a finite number of pages.

In the case of the Earth itself, the usual mappings - the ones you see in actual atlases - are typically Mercator projections. These are conformal mappings, which means that angles are correct, but they aren't isometric, which means that the Arctic and Antarctica look way too big. This distortion happens because the Earth itself is not flat but rather spherical. If we want to be accurate about measuring distances between things on a spherical surface, then a flat geometry - the kind of Euclidean geometry you may have learned in school - is not going to work. Instead we would have to use spherical geometry. In spherical geometry the most natural "lines" aren't flat lines that go on forever, but rather great circles - such as lines of latitude and longitude - that eventually circle around to their beginning. An interesting fact of spherical geometry is that the sum of angles in a triangle is more than 180 degrees.

There is only one compact 2-dimensional manifold with a spherical geometry, and that is the sphere. And there is only one compact 2-dimensional manifold with the familiar Euclidean geometry that you learned in school, where the sum of angles in a triangle is exactly 180 degrees. That particular 2-dimensional world is a torus... a donut shape. If you lived in a world shaped like a torus, you could create an atlas that used flat pages without introducing any distortion into the depicted distances, because the torus can be given the same Euclidean geometry that flat pages have.

But there are infinitely many other 2-dimensional worlds. There is the double torus and the triple torus and so on (more and more surfaces with more and more "holes"). All of these others can only be given hyperbolic geometries. In a hyperbolic geometry, the sum of the angles is less than 180 degrees.

In three dimensions there are five other geometries that are possible, but honestly, hyperbolic kind of runs away with the show. It's a pity this isn't better known.

posted by twoleftfeet at 1:02 AM on March 13, 2012 [2 favorites]

Maybe this is easier to understand one dimension down. Most 2-dimensional worlds are hyperbolic. Let me back up. By "world", I mean a compact manifold. By "compact manifold", I roughly mean a space that can be mapped with an atlas of charts using a book of maps with only finitely many pages. The idea is the same as mapping the surface of the Earth using an atlas that has a finite number of pages.

In the case of the Earth itself, the usual mappings - the ones you see in actual atlases - are typically Mercator projections. These are conformal mappings, which means that angles are correct, but they aren't isometric, which means that the Arctic and Antarctica look way too big. This distortion happens because the Earth itself is not flat but rather spherical. If we want to be accurate about measuring distances between things on a spherical surface, then a flat geometry - the kind of Euclidean geometry you may have learned in school - is not going to work. Instead we would have to use spherical geometry. In spherical geometry the most natural "lines" aren't flat lines that go on forever, but rather great circles - such as lines of latitude and longitude - that eventually circle around to their beginning. An interesting fact of spherical geometry is that the sum of angles in a triangle is more than 180 degrees.

There is only one compact 2-dimensional manifold with a spherical geometry, and that is the sphere. And there is only one compact 2-dimensional manifold with the familiar Euclidean geometry that you learned in school, where the sum of angles in a triangle is exactly 180 degrees. That particular 2-dimensional world is a torus... a donut shape. If you lived in a world shaped like a torus, you could create an atlas that used flat pages without introducing any distortion into the depicted distances, because the torus can be given the same Euclidean geometry that flat pages have.

But there are infinitely many other 2-dimensional worlds. There is the double torus and the triple torus and so on (more and more surfaces with more and more "holes"). All of these others can only be given hyperbolic geometries. In a hyperbolic geometry, the sum of the angles is less than 180 degrees.

In three dimensions there are five other geometries that are possible, but honestly, hyperbolic kind of runs away with the show. It's a pity this isn't better known.

posted by twoleftfeet at 1:02 AM on March 13, 2012 [2 favorites]

I have no idea how I've got this.

Love this slide.

posted by Barry B. Palindromer at 8:10 AM on March 13, 2012 [1 favorite]

Love this slide.

posted by Barry B. Palindromer at 8:10 AM on March 13, 2012 [1 favorite]

Mmmmmm, now THIS is some art I can enjoy. Can't afford, but can enjoy.

posted by BlueHorse at 1:59 PM on March 13, 2012

posted by BlueHorse at 1:59 PM on March 13, 2012

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Now I'm trying to figure out how I can justify ordering one of those gorgeous pendants to my partner, other than that I WANT ONE NOW. "But look, honey, it illustrates important mathematical theory three-dimensionally!" Yeah, that'll work.

posted by kinnakeet at 8:48 AM on March 12, 2012