It feels silly to reshare John’s posts because anyone following me is surely following him too, but I can’t help...
It feels silly to reshare John’s posts because anyone following me is surely following him too, but I can’t help myself when he posts about things I’ve worked on!
Originally shared by John Baez
The beauty of hyperbolic heptagons
Check out Roice Nelson's new picture! This picture lives in hyperbolic space, which been squashed down to a ball. The 'dents' are hyperbolic planes tiled by regular heptagons, each subdivided into 7 red and 7 blue triangles.
These triangles don't look like they have the same size - but in 3d hyperbolic space they do! The problem is that we've squashed hyperbolic space down to a ball. It's impossible to fit hyperbolic space in ordinary Euclidean space without doing violence to it.
Hyperbolic space has a group of symmetries called the Lorentz group. This is famous in special relativity: it's the group containing rotations and also Lorentz transformations.
The Lorentz group has symmetries that can map any of the red or blue triangles to any other one. If you include reflections in the Lorentz group, you also get symmetries that map red triangles to blue triangles. Let's do that.
If we then pick one triangle and call it our 'favorite', for any other triangle there's exactly one symmetry in Lorentz group mapping our favorite to that other triangle. So, the triangles here are actually a picture of a group contained in the Lorentz group!
This group is called {7,3,3}. Why? Because we're starting with the hyperbolic plane tiled by 7-gons with 3 meeting at each vertex. This called the {7,3} tiling. But then, to form the picture here, we take lots of these tilings, and make sure 3 meet along each edge of each heptagon. So, the picture here is called the {7,3,3} honeycomb - and its symmetry group is also called {7,3,3}.
You can buy a 3d model of this shape from Roice Nelson! Check it out:
https://plus.google.com/u/0/+RoiceNelson/posts/dMd8MvPmG6U
You can learn more about the math here:
https://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/
#geometry
Paraphrasing Oscar Wilde, "the only thing you should never resist is temptation."
ReplyDeleteWould you be willing to give me permission to add your picture to this page?
blogs.ams.org - {7,3,3} Honeycomb
That blog is dormant but your picture would help explain the Coxeter group {7,3,3}.
John Baez, ha! And of course... You have a permanent “knock yourself out” license on any math images I make :)
ReplyDeleteOne thing that's not obvious to me is how you determine the positions of the hyperbolic planes you're tiling. I guess that's the step involved in going from the 7,3 tiling to the 7,3,3 honeycomb?
ReplyDeleteMiguel Carrion Alvarez, I coathored a paper that describes the construction.
ReplyDeletehttps://arxiv.org/abs/1511.02851
In particular, see appendix A.2. The construction comes down to defining the geometry of a fundamental simplex. Then that simplex is used to generate the honeycomb via recursive reflections in its 4 facets.
One thing I thought I'd note is that the {7,3} tilings in this image don't live on geodesic planes (what probably comes to mind by the wording “hyperbolic plane”). They behave more like polyhedra, where adjacent heptagons meet with a dihedral angle of 120 degrees. All the vertices of a given {7,3} tiling live on a surface that is the analogue of a "hypercycle", i.e. a surface equidistant from a geodesic plane.
https://en.wikipedia.org/wiki/Hypercycle_(geometry)