Wacky Conformal Transformation



Wacky Conformal Transformation

I was trying to conformally stretch out the band model to more of the euclidean plane, and ran across this strange thing.

I don't understand all the features, but it's pretty.  It was generated by first applying the band model transformation to the disk, rotating that 90 degrees and scaling, then applying the band model transformation again.

Anybody know of a conformal transformation of the Poincaré disk that maps to a large area of the euclidean plane while keeping hyperbolic tiles approximately the same size?  I suspect it may not be possible, since the amount of material one needs to fit in grows exponentially.  If it is possible, I bet such a thing would need to be undulating, with some areas of smaller tiles and other areas with larger tiles.

Reference
Vladimir Bulatov's work, "Conformal Models of Hyperbolic Geometry", bulatov.org/math/1001/

Comments

  1. Owen MareshApril 20, 2016 at 4:00 PM

    those are essential singularities

    ReplyDelete
  2. Owen MareshApril 20, 2016 at 8:21 PM

    And, on further thinking about this -- it's kind of striking how the upper and lower bands look like fundamental domains from the Dedekind Tesselation/modular group/what have you.

    ReplyDelete
  3. Vladimir BulatovApril 21, 2016 at 6:07 AM

    Roice Nelson this disk to band transform actually maps the complex plane with 2 horizontal cuts (1,oo) and (-oo, -1) into infinite horizontal band (-pi < y < pi). Try to apply that transform to some of your hyperbolic tessellations. Points -1 and 1 will be stretched to +-infinity, so make sure to move  tessellation that it has limit point in -1 and 1.

    ReplyDelete

Post a Comment

Popular posts from this blog

Hyperbolic Hopf Fibrations

Image
Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful.  Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3.  In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case.  There is a 2-parameter family of them, plus one additional fibration that does not fit the family.  As with S^3, fibers in the H^3 cases are geodesics.  They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration.  I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations belo...
Read more

76 Unique Honeycombs

76 Unique Honeycombs Last weekend, Tom Ruen and I hit the milestone of uploading to wikipedia at least one image for 9 families of compact, Wythoffian, uniform H3 honeycombs, a total of 76 unique honeycombs. http://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space You can easily browse all the images on my wiki user page: http://commons.wikimedia.org/wiki/User:Roice3 - Compact means the cells are finite in extent.  - Wythoffian means we can generate them using a kaleidoscopic construction, that is by reflecting in mirrors.    - Uniform means they are vertex transitive and have uniform polyhedral cells. There may even be more honeycombs that meet all these criteria, I don't know. ( update: see Tom's comment below! )  I do know there are hundreds more which don't meet one or more of these criteria, many undiscovered.  In fact, there are infinitely more because there are some infinite families of honeycombs.  wendy krieger continues to discover and enumerate more...
Read more