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/
those are essential singularities
ReplyDeleteAnd, 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.
ReplyDeleteRoice 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.
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