Stretched Model of the Hyperbolic Plane

Stretched Model of the Hyperbolic Plane

David Eppstein shared some very cool fisheye disk models of euclidean geometry recently.

I generated some images of euclidean tilings in one of the models, as well as images for spherical tilings in a similar disk model.

Then I realized we could apply exactly the opposite transformation to the Poincaré disk model of the hyperbolic plane to get a model covering the entire complex plane. That's what these images show for the {7,3} hyperbolic tiling (the dual {3,7} tiling is overlaid as well). Tiles stretch out dramatically in the radial direction, but seem to retain constant tangential width.

This non-conformal model may not be terribly useful, but there are some interesting patterns in the pictures. Clear fractal structure shows up in radial slices, and zooming the entire image by certain ratios makes this especially evident. The Fibonacci numbers appeared when I started counting visual groups of tiles spiraling out from the origin.

One image renders the tiling out to 25x the size of the unit disk, the other to 100x. The coloring and line width is slightly different in the two, and by playing with these settings different features tend to stick out. These were just a couple of my favorites. The lightened hyperbolas struck me. Can you see them?

One challenge with rendering these is unintended aliasing because the features become tiny as you render more of the plane. That's why the 100x image is so large, 64 megapixels and 30MB.

Anyone know of this model being described in the literature? Maybe Saul Schleimer or Henry Segerman?

Update: I had trouble getting G+ to post these how I wanted, and the images seemed to be downsampled. The full res image are available in the following album, and zooming works better there for me:


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