Once again this summer, I am working with three students in Marshall University's Research Experience for Undergraduates to study the Ising Model on the hyperbolic plane, especially the {3,7} regular tiling shown above. In the hyperbolic plane, all these triangles are identical, equilateral triangles with straight sides; the Poincare disk model, shown above, maps the entire infinite hyperbolic plane into a finite circle at the expense of severe distortion. The same thing happens when we try to represent the spherical Earth on a flat map.
The severe distortion can be postponed by allowing the "map" to curl up. The animation below was created by one of my students, Jesse Raffield, to represent a part of the hyperbolic plane. He used simulated annealing to try to keep the edges of the triangles as nearly as possible equal. The result is quite pleasing, and reminiscent of some biological shapes.
In a few weeks we should be able to use this or similar figures to visualize the processes we will be simulating on the hyperbolic plane.
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