Consider the dual of a high-order geodesic dome: 12 pentagons (a dome based on the regular dodecahedron) and lots and lots of hexagons. Because it is large, small regions of the surface are nearly flat, easy to project into a plane. What does a small region in the neighborhood of a pentagon look like? The hexagons immediately adjacent to the pentagon are highly distorted compared to a regular hexagon. What about the next ring of hexagons? How quickly does the distortion dissipate? On different sized domes, is it a function of hexagon count from a pentagon, or a function of angle measured from the center of the sphere? The region has 5-fold symmetry. If you were an ant on such a dome, how easy would it be to determine where you are (modulo symmetry) based on local measurements (lengths, angles) of the distortion of the hexagons in your neighborhood?
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