Hexagonal tiling of the two-dimensional sphere

A. V. Akopyan, J. Crowder, H. Edelsbrunner, R. Guseinov
(Institute of Science and Technology Austria)
Submitted to Nature Mathematics on April 1, 2015

It is well known that the Euler characteristic of the boundary complex of every even-dimensional convex polytope vanishes. In other words, there are equally many odd-dimensional faces as there are even-dimensional faces. This opens the possibility of a symmetric design on high-dimensional spheres. Searching through the dimensions 4 to 24, we have found slices of the well-known Leech lattice that give a multi-cover of the two-dimensional sphere into hexagons. Using extensive computer calculations, we found the parameters under which the layers of the multi-cover match up. In other words, the layers converge to glue to a regular hexagonal tiling of the sphere: the displayed hexasphere design in three dimensions. Please do not rotate the hexasphere with your mouse unless you are not convinced that there are no hidden pentagons in the design.

Computer models show that the hexasphere can be physically realized with carbons forming a mechanically stable configuration that is capable to withstand extreme pressure and temperature. Further applications of the patent protected invention include aero-dynamically superior golf balls and symmetric laser-light distributions, which we hope is a modest step towards the dream of functioning atomic fusion chambers.
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Interactive model of hexasphere.