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The
number of convex tilings of the sphere by triangles, squares,
or hexagons
Philip Engel and Peter Smillie |
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Geometry & Topology 22 (2018) 2839–2864
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Abstract |
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A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most , , or polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main results are explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these formulas, we build on work of Thurston, who showed that the convex triangulations correspond to orbits of vectors of positive norm in a Hermitian lattice . First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice . Next, we integrate the Siegel theta function for to produce a modular form whose Fourier coefficients encode the weighted number of tilings. Finally, we determine the formulas using finite-dimensionality of spaces of modular forms. |
Keywords
triangulations, nonnegative curvature, sphere, tiling,
modular form, Thurston, polyhedra, shapes of polyhedra
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Mathematical Subject Classification 2010
Primary: 05C30, 32G15, 53C45
Secondary: 11F27
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References |
Publication
Received: 27 February 2017
Revised: 16 January 2018
Accepted: 4 March 2018
Published: 1 June 2018
Proposed: Ian Agol
Seconded: John Lott, Anna Wienhard
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Authors |
| Philip Engel | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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| Peter Smillie | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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