The number of convex tilings of the sphere by triangles, squares, or hexagons

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most $6$ , $4$ , or $3$ 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 $Λ \subset ℂ^{1, 9}$ . First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice $Λ$ . Next, we integrate the Siegel theta function for $Λ$ 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.

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Keywords

triangulations, nonnegative curvature, sphere, tiling, modular form, Thurston, polyhedra, shapes of polyhedra

Mathematical Subject Classification 2010

Primary: 05C30, 32G15, 53C45

Secondary: 11F27

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

Authors

Philip Engel
Department of Mathematics Harvard University Cambridge, MA United States

Peter Smillie
Department of Mathematics Harvard University Cambridge, MA United States

Volume 22, issue 5 (2018)

Philip Engel and Peter Smillie

Abstract