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

Engel, Philip¹ ; Smillie, Peter¹

¹ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 22 (2018) no. 5, pp. 2839-2864.

Résumé

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 $\Lambda \subset {ℂ}^{1,9}$. First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice $\Lambda$. Next, we integrate the Siegel theta function for $\Lambda$ 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.

Détail

DOI : 10.2140/gt.2018.22.2839

Classification : 05C30, 32G15, 53C45, 11F27
Keywords: triangulations, nonnegative curvature, sphere, tiling, modular form, Thurston, polyhedra, shapes of polyhedra

Bibliographie

[1] A D Alexandrov, Convex polyhedra, Springer (2005) | DOI

[2] D Allcock, The Leech lattice and complex hyperbolic reflections, Invent. Math. 140 (2000) 283 | DOI

[3] D Allcock, New complex- and quaternion-hyperbolic reflection groups, Duke Math. J. 103 (2000) 303 | DOI

[4] T Basak, The complex Lorentzian Leech lattice and the bimonster, J. Algebra 309 (2007) 32 | DOI

[5] R E Borcherds, Automorphic forms on Os+2,2(R) and infinite products, Invent. Math. 120 (1995) 161 | DOI

[6] J H Bruinier, K Ono, R C Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008) 673 | DOI

[7] P Deligne, G D Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986) 5 | DOI

[8] R Laza, Triangulations of the sphere and degenerations of K3 surfaces, preprint (2008)

[9] E Looijenga, Uniformization by Lauricella functions — an overview of the theory of Deligne–Mostow, from: "Arithmetic and geometry around hypergeometric functions" (editors R P Holzapfel, A M Uludağ, M Yoshida), Progr. Math. 260, Birkhäuser (2007) 207 | DOI

[10] C T Mcmullen, The Gauss–Bonnet theorem for cone manifolds and volumes of moduli spaces, preprint (2013)

[11] G D Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986) 91 | DOI

[12] K Ono, Unearthing the visions of a master : harmonic Maass forms and number theory, from: "Current developments in mathematics" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2009) 347

[13] É Picard, Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, Ann. Sci. École Norm. Sup. 10 (1881) 305 | DOI

[14] R E Schwartz, Notes on shapes of polyhedra, preprint (2015)

[15] C L Siegel, Indefinite quadratische Formen und Funktionentheorie, II, Math. Ann. 124 (1952) 364 | DOI

[16] W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: "The Epstein birthday schrift" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. Publ. (1998) 511 | DOI

[17] W T Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962) 21 | DOI