Algebraic & Geometric Topology

The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

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A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of 2π3 but less than 2π. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere M satisfies the following properties:

(1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of M;

(2) it embeds isometrically in the 3–dimensional Euclidean space as the boundary of a tetrahedron; and

(3) there is a simple closed geodesic γ in M such that a fractional Dehn twist along γ converts M to the double of a parallelogram.

Article information

Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1323-1343.

Received: 31 October 2010
Revised: 26 January 2011
Accepted: 15 February 2011
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M15: Relations with graph theory [See also 05Cxx]

singular Euclidean surfaces moduli spaces


Cruz-Cota, Aldo-Hilario. The moduli space of hex spheres. Algebr. Geom. Topol. 11 (2011), no. 3, 1323--1343. doi:10.2140/agt.2011.11.1323.

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