By Ian Short
In 2018 Demonet et al observed that there is essentially only one three-dimensional positive integer tiling with the property that each cross-section is an SL2-tiling (in which all squares have determinant 1). Inspired by Bhargava’s work on binary quadratic forms, we describe a model for the more general class of three-dimensional integer tilings, which we call hypertilings, with the property that each cross-section is a two-dimensional integer tiling (in which all squares have the same determinant). This model comprises a Bhargava cube and a triple of paths in the weighted Farey graph. The hypertiling entries are encoded geometrically by lambda lengths between horocycles or arithmetically by data from the weighted Farey graph. This is joint work with Oleg Karpenkov, Matty van Son, and Andrei Zabolotskii.
By Cruz Godar
By Jonathan Love
By Robert Corless
By Sébastien Labbé
By Ian Short
By Bernat Espigule
By Vlerë Mehmeti
By Karl-Mikael Perfekt
