Splitting spaces and their representation: Tilings from graphs and vice versa

Ruggero Gabbrielli

Abstract

Enumerating partitions of spaces can be a challenging task, especially if no order relationship is given. Using Delaney symbols, we ran a combinatorial search based on the computational tiling theory developed by Delaney, Delgado-Friedrichs, Dress and Huson aimed at the construction of periodic simple tilings of increasing complexity. Periodic tilings containing only tiles with 12 to 16 faces and 4, 5 and 6-sided faces have been considered. All Euclidean tilings with up to 11 crystallographically distinct kinds of vertices have been enumerated.