We discuss criteria for an orbifold to carry a flat geometry, i.e., one which is modelled on euclidean geometry, and show how these lead to a practical flatness test in the three-dimensional case. This has immediate applications in combinatorial tiling theory, namely in the classification of three-dimensional periodic tilings up to equivariant equivalence, and in the material sciences.
Received June 20, 2000, and in revised form October 25, 2000. Online publication May 4, 2001.
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Delgado-Friedrichs, O. Recognition of Flat Orbifolds and the Classification of Tilings in R 3 . Discrete Comput Geom 26, 549–571 (2001). https://doi.org/10.1007/s00454-001-0022-2