The plane and space groups are groups that specify how to tile two- or three-dimensional Euclidean space with a shape: They enumerate all possible waysin which a shape can be isometrically replicated across the space. I will describe how to explicitly compute approximate eigenfunctions of the Laplace-Beltrami operator on the orbifold defined by any such group.

These eigenfunctions provide a complete L2 basis of all functions on two- or three-dimensional space that are (i) continuous and (ii) periodic with respect to the group. The basis allows us to represent functions that arise as quantum observables of crystalline solids or in mechanical meta-materials, to generate random functions respecting the group symmetry, and to compute a form Fourier transform defined by the group. I will also explain how to construct an approximation to the orbifold in a higher-dimensional space and a map onto this embedding. Composing this map with function representations used in machine learning (say a neural network or kernel function) results in machine learning models that respect the group symmetry. This is joint work with Peter Orbanz.