Topological Data Analysis quantifies persistent structures hidden in unorganized data such clouds of points given by only pairwise distances. Our group in the Materials Innovation Factory at the University of Liverpool (UK) is developing new methods to quantify a similarity between solid crystalline materials (briefly, crystals). A periodic crystal is an infinite arrangement of atoms or molecules obtained by translating a unit cell (a non-rectangular box) in 3 linearly independent directions. The Crystal Structure Prediction aims to discover new crystals that have a given chemical composition and several target properties, most importantly the energy of a thermodynamic stability. The key data problem is the enormous ambiguity of crystal representations via conventional unit cells, because many different unit cells can define identical (up to a rigid motion) or nearly identical crystals. We propose a new classification of crystals by geometric invariants that continuously change under perturbations (atomic vibrations of atoms). These invariants will provide a well-defined distance between crystals for visualising large datasets of simulated crystals by varying a threshold for similarity.