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We show how many mesomorphic states illustrate the following general scheme: The symmetry group of an equilibrium state of Euclidean-invariant quantum statistical mechanics is a subgroup H of the Euclidean group E such that the orbit E/H is compact. Moreover, the homotopy groups of E/H yield a classification of the topologically stable defects and configurations of these ordered media. This suggests a predictive value of this scheme for yet unobserved media and for defects.

Spontaneous Breaking of Euclidean Invariance and Classification of Topologically Stable Defects and Configurations of Crystals and Liquid Crystals