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Melting is well understood in terms of the Lindemann criterion, essentially stating that crystalline materials melt when the thermal vibrations of their atoms become such vigorous that they shake themselves free of the binding forces. However, how about another common type of solids: glasses, where the nature of the solid-liquid crossover is highly controversial? The Lindemann criterion implies that the thermal expansion coefficients alpha of crystals are inversely proportional to their melting temperatures. Here we find that, unexpectedly, alpha of glasses decreases much stronger with increasing glass-transition temperature Tg marking the liquid-solid crossover in this material class. However, scaling alpha by the fragility m, a measure of particle cooperativity, restores the proportionality, i.e., alpha/m ~ 1/Tg. Obviously, for a glass to become liquid, it is not sufficient to simply overcome the interparticle binding energies. Instead, more energy has to be invested to break up the typical cooperative particle network which is considered a hallmark feature of glassy materials. Surprisingly, alpha of the liquid phase reveals similar anomalous behaviour and is universally enhanced by a constant factor of ~3. The found universalities allow estimating glass-transition temperatures from thermal expansion and vice versa.