  Glass transition is accompanied by a rapid growth of the structural
relaxation time and a concomitant decrease of configurational entropy. It
remains unclear whether the transition has a thermodynamic origin, and whether
the dynamic arrest is associated with the growth of a certain static order.
Using granular packing as a model hard-sphere glass, we show the glass
transition as a thermodynamic phase transition with a "hidden" polytetrahedral
order. This polytetrahedral order is spatially correlated with the slow
dynamics. It is geometrically frustrated and has a peculiar fractal dimension.
Additionally, as the packing fraction increases, its growth follows an
entropy-driven nucleation process, similar to that of the random first-order
transition theory. Our study essentially identifies a long-sought-after
structural glass order in hard-sphere glasses.
