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Quasi-crystals are aperiodic structures with crystallographic properties that are not compatible with that of a single unit cell. Their discovery in a metallic alloy more than four decades ago has required a full reconsideration of our definition of a crystal structure. Quasi-crystalline structures have also been discovered at much larger length scales in different microscopic systems for which the size of the elementary building blocks ranges from the nanometre to the micrometre scale. Here we report the first experimental observation of spontaneous quasi-crystalline self-assembly at the millimetre scale. This result is obtained in a fully athermal system of macroscopic spherical grains vibrating on a substrate. Starting from a liquid-like disordered phase, the grains begin to locally arrange into three types of square and triangle tile that eventually align, forming an eight-fold symmetric quasi-crystal that has been predicted in simulation but not yet experimentally observed in non-atomic systems. These results not only demonstrate an alternative route for the spontaneous assembly of quasi-crystals but are of fundamental interest for the connection between equilibrium and non-equilibrium statistical physics.

We discuss the situation where attractive and repulsive portions of the interparticle potential both contribute significantly to glass formation. We introduce the square-well potential as prototypical model for this situation, and rejectthe Baxter model as a useful model for comparison to experiment on glasses, based on our treatment within mode coupling theory. We present explicit results for various well widths, and show that, for narrow wells, there is a useful analytical formula that would be suitable for experimentalists working in the field of colloidal science. We raise the question as to whether, in a more exact treatment, the sticky-sphere limit might have an infinite glass transition temperature or a high but finite one.