  All hard, convex shapes are conjectured by Ulam to pack more densely than
spheres, which have a maximum packing fraction of {\phi} = {\pi}/\sqrt18 ~
0.7405. For many shapes, simple lattice packings easily surpass this packing
fraction. For regular tetrahedra, this conjecture was shown to be true only
very recently; an ordered arrangement was obtained via geometric construction
with {\phi} = 0.7786, which was subsequently compressed numerically to {\phi} =
0.7820. Here we show that tetrahedra pack much better than this, and in a
completely unexpected way. Following a conceptually different approach, using
thermodynamic computer simulations that allow the system to evolve naturally
towards high-density states, we observe that a fluid of hard tetrahedra
undergoes a first-order phase transition to a dodecagonal quasicrystal, which
can be compressed to a packing fraction of {\phi} = 0.8324. By compressing a
crystalline approximant of the quasicrystal, the highest packing fraction we
obtain is {\phi} = 0.8503. If quasicrystal formation is suppressed, the system
remains disordered, jams, and compresses to {\phi} = 0.7858. Jamming and
crystallization are both preceded by an entropy-driven transition from a simple
fluid of independent tetrahedra to a complex fluid characterized by tetrahedra
arranged in densely packed local motifs that form a percolating network at the
transition. The quasicrystal that we report represents the first example of a
quasicrystal formed from hard or non-spherical particles. Our results
demonstrate that particle shape and entropy can produce highly complex, ordered
structures.
