  An ellipsoid, the simplest non-spherical shape, has been extensively used as
models for elongated building blocks for a wide spectrum of molecular,
colloidal and granular systems. Yet the densest packing of congruent hard
ellipsoids, which is intimately related to the high-density phase of many
condensed matter systems, is still an open problem. We discover a novel dense
crystalline packing of ellipsoids containing 24 particles with a
quasi-square-triangular (SQ-TR) tiling arrangement, whose packing density
{\phi} exceeds that of the SM2 crystal for aspect ratios {\alpha} in (1.365,
1.5625), attaining a maximal {\phi} about 0.75806 at {\alpha} = 93/64. We show
that SQ-TR phase is thermodynamically stable at high densities over the
aforementioned {\alpha} range and report a novel phase diagram for self-dual
ellipsoids. The discovery of SQ-TR crystal suggests novel organizing principles
for non-spherical particles and self-assembly of colloidal systems.
