  We present the construction of a dense, quasicrystalline packing of regular
tetrahedra with icosahedral symmetry. This quasicrystalline packing was
achieved through two independent approaches. The first approach originates in
the Elser-Sloane 4D quasicrystal. A 3D slice of the quasicrystal contains a few
types of prototiles. An initial structure is obtained by decorating these
prototiles with tetrahedra. This initial structure is then modified using the
Elser-Sloane quasicrystal itself as a guide. The second approach proceeds by
decorating the prolate and oblate rhombohedra in a 3-dimensional Ammann tiling.
The resulting quasicrystal has a packing density of 59.783%. We also show a
variant of the quasicrystal that has just 10 "plane classes" (compared with the
190 of the original), defined as the total number of distinct orientations of
the planes in which the faces of the tetrahedra are contained. This small
number of plane classes was achieved by a certain "golden rotation" of the
tetrahedra.
