  Exploring nonminimal-rank quasicrystals, which have symmetries that can be
found in both periodic and aperiodic crystals, often provides new insight into
the physical nature of aperiodic long-range order in models that are easier to
treat. Motivated by the prevalence of experimental systems exhibiting aperiodic
long-range order with hexagonal and trigonal symmetry, we introduce a generic
two-parameter family of 2-dimensional quasiperiodic tilings with such
symmetries. We focus on the special case of trigonal and hexagonal Fibonacci,
or golden-mean, tilings, analogous to the well studied square Fibonacci tiling.
We first generate the tilings using a generalized version of de Bruijn's dual
grid method. We then discuss their interpretation in terms of projections of a
hypercubic lattice from six dimensional superspace. We conclude by
concentrating on two of the hexagonal members of the family, and examining a
few of their properties more closely, while providing a set of substitution
rules for their generation.
