  Symmetry sharing facilitates coherent interfaces which can transition from
periodic to quasiperiodic structures. Motivated by the design and construction
of such systems, we present hexagonal quasiperiodic tilings with a single
edge-length which can be considered as decorations of a periodic lattice. We
introduce these tilings by modifying an existing family of golden-mean trigonal
and hexagonal tilings, and discuss their properties in terms of this wider
family. Then, we show how the vertices of these new systems can be considered
as decorations or sublattice sets of a periodic triangular lattice. We conclude
by simulating a simple Ising model on one of these decorations, and compare
this system to a triangular lattice with random defects.
