  There seems to be a one to one correspondence between the phases of atomic
and molecular matter (AMOM) and vortex matter (VM) in superfluids and
superconductors. Crystals, liquids and glasses have been experimentally
observed in both AMOM and VM. Here, we propose a vortex quasi-crystal state
which can be stabilized due to boundary and surface energy effects for samples
of special shapes and sizes. For finite sized pentagonal samples, it is
proposed that a phase transition between a vortex crystal and a vortex
quasi-crystal occurs as a function of magnetic field and temperature as the
sample size is reduced.
