  Quasicrystals provide a fascinating class of materials with intriguing
properties. Despite a strong potential for numerous technical applications, the
conditions under which quasicrystals form are still poorly understood.
Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry
but no single example with 7 or 9-fold symmetry has ever been observed. Here we
report on geometrical constraints which impede the formation of quasicrystals
with certain symmetries in a colloidal model system. Experimentally, colloidal
quasicrystals are created by subjecting micron-sized particles to
two-dimensional quasiperiodic potential landscapes created by n=5 or seven
laser beams. Our results clearly demonstrate that quasicrystalline order is
much easier established for n = 5 compared to n = 7. With increasing laser
intensity we observe that the colloids first adopt quasiperiodic order at local
areas which then laterally grow until an extended quasicrystalline layer forms.
As nucleation sites where quasiperiodicity originates, we identify highly
symmetric motifs in the laser pattern. We find that their density strongly
varies with n and surprisingly is smallest exactly for those quasicrystalline
symmetries which have never been observed in atomic systems. Since such high
symmetry motifs also exist in atomic quasicrystals where they act as
preferential adsorption sites, this suggests that it is indeed the deficiency
of such motifs which accounts for the absence of materials with e.g. 7-fold
symmetry.
