Field
Order without repetition
Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.
Penrose tilings
Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.
Inflation, and where the golden ratio comes from
A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio.
Order is not periodicity
For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.