Theme
The thread: Order without repetition
Periodicity and order are not the same thing, and separating them is what quasicrystals forced. A pattern can be perfectly determined and never repeat.
Order without repetition
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.
Order without repetitionInflation, 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 without repetitionOrder 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.