Aperiodic — the ladder
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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.
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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.
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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.
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Matching rules, and what actually forces aperiodicity
The two Penrose rhombs are usually said to tile the plane only aperiodically. They tile it periodically without difficulty. What cannot be done periodically is tiling them according to the decoration, and the distinction is the whole result.
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Cut and project
Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.