New geometric patterns which lie between boredom and confusion.

There are a number of classical geometric patterns in the plane which
are intriguing (mathematically and aesthetically) for their repetitive
and predictable nature. About forty years ago, new patterns were
discovered which displayed the same kind of high degree of order, without
the same predictability. I will discuss the subtle differences and
similarities between these new patterns and the old, and
how they are created. Investigations of these mathematical patterns was followed
by the remarkable discovery that they actually appear as the atomic arrangement
in physical materials, now called quasicrystals.