What happens if we break some of the rules of crystallography?

A traditional crystal is a repeating pattern of units in three dimensions.  If you take shapes like triangles, squares and hexagons, you can tessellate these into two dimensions without any gaps.  What happens if you take a pentagon?  It then becomes impossible to create a tessellating pattern without gaps.  Triangles represent three-fold symmetry, squares four-fold symmetry and hexagons six-fold symmetry.  For a long time, these were the only types of symmetry that it was believed could exist in crystals.  This was before Dan Shechtman broke the rules and discovered a new form of crystals, quasicrystals!  This special class of crystals have five-fold symmetry (or pentagons) and their discovery resulted in a Nobel Prize for Dan Shechtman in 2011 and a new definition of a crystal!