The very definition of a crystal assumes a precisely symmetrical ordering of atoms in periodic patterns that repeat over and over in a 3D lattice. The patterns look the same no matter which direction you look at them, but quasicrystals are different. They clearly follow mathematical rules, but each cell has a slightly different configuration of cells nearby, rather than repeating in an identical pattern. It’s that unique structure that gives quasicrystals their unusual properties.
Think about tiling a bathroom floor. The tiles can only be in certain symmetrical shapes (triangles, squares, or hexagons), otherwise you won’t be able to fit them together without leaving gaps or having some of the tiles overlap. Pentagons, icosahedrons, and similar shapes with different symmetries that never precisely repeat just won’t work—except in the case of quasicrystals, where nature decided they could work. The trick is to fill the gaps with other kinds of atomic shapes to create the unlikely aperiodic structure.