Quasicrystal
Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. They were first observed by Dan Shechtman in 1982.
In a normal crystalline solid the positions of atoms are arranged in a periodic crystal lattice of points, which repeats itself in space the same way that a honeycomb structure repeats itself in the plane: each cell has an identical pattern of cells surrounding it. In a quasicrystal, the pattern of atoms is only quasiperiodic. The local arrangements of atoms are fixed, and in a regular pattern, but are not periodic throughout the entire material: each cell has a different configuration of cells surrounding it.
Quasicrystals are remarkable in that some of them display five-fold symmetry. In an ordinary crystal, only 1-, 2-, 3-, 4-, and 6-fold symmetries are possible. This is a geometrical consequence of filling space with congruent solids—these are the only symmetries that can fill space. Prior to the discovery of quasicrystals, it was thought that five-fold crystal symmetry could never occur, because there are no space-filling periodic tilings, or space groups, which have five-fold symmetry. Quasicrystals helped to redefine the notion of what makes a crystal, since they do not have a repeating unit cell but do display sharp diffraction peaks.
There is a strong analogy between the quasicrystal and the Penrose tiling of Roger Penrose. In fact, some quasicrystals can be sliced such that the atoms on the surface follow the exact pattern of the Penrose tiling.

For a periodic pattern, if you fill all of space with the pattern, you can slide the pattern a certain distance in a certain direction, and every atom will lie exactly where an atom lay in the original pattern.
For a quasiperiodic pattern, if you fill space with it, there is no distance you can slide the pattern to make every atom lie exactly where an atom lay in the original pattern. However, you can take a bounded region, no matter how large, and slide it to match up exactly with some other part of the original pattern.
There is actually a simple relationship between periodic and quasiperiodic patterns. Any quasiperiodic pattern of points can be formed from a periodic pattern in some higher dimension.
For example, to create the pattern for a three-dimensional quasicrystal, you can start with a regular grid of points in six-dimensional space. Let the 3D space be a linear subspace that passes through 6D space at an angle. Take every point in the 6D space that is within a certain distance of the 3D subspace. Project those points into the subspace. If the angle is an irrational number such as the golden mean, the pattern will be quasiperiodic.
Every quasiperiodic pattern can be generated this way. Every pattern generated this way will be either periodic or quasiperiodic.
This geometric approach is a useful way to analyze physical quasicrystals. In a crystal, flaws are locations where the pattern is interrupted. In a quasicrystal, flaws are locations where the 3D "subspace" is bent, or wrinkled, or broken as it passes through the higher-dimensional space.

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