Quasicrystals

Quasi-crystals represent a newly discovered state of matter

Most crystals in nature, such as those in sugar, salt or diamonds, are symmetrical and all have the same orientation throughout the entire crystal.  Quasicrystals represent a new state of matter that was not expected to be found, with some properties of crystals and others of non-crystalline matter, such as glass. 

With five-fold symmetry, once thought to be impossible, they were first observed in 1984 in an aluminiun-manganese alloy (Al₆Mn).  Since then, quasicrystals have been found in other substances.

Quasi-crystals fill space with five-fold symmetry based on phi

Penrose tiles allow a two-dimensional area to be filled in five-fold symmetry, using two shapes based on phi.  It was thought that filling a three-dimensional space in five-fold symmetry was impossible, but the answer was again found in phi.

Where the solution in 2D required two shapes, this can be accomplished in 3D with just one shape.  The shape has six sides, each one a diamond whose diagonals are in the ratio of phi: