Quasicrystals
Quasicrystals 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
noncrystalline matter, such as glass.
With fivefold symmetry, once thought to be impossible, they
were
first observed in 1984 in an aluminiunmanganese alloy (Al_{6}Mn).
Since then, quasicrystals have been found in other substances.
Quasicrystals fill space with fivefold
symmetry based on phi
Penrose tiles allow a
twodimensional area to be filled in fivefold symmetry, using two shapes
based on phi. It was thought that filling a threedimensional space in
fivefold 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:
The resulting solid looks like this: 
Download and print this to make your own! 


