Penrose Tiling
Tiling in 5fold symmetry was thought impossible!
Areas can be filled completely and symmetrically with tiles
of 3, 4 and 6 sides, but it was long believed that it was impossible to fill
an area with 5fold symmetry, as shown below:




3 sides 
4 sides 
5 sides leaves gaps 
6 sides 
The solution was found in Phi
In the early 1970's, however, Roger Penrose discovered that
a surface can be completely tiled in an asymmetrical, nonrepeating manner
in fivefold symmetry
with just two shapes based on phi, now
known as "Penrose tiles."
This is accomplished by creating a set of two symmetrical
tiles, each of which is the combination of the two triangles found in the geometry of
the pentagon.
Phi plays a pivotal role in these constructions. The relationship of the sides of the pentagon, and
also the tiles, is Ø, 1 and 1/Ø.
The triangle shapes
found within a pentagon are combined in pairs. 
One creates a set
of tiles, called "kites" and "darts" like this: 
The other creates a
set of tiles like this: 



The ratio of the two types of tiles in the resulting patterns
is always phi
As you expand the tiling to cover greater
areas, the ratio of the quantity of the one type of tile to the other always approaches phi, or 1.6180339..., the Golden
Ratio.
Within this tiling there can be small areas of fivefold
symmetry. Decagons can also occur, which when grouped together can look like
pentagons from a distance.

Phi gives 5fold symmetry in 3D with a single shape,
known as a quasicrystal. 
Phi is intrinsically related to the number 5
The appearance of the golden ratio in examples of fivefold
symmetry occurs because phi itself is intrinsically related to the number 5, mathematically and trigonometrically.

A 360 degree circle divided into five equal sections
produces a 72 degree angle, and the cosine of 72 degrees is
0.3090169944, which is exactly one half of phi, the reciprocal of Phi, or 0.6180339887.
5
^ .5 * .5 + .5 = Ø
In this mathematical construction, "5 ^ .5" means
"5 raised to the 1/2 power," which is the square root of 5, which
is then multiplied by .5 and to which .5 is then added.
