## Amman Bars

The tiles that are forced by a given patch of
tiles can be determined by decorating the tiles as shown below, and then
insisting that tiles be placed so the extra markings form straight lines.
These lines, known then as "Amman Bars", are then extended across the
plane, and any other tile must have its markings on an Amman Bar.

At
left we have a tiling with Amman bars in one direction. Note that adjacent
bars are always a longer (L) of a shorter (S) distance apart. The ratio of
these distances is the golden ratio. Note also that there are 6 L's and 4
S's. If you take a big enough piece of the tiling, this ratio tends as
well to the golden ratio. The sequence that a tiling generates will never
have SS or LLL. These sequences can be deflated much like a Penrose
tiling: replace every S by L, LL by S, drop all single L's. To
inflate, replace L by S, S by LL, and add L between a pair of S's. If you
deflate or inflate a Penrose tiling, the Amman sequences are also deflated or
inflated.

If two families of parallel bars are placed at the proper angle to each other, a
Penrose tiling is determined. This raises the question: Is the
tiling or the configuration of Amman bars the essential item?

In
the image at left, Amman bars have been drawn in 3 directions. If you
wanted to, you could draw them in two more directions.

The Amman sequences exhibit the same sort of local isomorphism properties at the
tilings do: If you find a string, say LSLLS, you never have to go very far
in the sequence to find an identical string. This is a great
contrast to many other nonperiodic phenomena. For instance, the digits of
an irrational number don't have this property.

The ratio of kites to darts is phi
Aperiodic Tiling