Below
are the famous "Penrose aperiodic rhombi",
named after Roger Penrose,
famed Oxford mathematical physicist and collaborator of
Stephen Hawking.
There is a "fat" rhombus (skewed box with
equal, parallel sides) and a
"skinny" rhombus which are marked with colored
bands as shown. The only
rule for using these tiles - you have an unlimited supply
of both types -
is that any two tiles must be placed exactly edge-to-edge
so that a colored
band of one tile lines up with the colored band from the
other, like so:
or a white edge lines up with another white edge. You
can combine fat tiles
with fat tiles, skinny tiles with skinny tiles,or any
mixed combination just as
long as the colored region of one iscontinued into the
colored region of the
other when the tiles are placed edge-to-edge. The
problem is to cover the
infinite two-dimensional plane without gaps using these
two types of tiles,
subject to the color matching rule.
The reason that the tiles are called aperiodic
is that any (and every)
successful covering of the plane will result in a tiling
which lacks the
repetitive structure of ordinary tilings. For a
successful covering of the
entire plane with Penrose tiles the trial-and-error
approach is out of the
question. However, there is mathematical "inductive"
approach (simple,
but nonobvious) which is successful. Click "Inductive Tiling of the Plane
by Penrose Aperiodic Rhombi,"
by Dean Clark and E.R. Suryanarayan,
The College Mathematics Journal, 26, pp. 266-267 (1995),
to see how
the algorithm works. Notice the unexpected band-effects
in the tiling,
below, which was constructed using this method..
