**Penrose Tiling****Author(s): **Andrew Glassner.**Article: **IEEE Computer Graphics and Applications, Vol. 18, No. 4, pp. 78--86, July/August,
1998.

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**Abstract: **

Theme and variation are part of nature. Birdwatchers
identify the species of a bird by its distinctive
markings, even though the specific colors and shapes
vary from one bird to the next. Every thunderclap
sounds roughly like thunder, but each one is different.
And of course snowflakes are beloved for the hexagonal
symmetry they all share, as well as the delicate patterns
unique to each one.
In the May/June 1998 issue of IEEE CG&A I discussed
the topic of aperiodic tiling for the plane. This technique
helps us create patterns with lots of theme and variation,
like the leaves on a tree. I’ll continue discussing the
subject, so let’s first briefly summarize the main ideas
from last time.
The basic approach is to take a bunch of 2D shapes
and impose rules on how they can connect, like the
pieces of a jigsaw puzzle. Suppose that you have an infi-
nite supply of these shapes, or tiles, and you cover the
plane with them, out to infinity in all directions. You
might be able to find a region of the pattern that you
could pick up and use as a rubber stamp, and by stamping
it out an infinite number of times (without rotating
or scaling it), fill the plane with the identical design that
you started with. So in essence you’ve reduced the original
set of pieces and their interlocking rules to just many
translated copies of a single, larger piece. If you can do
this, then the overall pattern—and the set of tiles used
to make it— are periodic.
If you can’t find such a single, big piece that replicates
the pattern by translation, the pattern is called nonperiodic.
Many tiles can create both periodic and nonperiodic
patterns, depending on how they’re laid down.
If you happen to have a set of tiles that can only make
nonperiodic patterns, and you can prove that’s the case,
then you have an aperiodic set of tiles.
The quest for aperiodic tiles has gone on for several
decades. The first set discovered involved 26,000 tiles.
In the last issue I discussed two far smaller sets of aperiodic
tiles, named after their creators. These Robinson
and Ammann tiles are based on squares, which make
them very convenient for computer graphics applications
like texture mapping and sampling. Here I’ll look
at perhaps the most famous set of aperiodic tiles, discovered
by Roger Penrose. Actually, Penrose found two
distinct sets of aperiodic tiles, one called “kites and
darts” and the other called “rhombs.” Each set is made
of only two tiles. Let’s look at the kites and darts first.

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