What do we mean when we say
the word "pattern" ? A simple definition is that
a
pattern is a repeating array of identical
units (typically with some symmetry).
In this course, we want to take
a broader perspective on the concept of a pattern,
by allowing the basic units of
a pattern to be similar but not necessarily identical,
and to repeat but not necessarily
with perfect regularity or symmetry. For instance,
in the natural patterns discussed
in Lecture
1, we allowed the presence of
Regular polygons
are ones in which the sides have the same length and the angles are all
equal. One can show that the only
regular polygonal tilings of the plane are by triangles,
squares, and hexagons.
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In particular, if you try to tile the plane with regular pentagons, or with any regular polygon other than those listed above, here is what happens: gaps are left. |
You are all familiar with other forms of tilings, such
as wallpaper
tilings. These are
periodic
tilings of the plane which are built by translating or applying an
orientation
reversing symmetry to a tile that serves as a building
block of the pattern. There are
exactly 17 such independent patterns for the plane. Many
of these are represented in
the
artwork of M.C. Esher. You can construct and examine these wallpaper
patterns
using the program Kali.
Finally, let us mention that there are also tilings which
are not periodic. For instance, a
quasi-tiling is one in which
a finite number (greater than one) of not necessarily regular
polygons can be used to tile the plane, but without having
any periodic structure. For this
reason, such tilings are sometimes also called
aperiodic
tilings. The QuasiTiler
web site
of the Geometry Center
(University of Minnesota) provides some excellent examples of
and tools for exploring quasi-tilings.
We can now go back to some of the patterns discussed in
Lecture
1 and try to classify
them as follows: