Natural Patterns

Lectures 2 & 3: tilings


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

To understand and classify these patterns, we need to look at the various ways of
tiling the plane. We will start with regular tilings, which are tilings with regular polygons.
Such perfect and idealized tilings are not observed in nature but we will see that we
can consider natural patterns as deformations of such perfect patterns.
 
 

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.
 
Triangles
Squares
Hexagons
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.
Of course, you can find ways of tiling the plane with pentagons, but such polygons
are not regular.
 
 

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:

As you can see, most natural patterns belong to one of the above three categories, which
are associated with regular tilings. This description will turn out to be the starting point
of the modeling of pattern formation in systems driven far from equilibrium.
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