What is pattern?

It is easy to see the role of pattern in art. But pattern plays a central but underappreciated role in fields as diverse as mathematics, physics, philosophy, mythology and religion.

As Hardy points out, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” (Hardy 1992, page 84)

Many define Mathematics as “the science of patterns.” (for example Lynne Steen 1988), Devlin in Mathematics: The Science of Patterns (Devlin 1994), Resnik in Mathematics as a Science of Patterns (1999) and the NCTM Principles and Standards (NCTM 2000). Ian Stewart often points out that “Mathematics is about patterns” (2012, page 109).

A pattern is a discernible regularity or order, some object regularly repeating, typically over time or space. When defining pattern, many emphasize that the regularity is based on a rule. Patterns that seem complex, like fractals, very often have underlying simple rules. Other patterns are simple by any standard. For example, a checkerboard pattern is simple and also is based on a simple rule that determines the regularity or order. A checkerboard is a very simple kind of pattern directly determined by rules, such as every other space is a different color. The rules can predict exactly what any size checkerboard would look like. Likewise with any exactly repeating sequence of numbers or shapes. Below we use the rule, each square gives rise to four neighbors of the opposite color. We start with a single red square.  Surround the four sides of that red square with green squares. Then surround the four sides of each of those green squares with red squares. (it makes no difference if a square already has a neighbor). The fourth picture shows the result after repeating this rule 14 times. Such patterns are exactly predictable by the rules.

Below we start with a single square, but the following rule: Each square gives rise to eight neighbors (includes diagonals). Then any square that is not next to four white neighbors will die. Then again each square gives birth to eight neighbors, then any square that doesn’t have at least four white neighbors dies away.  Repeat this and you see the below patterns.
After repeating these two steps 12 times, we end up with the below pattern of 16 spread out squares. After two more iterations of these two rules, we have 64 squares, evenly spaced.
  • Hardy, G. H. A Mathematician’s Apology.   Cambridge: Cambridge University Press, 1992.
  • National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics.   Reston, VA.: NCTM, 2000.
  • Resnik, Michael D.   Mathematics as a Science of Patterns.  Oxford: Clarendon Press, 1999.
  • Steen, Lynn Arthur. “The Science of Patterns.” Science 240 (April 1988): 611-616.
  • Stewart, Ian. In Pursuit of the Unknown: 17 Equations that Changed the World. New York: Basic Books. 2012.