Patterns

When you hear the word “pattern” you might first think of regular arrangements of geometrically similar figures. More generally, a pattern is a consistent structure underlying a repetitive pattern, or, more figuratively, a sequence of actions or a way of thinking, designing, or behaving that is intended for uniform repetition (reproduction).

The initially mentioned regular arrangement of geometrically similar figures fits this definition in that segments of the overall figure repeat spatially at different locations, with this repetition following certain rules. However, entirely different examples also fit. For instance, consider the sequence of numbers 4, 16, 36, 64, 100, 144: the repetition initially consists of multiple comma-separated decimal numbers appearing in reading order. It can be observed that all the numbers are even and that they are all perfect squares — these are regularities that underlie the repeating elements.

A formal description of the pattern which highlights the underlying regularity can be provided in the form of a recipe for producing the repeating forms. In this case, one might take the even numbers from \(2\) to \(12\) as ingredients, from which the forms are produced by squaring, i.e., \((n^2)_{n\in\mathbb G_{\leq 12}}\), where \(\mathbb G\) stands for even natural numbers. Of course, this description is not unique. By altering the ingredients, one can still achieve the same result through a different production rule, as demonstrated by the alternative representations \((4\cdot n^2)_{n\in\mathbb N_{\leq 6}}\) or \((n)_{n\in\mathbb G_{\leq 144}\cap \{m^2|m\in\mathbb N\}}\).

Which representation is best depends on the intended use and cannot be generalized. However, it is undisputed that a recipe-like description designed for repeated application can be used for the formal representation of patterns: each application of the recipe produces a form, and with numerous applications involving different ingredients, the rules of the recipe become indirectly visible as patterns among the results. Functions are thus pattern generators, with the pattern becoming evident in the values of the function.