## Patterns & Sequences

Mathematics is sometimes called the science of patterns, meant as rules that can be found absolutely everywhere.

A pattern is an underlying relationship between the terms of a a sequence, namely a succession of elements.

Patterns can create repetition or recurrence. With repeating patterns, the elements are periodically displayed, as in ABCABCABC. With recurring patterns, the elements change with regularity, making it possible to predict what the next element will be by observing the previous ones. It is the case of the Fibonacci sequence, which is frequently found in the natural world, like in the petals of certain flowers.

Patterns are regularities in the natural world, in human-made design, or in abstract ideas. Their elements repeat in a predictable manner, and we can perceive them with all our senses.

Patterns are widespread in nature, where they can be hidden behind apparent chaos, often involving fractals. Natural patterns include spirals, waves, foams, tilings or tessellations, stripes and even cracks. They can also be found in starry skies, under the name of constellations or asterisms.

In art, a pattern may be a geometric or other repeating shape in a painting, tapestry, or ceramic tiling. Patterns are also essential in musical and literary compositions.

Patterns and structures are cornerstones of mathematics and art.

They help mathematicians and artists to spot when something interesting is going on, to identify the core of a problem and to generalise from a specific example to a wider understanding.