patternrepeating

Pattern repeating refers to the repetition of a motif across a domain to form a larger design. It can be one-dimensional (such as sequences), two-dimensional (such as images or textures), or higher-dimensional.

In mathematics, repeating patterns are linked to periodicity. A one-dimensional sequence is periodic if there exists a positive integer p such that a(n+p) = a(n) for all n beyond some starting point. In the plane, a motif translated across the plane creates a tiling, with a fundamental region that, when repeated by translations, covers the plane without gaps. Two-dimensional repeating patterns are classified by their symmetries, often described using wallpaper groups. A periodic function satisfies f(t+T) = f(t).

In design and art, pattern repetition is used to create rhythm, unity, and texture. Common motifs include stripes, checks, and tessellations. In textiles, wallpapers, and architecture, repeating motifs enable large surfaces to be formed from a single design block while maintaining visual coherence.

In computing and digital media, pattern repetition underlies textures and backgrounds. Tiling can introduce seams if not seamless, and algorithms exist to generate seamless tiles or to vary motifs procedurally while preserving repetition constraints.

See also related concepts such as repetition, tessellation, tiling, periodic function, and wallpaper group.