Nontessellations

Nontessellations refer to geometric patterns or designs that do not tile the plane without gaps or overlaps when repeated infinitely. Unlike tessellations, which cover a surface completely with congruent shapes without any spaces or overlaps, nontessellations fail to achieve this coverage due to their inherent geometric properties.

A classic example of a nontessellating shape is the regular pentagon. When regular pentagons are placed adjacent to one another, they leave noticeable gaps between them, preventing a seamless tiling of the plane. This occurs because the internal angles of a regular pentagon (108 degrees) do not divide evenly into 360 degrees, the full angle around a point. Similarly, other polygons with angles that do not sum to 360 degrees when repeated around a vertex—such as regular heptagons (128.57°) or nonagons (140°)—also fail to tessellate.

Nontessellations can also arise from irregular shapes or combinations of shapes that do not align perfectly when repeated. For instance, certain star polygons, like the five-pointed star (pentagram), do not tessellate because their angles and intersections do not allow for continuous coverage. Some non-polygonal curves or fractal-like designs may also exhibit nontessellating properties due to their complex, non-repeating structures.

The study of nontessellations is relevant in fields such as mathematics, particularly in geometry and tiling theory, as well as in art and design, where understanding the limitations of tessellation can inspire creative solutions. Researchers often explore modified versions of nontessellating shapes—such as truncated or adjusted polygons—to create semiregular or aperiodic tilings that approximate tessellation while accommodating irregularities.