Nontessellation

Nontessellation refers to a shape or collection of shapes that cannot perfectly tile a plane without gaps or overlaps. Tessellation, in contrast, involves repeating a shape or a set of shapes to cover a surface completely. The concept of nontessellation is fundamental to understanding geometric constraints and the properties of different shapes.

For a shape to be considered a tessellating shape, its angles at any vertex where shapes meet must add up to 360 degrees. This ensures a seamless fit. If the angles do not sum to 360 degrees, or if the shape's edges do not align perfectly with adjacent shapes, it will result in gaps or overlaps, thus exhibiting nontessellation.

Many common shapes are nontessellating. For example, a regular pentagon cannot tile a plane by itself. While it's possible to arrange pentagons in various patterns, there will always be interstitial spaces that cannot be filled by another pentagon without overlapping. Similarly, irregular polygons can be nontessellating depending on their angles and side lengths. The study of nontessellation has implications in fields such as architecture, design, and even in understanding the packing of molecules.