Tiling in Geometry

Tiling, also known as tessellation, is a concept in geometry that involves covering a surface with a pattern of flat shapes such that there are no overlaps or gaps. This can be done with a variety of shapes, such as polygons, and can occur in one, two, or even three dimensions. Tiling is a fundamental aspect of both mathematical study and practical applications, such as in architecture and art.

Types of Tilings

Regular Tiling

Regular tiling involves covering a surface using congruent regular polygons. In Euclidean geometry, only three regular tessellations exist: the square tiling, the hexagonal tiling, and the triangular tiling. For example, the square tiling consists of squares meeting at each vertex, forming a grid-like pattern.

Semiregular Tiling

Semiregular tiling, or Archimedean tiling, is characterized by two or more types of regular polygons meeting in identical vertices. An example is the rhombitrihexagonal tiling, which combines triangles, squares, and hexagons.

Aperiodic Tiling

A remarkable kind of tiling is aperiodic tiling, which does not repeat regularly. Penrose tiling is a well-known example, showcasing a pattern that never repeats yet covers the plane completely.

Non-Euclidean Tilings

In non-Euclidean geometries, such as hyperbolic geometry, tilings can take on unique forms. For instance, the heptagonal tiling is a regular tiling of the hyperbolic plane using seven-sided polygons.

Mathematical Considerations

Tessellations are not limited to the plane; they can be extended into higher dimensions. In three dimensions, tiling is referred to as space-filling or tessellating the space with polyhedra. Concepts like the Voronoi diagram provide a framework for understanding tessellations in terms of partitioning space based on distance to a set of given objects.

Applications

Tessellations have numerous applications in various fields:

Art: Artists have used tessellations to create visually stimulating patterns, famously explored by M.C. Escher who used mathematical principles to create impossible constructions and intricate patterns.
Architecture: Tessellations feature prominently in the design of buildings and pavements, utilizing patterns for both aesthetic appeal and structural integrity.
Science and Technology: In computational geometry, tessellation is crucial for tasks such as mesh generation in finite element methods and computer graphics where surfaces are divided into simpler shapes for rendering.
Patterns in Nature: Natural tessellations include honeycombs and crystal structures, where efficiency in space utilization is critical.

Tiling In Geometry