Tessellations

Tessllations

is an arrangement of shapes that covers a plane such that no shapes overlap and there are no gaps.

Vertex configuration

is a notation that describes the arrangement of regular polygons around each vertex in the tiling. It specifies the types and order of polygons that meet at a single vertex. Each number in the configuration represents a regular polygon with that many sides. The numbers are listed in cyclic order, either clockwise or counterclockwise, around the vertex. For Archimedean tessellations, the vertex configuration is the same at every vertex.

The tessellation of an equilateral triangle has a vertex configuration of 3.3.3.3.3.3. It involves six equilateral triangles meeting at each vertex, forming a pattern where each triangle has three sides, represented by the number 3.

Regular Tessellations

use one type of regular polygon. However, not all regular polygons tessellate. To see which regular polygons can tessellate, check their interior angles, and see if duplicates of them can fit evenly around a point called the vertex of a tessellation. The vertex of a tessellation is a point where three or more corners of the tessellating shapes are joined.

Semiregular Tessellation

Tessellations can be formed using more than one type of polygon. These tessellations are semiregular tessellations.

Irregular Tessellation

is a tiling of the plane using nonregular polygons, or even non-polygonal shapes, that fit together without gaps or overlaps.

Translation

is the simplest transformation to create tessellations. It involves sliding a shape in a straight line without rotating or flipping it. Many common tilings in our surroundings, like brick walls, tiled floors, or rectangular windows, are formed by translations.

Reflection

or a flip, is a transformation that creates a mirror image of a shape across a line, known as the line of reflection. In tessellations, reflections often produce patterns with bilateral symmetry, making designs more artistic and visually appealing. Many mosaic patterns and artworks use reflections to create balance and harmony.

Glide reflection

is a combination of two transformations: a reflection (Aip) followed by a translation (slide) along the line of reflection. It produces patterns with flowing or alternating symmetry, making it useful in artistic designs and real-world patterns like footprints or textile weaves. Among all transformations, glide reflection produces the most dynamic-looking tessellations.

Rotation

is a transformation that turns a figure around a fixed point by a given angle. When used in tessellations, rotations often create star-like or circular designs that spread evenly around a center. Because certain angles divide 360° evenly, polygons like equilateral triangles, squares, and hexagons are often used for rotational tessellations.

Dutch artist, Maurits Cornelis "M.C." Escher (1898-1972)

transformed mathematical tessellations into captivating works of art by replacing abstract geometric shapes with recognizable, interlocking figures of animals, humans, and fantastical creatures that seamlessly tile the plane without gaps or overlaps. His work sits at the intersection of art, geometry, symmetry, and topology. Each Escher tessellation describes combinations of translations, rotations, reflections, and glide reflections.