A tessellation is formed when a shape is repeated over and over again, completely covering a plane with **no gaps** or overlaps. Tessellation is sometimes referred to as tiling. The figure below cannot be used because it contains gaps. However, many other figures could be created using similar techniques and would therefore also be considered tessellations.

In mathematics, a tessellation is an infinite set of non-overlapping identical polygons that cover a surface without leaving any gaps. Polygons are examples of surfaces that can be covered by tessellations. A triangular grid is an example of a tessellation that can be constructed in the plane. There are several ways of constructing regular tilings of the plane by triangles, such as hexagons and squares. Some types of polyhedra (such as truncated cubes) can only be constructed with finite sets of faces, but they too can be said to tile the plane.

In physics, a tessellation is the partitioning of space into cells or regions which are then divided into groups according to **some criterion** or criteria. In this context, "cells" do not have to be geometric objects but can be any region in space for which there is no overlap with any other cell.

A tessellation is a tiling of one or more figures across a plane so that the figures fill the plane with no overlaps or gaps. You've most likely encountered tessellations before. A tessellation is anything like **a tile floor**, a brick or block wall, a checker or chess board, or a fabric design. Tessellations are found in nature and can be seen in crystals, seashells, snowflakes, and bird feathers.

There are many types of tessellations. They include: honeycomb, grid, herringbone, parquet, checkered, and striped.

Honeycombing is the name given to the hollow spaces within a rock formation caused by the interlocking of very small grains of sand or other material. Honeycombs appear as woven or folded fabrics of fine threads. The word comes from the Greek for "to weave together" and "a nest for bees". Although honeycombs are often thought of as natural formations, they are actually the result of human activity! The Romans used the hollowed-out hearts of trees as a place to store wine because it prevented air contact which would have spoiled the drink. Today, honeycombs are used to stabilize collapsing structures such as buildings and bridges by allowing the insertion of rods to provide extra support.

Grid patterns are regular arrangements of squares or rectangles. They can be found in bricks, flagstones, carpets, and wallpaper.

Tessellations are continuous patterns composed of repeated forms that completely cover a surface without overlapping or leaving any gaps. They can be geometric or abstract, but they all have one thing in common: each part of the pattern is identical to every other part.

MC Escher was a famous Dutch artist who painted many images with a distinctive look. One of his best-known works is called "Rainbow Landscape". It shows an outdoor scene with a rainbow as a main subject. Below the landscape are two drawings that resemble windows looking out onto the world from inside the picture.

In mathematics and physics, a tessellation is the division of a plane into regular polygons, triangles, or other shapes. Many types of tessellations exist. In architecture, the word is used mainly for **floor plans** made up of squares or rectangles, but it also applies to other regular polygonal spaces such as cubes, cylinders, and spheres. The term is used in **land surveying** to describe the partition of a piece of property into lots that are generally rectangular in shape. Tessellations can also be found in nature, for example in the honeycomb structure of some bees' nests.

Tessellation A pattern created by transforming geometric shapes in various ways. If a form is a regular geometric figure and the sides all fit together exactly with no gaps, it will tessellate. For example, the square and the pentagon are regular figures that tessellate.

There are two ways to know if a given shape tessellates: experimentally or analytically. In experiments, we try different combinations of copies of the shape and see which ones tile space. This is how we know that the octagon and the decahedron tessellate but not the dodecahedron. Analytically, we can use formulas to determine **whether a given shape tiles**. For example, there are formulas for triangles, squares, and hexagons that tell us whether they tessellate. These formulas work because any shape that does tessellate must be split into identical pieces that fit together in the same way they overlap. Using **these formulas**, we can also find out that the dodecahedron does not tessellate.

All polygonal numbers greater than or equal to 5 are tessellating (except the dodecahedron). There are many proofs that use **only basic geometry** and some algebraic techniques to show that a given polygonal number tends to be tessellating.

Useful Information Some forms tessellate, but others do not. Hexagons are an example of the former. Many hexagons can fit together seamlessly to form a seamless, repeating pattern. Hexagon tessellations are found naturally in honeycombs and are used in many tiled flooring. Triangles are the most common shape in nature and geometry; therefore, triangular tessellations occur frequently in art and architecture. Examples include the rock formations known as tetrahedrons-four triangles that meet at each vertex or point-and octahedrons-eight triangles that also meet at each vertex.

Human beings create triangular and hexagonal patterns intentionally for decorative purposes. The best-known example is the work of Pieter Mondrian, who used repeated arrangements of simple shapes such as squares and triangles to create complex paintings with strong geometric elements. Mondrian's work was influential in early 20th-century modernism.

In science, mathematics, and technology, tessellations provide insight into how the universe works on large scales and what effects certain numbers have within **this structure**. They also play **an important role** in physics: for example, quantum mechanics requires us to describe particles as both waves and particles, and this duality is expressed through **wave/particle tessellations**.

In computer graphics, tessellations are used to generate more detail than would otherwise be possible without sacrificing image quality.