Tessellation in the context of Triangular prismatic honeycomb

The mosaics of Delos are a significant body of ancient Greek mosaic art. Most of the surviving mosaics from Delos, Greece, an island in the Cyclades, date to the last half of the 2nd century BC and early 1st century BC, during the Hellenistic period and beginning of the Roman period of Greece. Hellenistic mosaics were no longer produced after roughly 69 BC, due to warfare with the Kingdom of Pontus and the subsequently abrupt decline of the island's population and position as a major trading center. Among Hellenistic Greek archaeological sites, Delos contains one of the highest concentrations of surviving mosaic artworks. Approximately half of all surviving tessellated Greek mosaics from the Hellenistic period come from Delos.

The paved walkways of Delos range from simple pebble or chip-pavement constructions to elaborate mosaic floors composed of tesserae. Most motifs contain simple geometric patterns, while only a handful utilize the opus tessellatum and opus vermiculatum techniques to create lucid, naturalistic, and richly colored scenes and figures. Mosaics have been found in places of worship, public buildings, and private homes, the latter usually containing either an irregular-shaped floor plan or peristyle central courtyard.

A unit of real estate or immovable property is limited by a legal boundary (sometimes also referred to as a property line, lot line or bounds). The boundary (in Latin: limes) may appear as a discontinuation in the terrain: a ditch, a bank, a hedge, a wall, or similar, but essentially, a legal boundary is a conceptual entity, a social construct, adjunct to the likewise abstract entity of property rights.

A cadastral map displays how boundaries subdivide land into units of ownership. However, the relations between society, owner, and land in any culture or jurisdiction are conceived of in terms more complex than a tessellation. Therefore, the society concerned has to specify the rules and means by which the boundary concept is materialized and located on the ground.

A tile-based game is a game that uses tiles as one of the fundamental elements of play. Traditional tile-based games use small tiles as playing pieces for gambling or entertainment games. Some board games use tiles to create their board, giving multiple possibilities for board layout, or allowing changes in the board geometry during play.

Each tile has a back (undifferentiated) side and a face side. Domino tiles are usually rectangular, twice as long as they are wide and at least twice as wide as they are thick, though games exist with square tiles, triangular tiles and even hexagonal tiles. Modern games may use unconventional non-tileable shapes such as the curved-shaped Bendominoes, or use many different shapes that together tile a surface such as the polyominoes in Blokus.

A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable and logical manner. There exists countless kinds of unclassified patterns, present in everyday nature, fashion, many artistic areas, as well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper design.

Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world.

In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal (from Greek τόξον  'arc') or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope acts transitively on its vertices, or that the vertices lie within a single symmetry orbit.

In geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces.

A lozenge (/ˈlɒzɪndʒ/ LOZ-inj; symbol: ◊), often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and the word is sometimes used simply as a synonym (from Old French losenge) for rhombus. Most often, though, lozenge refers specifically to a thin rhombus, especially one with two acute angles of 45° and two obtuse angles of 135°.

The lozenge shape is often used in parquetry (with acute angles that are 360°/n with n being an integer higher than 4, because they can be used to form a set of tiles of the same shape and size, reusable to cover the plane in various geometric patterns as the result of a tiling process called tessellation in mathematics) and as decoration on ceramics, silverware and textiles. It also features in heraldry and playing cards.

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.

There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

The arabesque is a form of artistic decoration consisting of "surface decorations based on rhythmic linear patterns of scrolling and interlacing foliage, tendrils" or plain lines, often combined with other elements. Another definition is "Foliate ornament, used in the Islamic world, typically using leaves, derived from stylised half-palmettes, which were combined with spiralling stems". It usually consists of a single design which can be 'tiled' or seamlessly repeated as many times as desired. Within the very wide range of Eurasian decorative art that includes motifs matching this basic definition, the term "arabesque" is used consistently as a technical term by art historians to describe only elements of the decoration found in two phases: Islamic art from about the 9th century onwards, and European decorative art from the Renaissance onwards. Interlace and scroll decoration are terms used for most other types of similar patterns.

Arabesques are a fundamental element of Islamic art. The past and current usage of the term in respect of European art is confused and inconsistent. Some Western arabesques derive from Islamic art, however others are closely based on ancient Roman decorations. In the West they are essentially found in the decorative arts, but because of the generally non-figurative nature of Islamic art, arabesque decoration is often a very prominent element in the most significant works, and plays a large part in the decoration of architecture.

Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures.

The geometric designs in Islamic art are often built on combinations of repeated squares and circles, which may be overlapped and interlaced, as can arabesques (with which they are often combined), to form intricate and complex patterns, including a wide variety of tessellations. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, or may retreat into the background around other motifs. The complexity and variety of patterns used evolved from simple stars and lozenges in the ninth century, through a variety of 6- to 13-point patterns by the 13th century, and finally to include also 14- and 16-point stars in the sixteenth century.

Generative art is post-conceptual art that has been created (in whole or in part) with the use of an autonomous system. An autonomous system in this context is generally one that is non-human and can independently determine features of an artwork that would otherwise require decisions made directly by the artist. In some cases the human creator may claim that the generative system represents their own artistic idea, and in others that the system takes on the role of the creator.

"Generative art" often refers to algorithmic art (algorithmically determined computer generated artwork) and synthetic media (general term for any algorithmically generated media), but artists can also make generative art using systems of chemistry, biology, mechanics and robotics, smart materials, manual randomization, mathematics, data mapping, symmetry, and tiling.

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k − 1)-polytopes in common.

Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.

An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.

The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry.

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonice Mundi (Latin: The Harmony of the World, 1619).