What Is A Tessellating Pattern?

A tessellating pattern (a tessellation, if you want the math word) is just shapes that fit together perfectly: no gaps, no overlaps, covering a flat plane. Every tile shares its edges with its neighbors, and the whole thing could in theory run on forever. People have been at this a long time. Ancient Sumer tiled floors with clay, Rome laid mosaic floors, Islamic art turned it into some of the most intricate pattern work ever made, and modern op art kept it going. It is still a foundational idea in geometry, tile work, and wallpaper design.

Below we cover what counts as a tessellation, the rules behind it, the main types, examples from art and nature, where it came from, and how it gets used today.

What is a tessellating pattern?

A tessellating pattern is a repeating pattern of shapes (called tiles) that fit together edge to edge to fill a flat plane without any gaps or overlaps. Each tile in the pattern shares full edges with its neighbors, and the same combination of shapes repeats across the surface in a regular or semi-regular way. The pattern can in principle continue forever in every direction.

To tessellate means to cover a surface with a repeating pattern of one or more shapes. The verb form ("to tessellate") and the noun form ("a tessellation") both come from the Latin word "tessella" (a small cube or tile, originally a piece of a Roman mosaic). The same word root gives us "tessellated" (the past participle, used as an adjective).

The shapes used in a tessellation are most often regular polygons (equilateral triangles, squares, hexagons) or combinations of regular polygons. But many other shapes can also tessellate, including some irregular polygons, certain curved shapes, and complex shapes that interlock in specific ways. The defining requirement is that the shapes fit together without gaps or overlaps; the specific shapes used can vary widely.

Every tessellation has a vertex (or vertices, plural), the point of intersection where three or more bordering tiles meet. In a regular tessellation, every vertex point is identical (the same combination of shapes meets at every vertex). In a semi-regular tessellation, every vertex is identical but the tessellation uses two or more types of regular polygons. In an irregular tessellation, the vertices can vary across the pattern.

Tessellating patterns appear naturally (honeycomb, snake scales, certain crystal patterns), in art (Islamic mosaics, M.C. Escher prints, modern op art, William Morris wallpapers), and in mathematics (the study of tessellations is a major topic in geometry and discrete mathematics).

What are the rules of tessellation?

A pattern qualifies as a tessellation if it follows three core rules: the shapes fit together edge to edge, there are no gaps between shapes, and there are no overlaps between shapes. Any pattern that satisfies these three conditions is a tessellation.

Edge-to-edge fitting means that adjacent tiles share full edges (not partial edges or just corners). A pattern in which tiles share only corners (without full edge contact) is not a strict tessellation; it might be a related arrangement called a non-edge-to-edge tiling. Most decorative tessellations use full edge sharing.

No gaps means that the tiles cover the plane completely, with no empty space between shapes. A pattern with gaps between tiles is a partial tiling or a sparse arrangement, not a tessellation.

No overlaps means that the tiles do not stack on top of each other. Each point on the surface is covered by exactly one tile. A pattern with overlapping tiles is a layered design, not a tessellation.

For the simplest tessellations, regular polygons are the building blocks. Three regular tessellations exist when copies of a single shape are used to tile the plane without any gaps or overlaps: the equilateral triangle, the square, and the hexagon. Regular polygons are arranged so that the interior angles at every vertex add up to 360 degrees. No other regular polygon (pentagon, heptagon, octagon, etc.) can form a tessellation by itself with a single shape, because the interior angles of those polygons do not divide evenly into 360 degrees. The three regular tessellations are made up of regular polygons that meet this requirement.

For semi-regular tessellations (which combine two or more types of regular polygons), eight distinct patterns exist. Each one uses a specific combination of regular polygons (square and equilateral triangle, hexagon and equilateral triangle, square and octagon, etc.) in a repeating arrangement where every vertex is identical.

For irregular tessellations, the rules are less restrictive: any combination of shapes that fits together without gaps or overlaps qualifies, even if the tiles differ in shape or size, and even if the vertices are not all identical. Irregular tessellations include most decorative tile patterns, William Morris wallpaper repeats, and many natural patterns.

What are the main types of tessellations?

Tessellations fall into several major types based on the shapes used and the vertex structure.

Regular tessellation: a tessellation using copies of a single regular polygon. Only three exist: triangle tessellation (six equilateral triangles meeting at each vertex), square tile tessellation (four squares meeting at each vertex), and hexagon tessellation (three hexagons meeting at each vertex, the honeycomb pattern).

Semi-regular tessellation: a semi-regular tessellation is made of two or more types of regular polygons fitted together to form a tessellation, with every vertex identical. Eight semi-regular tessellations exist, including the truncated square (squares plus octagons) and the triangular-hexagonal patterns. These are sometimes called Archimedean tessellations after Archimedes.

Irregular tessellation: a tessellation using shapes that are not all regular polygons, or with vertices that are not all identical. Most decorative tile patterns, brick patterns, fish-scale patterns, and many wallpaper repeats are irregular tessellations.

Aperiodic tessellation: a tessellation that fills the plane but does not repeat in a regular periodic way. Penrose tilings (discovered by Roger Penrose in the 1970s) are the most famous aperiodic tessellations; they use two specific tile shapes that can tile the plane forever without ever repeating the same arrangement.

Three-dimensional tessellations (sometimes called space-filling tessellations or honeycombs) extend the concept into three dimensions, using three-dimensional shapes to fill space without gaps or overlaps. Cubes, truncated octahedra, and certain other polyhedra fill space in this way.

Curved tessellations use curved shapes rather than straight-sided polygons. These can tessellate if the curves interlock cleanly. M.C. Escher's famous prints often use curved tessellating shapes (fish, birds, lizards) that fit together without gaps or overlaps.

What are examples of tessellating patterns?

Tessellated patterns appear everywhere once you start looking, from natural patterns to jigsaw puzzles to ancient mosaic floors. Some of the most famous examples of tessellation in art and design include:

Honeycomb: the hexagonal pattern of bee honeycomb is a regular tessellation of hexagons. Bees produce this pattern naturally because the hexagon is the most efficient regular shape for filling space with the least amount of wax.

Square tile floors: the standard square tile floor is a regular tessellation of squares. The same pattern appears on chessboards, on city sidewalks, and on many bathroom and kitchen floors.

Brick walls: a standard brick wall is a tessellation of identical brick rectangles arranged in a running bond or stack bond pattern. The pattern is irregular by the strict definition (vertices are not all identical), but it is a valid tessellation.

Islamic mosaics: Islamic art and architecture produced some of the most complex tessellation art in history. Tessellation in art has rarely reached the level found in Islamic art and architecture, where artists used mathematical relationships between regular polygons to design endlessly varied surface decoration. The Alhambra in Granada, Spain, contains many of the most famous Islamic tessellations, with intricate interlocking geometric shapes covering walls and floors throughout the palace complex.

Roman mosaic floors: ancient Roman buildings used mosaic patterns extensively. The Roman pattern catalog includes many tessellating designs in stone, ceramic, and glass tile, often using complex multi-shape combinations.

M.C. Escher prints: the Dutch artist M.C. Escher (1898-1972) produced many famous prints featuring tessellating shapes of animals, people, and abstract forms. Escher used curved interlocking tessellations to create his characteristic shifting-image graphic art.

Op art: 1960s op art (optical art) used tessellating patterns to create visual illusions of motion, depth, and shifting perception. Bridget Riley and other op art artists used regular and semi-regular tessellations as the basis for their compositions.

Penrose tilings: Roger Penrose's aperiodic tessellations use two specific "kite" and "dart" tile shapes (or alternatively two "rhombus" tile shapes) that cover the plane without ever repeating the same arrangement. Penrose tilings have become a famous example in mathematics and decorative arts.

Wallpaper repeats: most repeating wallpaper patterns are tessellations of some kind. A William Morris wallpaper pattern is an irregular tessellation where each repeat tile contains a complex floral or geometric design that interlocks with neighboring tiles cleanly.

Patterns in nature also include many natural tessellations: snake skin scales, pine cone scales, sunflower seed heads (Fibonacci tessellation), giraffe spots, and certain crystal patterns all show natural tessellating arrangements.

What is the history of tessellation?

The history of tessellations runs back to ancient civilizations and forward through every period of art and mathematics. The earliest tessellations appear in Sumerian and Mesopotamian decorative tile work from around 4000 BCE; the Sumerians used clay tiles to cover walls and floors with simple geometric repeating patterns.

Ancient Egypt, ancient Greece, and ancient Rome all used tessellation patterns in decorative arts. Roman mosaic floors and walls used complex multi-color tessellating designs across the empire, with motifs ranging from simple geometric patterns to elaborate figural scenes built from tessellating tile units.

Islamic art and architecture developed tessellation to its highest level. From the seventh century onward, Islamic architects and artists produced increasingly complex tessellation patterns across mosques, palaces, mausoleums, and decorative objects. The Alhambra (built between 1238 and 1492) contains examples of all 17 mathematically possible wallpaper-group symmetries in tessellation art, a remarkable mathematical achievement made centuries before the underlying geometry was formally proved.

European medieval and Gothic architecture used tessellation in church tile floors, stained glass, and decorative carving. Gothic cathedral floors often included complex geometric tessellating patterns in stone and ceramic tile.

The history of mathematics and tessellations runs in parallel with the decorative tradition. Ancient Greek mathematicians studied regular polygons and their tessellations. Johannes Kepler (1571-1630) gave the first systematic mathematical treatment of tessellations, classifying regular and semi-regular tessellations and analyzing their properties.

The nineteenth and twentieth centuries brought tessellation into modern mathematics and art. The classification of wallpaper groups (the 17 possible symmetry types of two-dimensional repeating patterns) was completed in the late nineteenth century. M.C. Escher's tessellation prints made the concept widely known in twentieth-century art. Roger Penrose's discovery of aperiodic tessellations in the 1970s, and four new tessellations with pentagons discovered in the 2010s, kept the field actively developing.

Recent mathematical research has discovered additional pentagonal tessellations and continues to explore the properties of complex aperiodic and curved tessellations. The 2023 discovery of an "einstein" aperiodic monotile (a single tile shape that tessellates aperiodically) was a major mathematical achievement that updates the field.

Where are tessellating patterns used today?

Tessellating patterns appear widely in contemporary design, architecture, decorative arts, and mathematical study. The pattern's combination of structural strength, visual interest, and mathematical elegance keeps it in continuous active use.

Wallpaper and interior design use tessellation patterns extensively. Most repeating wallpaper designs are tessellations of some kind, and many contemporary wallpaper collections feature explicitly geometric tessellation patterns. The Wallpaper Trends 2026 guide covers tessellation-style and geometric wallpaper directions.

Tile work uses tessellations across kitchens, bathrooms, entryways, and floor and wall surfaces. Standard square tile, hexagon tile, fish-scale tile, and many other tile shapes produce tessellating floor and wall patterns. Designer tile work often uses complex multi-shape semi-regular tessellations as a statement design element.

Textile design uses tessellations across upholstery, drapery, dress fabric, and quilting. Many quilt patterns are explicit tessellations, with the quilt blocks designed to interlock in repeating patterns. Designer fabric often features op-art-style or Islamic-inspired tessellation designs.

Graphic art and digital design use tessellations in logos, illustrations, posters, and screen patterns. Tessellation generators (online tools that produce custom tessellations from user-drawn shapes) have made the concept widely accessible to non-mathematician designers.

Architecture uses tessellation patterns in facade design, paving, and decorative elements. Contemporary architects from many countries have used complex tessellation patterns as both structural and decorative features, often referencing the Islamic architectural tradition.

Mathematics education uses tessellations to teach geometry, symmetry, polygon properties, and pattern analysis. Tessellation activities are standard in elementary and secondary mathematics curricula in many countries.

How do you create a tessellation?

To create a tessellation, start with a single shape with straight sides (or a small set of shapes) and test whether copies of the shape will tile the plane without any gaps or overlaps. Many tessellated patterns can be designed in this way; the trick is to find shapes whose edges follow the principles of tessellation cleanly.

For a regular tessellation, start with one of the three regular polygons that tessellate (equilateral triangle, square, or hexagon) and tile a flat surface with copies of that shape. The pattern will tessellate by definition.

For a semi-regular tessellation, combine two or more regular polygons that fit together at a vertex without gaps. The eight possible semi-regular tessellations are documented; pick one of those combinations and tile a surface using the appropriate shapes.

For an irregular tessellation, start with any shape and try to fit copies of it together. If the copies interlock without gaps or overlaps, the shape tessellates. Many irregular shapes will tessellate; the trick is finding shapes that produce visually interesting results.

For an Escher-style tessellation with curved shapes, start with a square (which tessellates by itself) and modify one side of the square by cutting a curve out of it. Translate the same curve to the opposite side of the square (so what's cut from one side is added to the other). The modified shape will still tessellate, but with curved interlocking edges. Continue modifying additional sides to create more complex tessellating shapes.

Penrose tilings and other aperiodic tessellations are more challenging to construct because the tile shapes follow specific matching rules. Penrose's "kite and dart" or "fat and thin rhombus" tile sets work, but only when assembled following the rules; random placement will not produce a Penrose tiling.

Computer tools and online tessellation generators can help with both the design and the testing of tessellations. Vector graphics software (Adobe Illustrator, Inkscape) lets you design and copy tile shapes precisely. Specialized math software (GeoGebra, Mathematica) lets you analyze the mathematical properties of a candidate tessellation.

What about tessellations in higher dimensions?

Tessellations extend naturally into three dimensions and beyond. A three-dimensional tessellation (sometimes called a space-filling tessellation or a honeycomb) covers three-dimensional space with copies of a three-dimensional shape, with no gaps or overlaps.

The simplest three-dimensional tessellation uses cubes; copies of a cube fill space exactly, producing the cubic honeycomb. The standard brick-wall pattern is a two-dimensional projection of a three-dimensional cubic honeycomb seen from one face.

More complex three-dimensional tessellations use other polyhedra. The truncated octahedron tessellates three-dimensional space; so does a combination of tetrahedra and octahedra. These tessellations appear in some crystal structures and in certain engineered materials.

Tessellations in non-Euclidean geometries (spherical, hyperbolic) follow different rules. On a sphere, only finitely many tessellations exist (corresponding to the five Platonic solids and a few related shapes). In hyperbolic geometry, infinitely many tessellations exist, including some that use shapes that cannot tessellate the Euclidean plane.

Higher-dimensional tessellations (in four, five, or more dimensions) exist mathematically and have applications in physics, materials science, and pure mathematics. Most everyday decorative and architectural tessellations stay in the two- or three-dimensional Euclidean cases.

Tessellating pattern questions

What is a tessellating pattern?

A tessellating pattern is a repeating arrangement of geometric shapes that fit together edge to edge to cover a flat plane without any gaps or overlaps. Tessellation patterns can use regular polygons (triangle, square, hexagon), combinations of regular polygons (semi-regular tessellation), or more complex shapes (irregular tessellation). Tessellating patterns appear in nature (honeycomb), in art (Islamic mosaics, Escher prints), in architecture (tile floors, brick walls), and in modern wallpaper design.

What is an example of a tessellating pattern?

Common examples include the honeycomb (regular tessellation of hexagons), a standard tile floor (regular tessellation of squares), a brick wall (irregular tessellation of rectangles), Islamic mosaic patterns (complex multi-shape tessellations), and M.C. Escher's tessellating animal prints (curved interlocking tessellations). Most repeating wallpaper patterns are tessellations of some kind.

What are the 3 rules of tessellation?

The three core rules of a tessellation are: the shapes fit together edge to edge (sharing full edges with neighbors), there are no gaps between shapes (the plane is completely covered), and there are no overlaps between shapes (each point on the plane belongs to exactly one tile). Any pattern that satisfies these three conditions is a tessellation.

What famous artist used tessellations?

The Dutch artist M.C. Escher (1898-1972) is the most famous tessellation artist. Escher produced many prints featuring tessellating shapes of fish, birds, lizards, and abstract forms, often with curved interlocking edges. Other notable tessellation artists include Bridget Riley (op art tessellations), Hokusai (Japanese tessellation prints), and the anonymous Islamic artists who produced the tessellation art of the Alhambra in Granada, Spain.

What does tessellation look like?

A tessellation looks like a repeating pattern of shapes that fit together perfectly to cover a surface. The visual appearance depends on the shapes used: regular hexagon tessellations look like honeycomb, square tessellations look like checkerboards, and complex semi-regular or irregular tessellations look like Islamic mosaic patterns or Escher prints. Every tessellation has a sense of rhythm and structural completeness from the way the shapes lock together.

What are the types of tessellations?

The main types of tessellations are regular tessellations (using a single regular polygon, only three exist: triangle, square, hexagon), semi-regular tessellations (using two or more regular polygons, eight exist), irregular tessellations (using shapes that are not all regular polygons), aperiodic tessellations (like Penrose tilings, which never repeat the same arrangement), and three-dimensional tessellations (space-filling tessellations using polyhedra).

Where are tessellations found in nature?

Tessellations appear naturally in honeycomb (hexagonal regular tessellation), snake skin scales, fish scales, pine cone scales, sunflower seed heads, giraffe spots, certain crystal patterns (mica, quartz), turtle shell scutes, and dragonfly wing veins. Patterns in nature follow the principles of tessellation when the shapes are constrained to fill space efficiently with minimal material, which is why hexagons appear so often in natural patterns.

What is the history of tessellations?

The history of tessellations runs from ancient Sumerian clay tile patterns (around 4000 BCE) through ancient Rome (mosaic flooring), through the highest development in Islamic art and architecture (especially the Alhambra in Granada), through medieval European church floors, through Kepler's seventeenth-century mathematical treatment, through M.C. Escher's twentieth-century art, through Penrose's 1970s discovery of aperiodic tessellations, to the 2023 discovery of the einstein aperiodic monotile.