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. Every tile shares whole edges with the tiles around it, and the same group of shapes comes back again and again across the pattern, in a regular or semi-regular way. Nothing tells it to stop. Run out of floor and the tessellation, on paper anyway, just carries on in every direction.

The word started on a floor. In Latin a tessella was a small cube or tile, originally one piece of a Roman mosaic, and English took the verb (to tessellate, meaning to cover a surface with a repeating pattern of one or more shapes) and the noun, a tessellation, from it. Tessellated is the past participle, and it works as an adjective.

Most of the time the tiles are regular polygons. Squares and hexagons are favorites, or equilateral triangles, alone or mixed. Plenty of other shapes can tessellate, though, from irregular polygons to curved shapes that interlock in very specific ways. What can't vary in a tessellation is the fit (no gaps, no overlaps, and you're in).

Then there are the corners. A vertex is the point where three or more bordering tiles meet, and every tessellation has them. When one regular shape meets the same way at every vertex point, that's a regular tessellation. A semi-regular tessellation keeps every vertex identical too, but with two or more types of regular polygon. An irregular tessellation lets its vertices vary.

Nature makes tessellating patterns without trying, in honeycomb and in snake scales. Certain crystals grow that way.

In art, Islamic mosaics are the old masters. M.C. Escher's prints and modern op art came much later, and William Morris designs belong on the list as well (we're biased). Mathematicians have their own claim, since 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. Meet all three and it's a tessellation. Miss one and it's some other kind of pattern.

Edge to edge is the strict one. Neighboring tiles have to share full edges, not a partial edge and not just a corner. When tiles meet only at corners, without full edge contact, you don't have a strict tessellation; you may have a related arrangement called a non-edge-to-edge tiling. Most decorative tessellations share full edges anyway.

The other two you can check by eye. Leave empty space between shapes and the tiles no longer cover the plane completely, so you've made a partial tiling (or a sparse arrangement). Let one tile sit on top of another and it's a layered design. In a real tessellation each point on the surface is covered by exactly one tile. Never two, never none.

Corners decide which shapes can form a tessellation alone. Where tiles meet, the interior angles have to add up to 360 degrees, and with copies of a single regular polygon only three shapes pass. The equilateral triangle is one. The square and the hexagon are the others. No other regular polygon forms a tessellation by itself, not even the pentagon, because those interior angles don't divide evenly into 360 (lay pentagons around a point on a table and something always falls short or overlaps).

Mixing shapes opens things up, but only to eight semi-regular tessellations, each with its own fixed recipe. One pairs squares with equilateral triangles, another hexagons with triangles. Squares and octagons make a third. Wherever you stop and check, the vertex looks the same.

Irregular tessellations are the relaxed cousins. Any combination of shapes qualifies if it fits together without gaps or overlaps, even when the tiles differ in shape or size and the vertices don't all match. That covers most decorative tile patterns. Many natural patterns belong here, and so do William Morris wallpaper repeats, the kind we spend most of our time with.

What are the main types of tessellations?

Tessellations fall into several major types based on the shapes used and the vertex structure. Two of the groups are short, closed lists. The rest are wide open.

Regular tessellation is the smallest club, with three members, each built from copies of a single regular polygon. The triangle tessellation has six equilateral triangles meeting at each vertex. In the square tile tessellation four squares meet at every vertex, and the hexagon tessellation has three hexagons, which is the honeycomb pattern (bees got there first).

Fit two or more types of regular polygon together with every vertex identical and you get a semi-regular tessellation. There are eight semi-regular tessellations. The truncated square (squares plus octagons) is the easiest to sketch, and the triangular-hexagonal patterns are among them too. Some people call them Archimedean tessellations, after Archimedes.

Everything left over is an irregular tessellation, with shapes that aren't all regular polygons or vertices that aren't all identical. Brick patterns sit here. So do fish-scale patterns, along with most decorative tile patterns and a lot of wallpaper repeats.

Aperiodic tessellations are the odd ones out, filling the plane without repeating in a regular periodic way. The most famous are Penrose tilings, discovered by Roger Penrose in the 1970s, with two specific tile shapes that can tile the plane forever and never repeat the same arrangement.

Some fill space instead of a plane. Three-dimensional tessellations (space-filling tessellations, or honeycombs) use solid shapes, again with no gaps or overlaps. Cubes do it. So do truncated octahedra and certain other polyhedra.

Curves work as well, provided they interlock cleanly, and those are the curved tessellations. M.C. Escher's famous prints often use them (his lizards are curved tiles, and so are the fish and birds).

What are examples of tessellating patterns?

Tessellated patterns appear everywhere once you start looking, from natural patterns to jigsaw puzzles to ancient mosaic floors. A few might be on your walls right now.

Honeycomb is the classic natural tessellation, a regular tessellation of hexagons. Bees aren't doing geometry for fun; the hexagon is the most efficient regular shape for filling space with the least amount of wax.

A standard square tile floor is a regular tessellation of squares, the same pattern you see on city sidewalks and on a lot of bathroom and kitchen floors. A chessboard is that pattern again.

Brick is a quieter example. A standard wall of identical brick rectangles, laid in a running bond or a stack bond, is a valid tessellation (running bond isn't edge-to-edge, which the strictest definitions want, but it still covers the wall with no gaps or overlaps) (not that a bricklayer would care).

Islamic art and architecture produced some of the most complex tessellation art in history. Artists there worked from the mathematical relationships between regular polygons to design endlessly varied surface decoration. If you see one place, make it the Alhambra in Granada, Spain, where interlocking geometric shapes cover walls and floors throughout the palace complex. Many of the most famous Islamic tessellations are there.

Rome was earlier. Roman buildings used mosaic patterns extensively, and the Roman pattern catalog has many tessellating designs, often complex multi-shape combinations, in stone or ceramic, sometimes glass tile.

Jump to the twentieth century. The Dutch artist M.C. Escher (1898-1972) made many famous prints from tessellating shapes of animals and people, with abstract forms mixed in, using curved interlocking tessellations for his characteristic shifting-image graphic art.

Op art arrived in the 1960s and pushed tessellating patterns toward illusion. Bridget Riley and other op art artists built compositions on regular and semi-regular tessellations that seem to move. They play with depth and shifting perception, too.

Penrose tilings come at it from the math side. Roger Penrose's aperiodic tessellations use two specific "kite" and "dart" tile shapes, or alternatively two "rhombus" shapes, to cover the plane without ever repeating the same arrangement. They're a famous example now, in mathematics and in the decorative arts.

Wallpaper is the one we know best. Most repeating wallpaper patterns are tessellations of some kind, and a William Morris wallpaper pattern is an irregular tessellation, each repeat tile holding a complex floral or geometric design that interlocks cleanly with its neighbors. Hang a dozen strips on a bedroom wall and you've tiled it with one pattern repeat (our favorite kind of tessellation, frankly).

Nature has plenty more. Snake skin scales and pine cone scales both tessellate. Sunflower seed heads pack their seeds in Fibonacci spirals, and giraffe spots show a natural tessellating pattern, as do certain crystal patterns.

What is the history of tessellation?

The history of tessellations runs back to ancient civilizations and forward through every period of art and mathematics. It starts in Mesopotamia, around 4000 BCE, where the earliest tessellations turn up in Sumerian and Mesopotamian decorative tile work. Sumerian builders covered walls and floors in clay tiles set in simple geometric repeating patterns.

Egypt used tessellation patterns in its decorative arts. So did ancient Greece, and Rome made them an empire-wide habit, laying mosaic floors and walls with complex multi-color tessellating designs. Some were simple geometric patterns. Others were elaborate figural scenes built from tessellating tile units.

Islamic art and architecture took tessellation to its highest level. From the seventh century onward, Islamic architects and artists made the patterns steadily more complex, on mosques and palaces, then on mausoleums and decorative objects as well. The Alhambra, built between 1238 and 1492, is often said to hold all 17 wallpaper-group symmetries, though mathematicians still argue over how many are really there. A remarkable mathematical achievement, too, since its designers got there centuries before the underlying geometry was formally proved.

In medieval Europe tessellation moved into churches. Gothic cathedral floors often carried complex geometric tessellating patterns in stone and ceramic tile, and medieval and Gothic builders used tessellation in stained glass and decorative carving as well.

The math ran alongside. Ancient Greek mathematicians studied regular polygons and their tessellations, but it was Johannes Kepler (1571-1630) who gave tessellations their first systematic mathematical treatment, classifying the regular and semi-regular tessellations and analyzing their properties.

In the late nineteenth century the classification of wallpaper groups, the 17 possible symmetry types of two-dimensional repeating patterns, was completed. M.C. Escher's tessellation prints then made the idea widely known in twentieth-century art. Roger Penrose found his aperiodic tessellations in the 1970s, and a fifteenth type of tiling pentagon turned up in 2015, so the field never sat still.

It still hasn't. Recent mathematical research has discovered additional pentagonal tessellations and keeps exploring the properties of complex aperiodic and curved tessellations. Then, in 2023, came the "einstein" aperiodic monotile, a single tile shape that tessellates aperiodically. A major mathematical achievement, and a long way from those Sumerian clay tiles.

Where are tessellating patterns used today?

Tessellating patterns appear widely in contemporary design, architecture, decorative arts, and mathematical study. Why do they stick around? Structural strength, for a start. Good looks help, and so does a mathematical elegance that designers keep coming back to.

Walls come first for us. Most repeating wallpaper designs are tessellations of some kind, and many contemporary wallpaper collections feature openly geometric tessellation patterns.

Tile work is the obvious home, on floors and walls in kitchens and bathrooms (entryways too). Many tile shapes make a tessellating pattern, from plain square tile to fish-scale, and designer tile work often reaches for complex multi-shape semi-regular tessellations as a statement.

Take a hexagon backsplash behind a stove. Three hexagons meet at every vertex, so it's a regular tessellation, the honeycomb in ceramic. Swap in octagons with small squares between them and you've moved to the truncated square, one of the eight semi-regular tessellations. Same wall, different math (in a small kitchen we'd pick the hexagons, they're calmer).

Upholstery and drapery carry tessellations, and so does dress fabric. Quilters take it literally, since many quilt patterns are explicit tessellations with blocks designed to interlock, and designer fabric often goes op-art-style or Islamic-inspired.

Graphic art and digital design put tessellations into logos and illustrations. Posters too, and screen patterns. Online tessellation generators build a custom tessellation from a shape you draw, so non-mathematicians can make one.

Architects use tessellation patterns in facades and paving, and contemporary architects from many countries have used complex tessellations as both structural and decorative features, often referencing the Islamic architectural tradition.

And then there's the classroom. Tessellation activities are standard in elementary and secondary mathematics curricula in many countries, where they teach geometry and symmetry. Polygon properties and pattern analysis come along with them.

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. Plenty of tessellated patterns are designed exactly that way. The trick is finding shapes whose edges follow the principles of tessellation cleanly.

A regular tessellation is close to cheating. Take a square and tile a flat surface with copies. An equilateral triangle or a hexagon works the same way (the only other regular polygons that tessellate alone), and the result tessellates by definition.

Semi-regular tessellations take a little more planning: two or more regular polygons that fit together at a vertex without gaps. Don't go hunting for new ones. The eight possible semi-regular tessellations are documented, so pick one of those combinations and tile with the shapes it calls for.

An irregular tessellation is trial and error. Start with any shape and try fitting copies together; if they interlock without gaps or overlaps, the shape tessellates. Many irregular shapes will. Getting one that produces a visually interesting result is the actual work.

Escher-style curved tiles are the fun ones to make, and a paper square and scissors will do. A square tessellates by itself, so it's a safe start. Cut a curve out of one side, slide that piece straight across to the opposite side and tape it on (what's cut from one side is added to the other). Trace it a few times to see the pattern. It still tessellates, only now the edges curve and lock like jigsaw pieces, and if you keep modifying the other sides a fish or a bird starts to appear.

Penrose tilings and other aperiodic tessellations are harder to construct, because the tile shapes follow specific matching rules. Penrose's "kite and dart" set works, as does the "fat and thin rhombus" set, but only when assembled by the rules. Random placement will not produce a Penrose tiling.

Software helps with the design and the testing, and so do online tessellation generators. For drawing and copying tile shapes precisely we'd use vector graphics software like Adobe Illustrator or Inkscape, while GeoGebra or Mathematica will analyze the mathematical properties of a candidate tessellation.

What about tessellations in higher dimensions?

Tessellations extend naturally into three dimensions and beyond. Add depth and the tiles turn into solids. A three-dimensional tessellation (you'll also see space-filling tessellation, or honeycomb) covers space with copies of a three-dimensional shape, no gaps or overlaps.

Cubes are the easy case. Stack copies and they fill space exactly, which gives you the cubic honeycomb. Look at any one face of that stack and you get a plain square grid.

Other polyhedra make it trickier. The truncated octahedron tessellates three-dimensional space on its own; tetrahedra and octahedra manage it in combination. Tessellations like these appear in some crystal structures and in certain engineered materials.

Non-Euclidean geometries, spherical or hyperbolic, follow different rules. On a sphere only five regular tessellations exist, one for each Platonic solid. Hyperbolic geometry goes the other way. There, infinitely many tessellations exist, including some built from shapes that cannot tessellate the Euclidean plane.

Tessellations in four, five or more dimensions exist mathematically, with applications in physics and materials science (and pure mathematics). For a floor or a wall, though, you're in the two- and three-dimensional Euclidean cases, like most everyday decorative and architectural tessellations. Which suits a wallpaper company fine.

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. Honeycomb is one tessellation, and a brick wall is another. Plenty of modern wallpaper design qualifies as well, while the famous cases are Islamic mosaics and Escher prints.

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, so there may be one on the wall beside you.

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). Fail even one, by a sliver, and it's no longer a tessellation.

What famous artist used tessellations?

The Dutch artist M.C. Escher (1898-1972) is the most famous tessellation artist. Bridget Riley's op art belongs on the list too, though the anonymous Islamic artists of the Alhambra in Granada, Spain, got there centuries before any of them.

What does tessellation look like?

A tessellation looks like a repeating pattern of shapes that fit together perfectly to cover a surface. Hexagons give you honeycomb and squares a checkerboard; go semi-regular or irregular and the tessellation starts to resemble an Islamic mosaic or an Escher print. To our eye the giveaway is rhythm, that locked-in sense of a surface that's complete.

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). Only the first two types of tessellation are closed lists, at three and eight.

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. The reason is thrift: when shapes are constrained to fill space efficiently with minimal material, patterns in nature follow the principles of tessellation, which is why hexagons keep appearing.

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 two-tile aperiodic sets of the 1970s, to the 2023 discovery of the einstein aperiodic monotile. Want to see the high point of tessellation in person? Go to Granada.