An irregular tessellation is a pattern of shapes that covers a flat surface completely, with no gaps and no overlaps, using shapes that aren’t regular polygons or that combine different shapes without a strict repeating arrangement. Unlike the neat grids of squares or honeycombs of hexagons you might picture, irregular tessellations can use polygons with unequal sides, curved figures, or even freeform shapes to tile a plane.
How Tessellations Work
Every tessellation, whether regular or irregular, follows the same basic rules. The shapes must fit together perfectly so that no part of the surface is left uncovered and no shapes overlap. At every point where shapes meet (called a vertex), the interior angles of the surrounding shapes must add up to exactly 360 degrees. This is what allows the pieces to sit flush against one another, like tiles on a floor.
Beyond that core requirement, the shapes can be combined using transformations: sliding, rotating, or flipping them into position. As long as the result is a seamless covering of the plane, it qualifies as a tessellation.
Regular and Semi-Regular Tessellations
To understand what makes a tessellation “irregular,” it helps to know what the regular versions look like. A regular tessellation uses only one type of regular polygon, meaning every side and angle on that polygon is equal. Only three regular polygons can tile a plane on their own: equilateral triangles, squares, and regular hexagons. That’s it. No other single regular polygon works, because the interior angles won’t divide evenly into 360 degrees at each vertex.
A semi-regular tessellation loosens the rules slightly. It uses two or more types of regular polygons, arranged so that every vertex has the same combination of shapes around it. There are exactly eight semi-regular tessellations, such as the pattern of octagons and squares you sometimes see on bathroom floors. The key constraint is that while the shapes differ, the arrangement at each vertex is identical throughout the pattern.
What Makes a Tessellation Irregular
An irregular tessellation breaks free from both of those constraints. You can use any polygon, any shape, or any figure to create one. The shapes don’t need equal sides or equal angles. The arrangement at each vertex doesn’t need to be the same across the whole pattern. The only rule that remains is the fundamental one: the shapes must cover the plane without gaps or overlaps.
This opens up enormous possibilities. You could tile a surface with oddly shaped quadrilaterals, with a mix of triangles that have different proportions, or with entirely freeform figures. The Dutch artist M.C. Escher famously exploited this freedom, creating tessellations from interlocking birds, fish, and lizards. Each of his figures was carefully designed so its edges fit perfectly against the edges of its neighbors, satisfying the no-gaps, no-overlaps rule while looking nothing like a geometric grid.
Some irregular tessellations still repeat. You might use an L-shaped piece that tiles the plane in a predictable, periodic way. Others are aperiodic, meaning the pattern never repeats no matter how far it extends. One well-known example is the Penrose tiling, which uses just two shapes of parallelogram (one smaller, one larger) to fill the plane in a pattern that looks orderly locally but never settles into a repeating cycle.
The 2023 “Einstein” Tile
For decades, mathematicians wondered whether a single shape could tile the plane but only in a non-repeating way. This hypothetical shape was nicknamed an “einstein,” from the German for “one stone.” In 2023, a team of researchers answered the question by discovering a 13-sided polygon they called the “hat.” The hat can cover an infinite plane, but it is mathematically impossible for it to produce a periodic, repeating pattern. A related shape called the “spectre” followed shortly after, refining the discovery further. These are irregular tessellations in the purest sense: a single, non-regular polygon that tiles the plane in a way no regular polygon ever could.
Irregular Tessellations in Nature
Nature is full of irregular tessellations, even if they don’t look like textbook geometry at first glance. The spots on a giraffe form a pattern where dark borders divide the skin into irregularly shaped polygons that cover the surface completely. Dried mud cracks into a tessellation as water evaporates and the surface contracts, producing a patchwork of uneven polygonal pieces. Dragonfly wings are divided by veins into irregular cells that tile the entire wing membrane. Leaf veins do something similar, creating a network that distributes fluid to every part of the leaf while subdividing its surface into small, irregular regions.
Many of these natural patterns closely resemble a mathematical structure called a Voronoi tessellation (sometimes called a Voronoi diagram). A Voronoi tessellation starts with a set of seed points scattered across a surface. Each point “claims” the region of space closer to it than to any other seed point, and the boundaries between these regions form a tiling of irregular polygons. When researchers place points in the centers of a giraffe’s spots and generate a Voronoi diagram from them, the result closely matches the animal’s actual pattern.
Practical Uses in Technology
Irregular tessellations aren’t just mathematical curiosities. They are essential tools in computing, science, and engineering.
In computer graphics and engineering simulations, complex surfaces and three-dimensional objects need to be broken into small, simple pieces so that a computer can calculate forces, temperatures, or light behavior across them. This process, called mesh generation, typically divides a surface into irregular triangles or tetrahedra. The most widely used approach is Delaunay triangulation, which connects a set of points into triangles following specific geometric rules. The result is an irregular tessellation tailored to the shape being modeled, with smaller triangles clustered in areas that need more detail and larger ones where less precision is needed.
Voronoi tessellations have an equally rich history of practical applications. One of the earliest documented uses dates to 1854, when physician John Snow mapped cholera deaths in London and showed that proximity to a contaminated water pump predicted who fell ill. His analysis was essentially a Voronoi diagram, assigning each death to its nearest pump. Since then, Voronoi tessellations have been applied in crystallography to study how atoms pack into space-filling structures, in ecology to model the territory each tree or plant effectively controls, in atmospheric science to build global climate models, and in biology to map capillary distribution in tissue.
How to Tell if a Shape Can Tessellate
Every triangle and every quadrilateral, no matter how irregular, can tessellate the plane. This is a proven mathematical fact. You can take any triangle with three completely different side lengths and angles, and copies of it will tile a flat surface perfectly. The same goes for any four-sided polygon, including concave ones (shapes with an inward-pointing angle).
For polygons with five or more sides, things get more complicated. Only certain families of pentagons can tessellate, and identifying all of them was an open problem for decades. For convex polygons with seven or more sides, tessellation is impossible. The interior angles become too large to combine to 360 degrees at a vertex. Hexagons sit at the boundary: some convex hexagons tessellate and others don’t, depending on their specific angles and side lengths.
This means irregular tessellations are, in a sense, far more common than regular ones. The three regular tessellations are special, constrained cases. The vast majority of shapes that can tile a plane are irregular, and the patterns they produce are endlessly varied.