Contents
Definition
A tessellation is a pattern of geometric shapes that fit together perfectly on a plane without any gaps or overlaps and can repeat in all directions infinitely. The word comes from the Latin word tessera, meaning a small square stone or tile used in ancient mosaics. Tessellations can be composed of one or more simple polygons, which are two-dimensional shapes with any number of straight sides. Examples of polygons include triangles, squares, rectangles, hexagons, and others.
Who
Tessellations (repeating shapes that fit together with no gaps or overlaps) are used by many different professionals to design structures, analyze data, and create art.
Architects and Engineers
- Design roofs and surfaces: They use geometric patterns to build strong, lightweight, and modern structures like domes and flat roofs.
- Improve structural efficiency: Repeating shapes spread weight evenly across a building.
Artists and Graphic Designers
- Create repeating patterns: Famous artists like Maurits Cornelis Escher used interlocking animal and shape patterns to make mesmerizing optical illusions and graphic art.
- Design textiles: Quilt makers, carpet weavers, and wallpaper designers use tessellations for repeating decorative prints.
Mathematicians and Computer Scientists
- Simplify complex equations: Statisticians use structures like Delaunay tessellations to break huge spaces into smaller cells so computers can solve equations faster.
- Render 3D graphics: Video game developers and animators use tiny tessellated triangles to build smooth surfaces on 3D models.
Origami Artists and Industrial Designers
- Fold compact shapes: Origami artists use crease-pattern tessellations to fold flat sheets of paper into complex 3D structures or collapsible medical stents and solar panels for space travel.
What
Types of Tessellations
There are three different types of tessellations: regular, semi-regular, and irregular tessellations.
Regular Tessellations
Regular tessellations are patterns made of regular polygons—shapes with sides of equal length and angles of equal size—that fit together perfectly without any gaps or overlaps.
Key Rules
- Edge-to-edge: The shapes meet edge-to-edge, meaning a side of one polygon rests completely against a side of another.
- Single shape type: Only one type of shape is used throughout the entire pattern.
- Divisible interior angles: The interior angle of the polygon must divide evenly into 360 degrees so the corners meet perfectly at a point without leaving open space.
The Only Three Types
Because of the math rules governing angles, only three regular polygons can form a regular [1-dimensional / 2-dimensional plane] tessellation:
- Equilateral triangles: Six triangles meet at each vertex (corner), where each angle is 60 degrees (6 × 60° = 360°).
- Squares: Four squares meet at each vertex, where each angle is 90 degrees (4 × 90° = 360°).
- Regular hexagons: Three hexagons meet at each vertex, where each angle is 120 degrees (3 × 120° = 360°). [2, 9]
Semi-Regular Tessellations
A semi-regular tessellation (also called an Archimedean tessellation) is a repeating flat pattern made of two or more different types of regular polygons where the exact same arrangement of shapes surrounds every single vertex. Semi-regular tessellations are made from two or more types of regular polygons such as triangles, squares, hexagons, and octagons.
Key Rules
- Edge-to-Edge: Shapes must meet edge-to-edge and corner-to-corner with no gaps or overlaps.
- Multiple Polygons: It must use at least two different types of regular polygons (like a mix of squares and triangles), unlike a regular tessellation which uses only one.
- 360-Degree Rule: The interior angles of the polygons meeting at any single corner must add up to exactly 360 degrees.
- Identical Vertices: The cyclic order of the shapes around every vertex must be identical.
How They are Named
Mathematicians name these patterns by listing the number of sides of the polygons in order around a vertex, usually starting with the smallest number and moving in a circle. For example, a vertex surrounded by three triangles and two hexagons in a row is written as 3.3.3.6.6.
How Many Exist?
There are exactly eight distinct semi-regular tessellations possible on a flat plane. You can explore visual examples and learn more about their properties on Semi-Regular Tessellations by NRICH or read the detailed overview on MathWorld’s Semiregular Tessellation Guide.
Irregular Tessellations
An irregular tessellation is a pattern that covers a flat surface using shapes that are not regular polygons, fitting together perfectly with no gaps or overlaps.
Key Characteristics
- Non-Regular Shapes: The individual tiles do not have equal sides and equal angles like squares or equilateral triangles.
- Infinite Variety: They can be made using any combination of irregular triangles, quadrilaterals, pentagons, or custom interlocking shapes.
- Complex Design: Unlike regular tessellations which repeat a single uniform shape, irregular patterns vary significantly in visual complexity.
Common Examples
- Art and Mosaics: The interlocking silhouettes created by artist M.C. Escher.
- Architecture: Stone floor tiles or brick pavements cut into non-uniform or mixed multi-sided shapes.
- Computer Graphics and GIS: Variable-size cells used in spatial data mapping to save memory by adapting to local details.
Einstein Tile
GOODMAN-STRAUSS – How Stuff Works
The einstein tile is an aperiodic monotile. An aperiodic monotile is a shape that tiles a plane, or an infinite two-dimensional flat surface, but only in a nonrepeating pattern. Your typical wallpaper or tiled floor is part of an infinite pattern that repeats periodically; when shifted, or translated, the pattern can be exactly superimposed on itself. An aperiodic tiling displays no such “translational symmetry,” and mathematicians have long sought a single shape that could tile the plane in such a fashion. This is known as the einstein problem.
Using a 13-sided “hat” shape, David Smith, a retired print technician and an aficionado of jigsaw puzzles, fractals and road maps, along with computer scientist Craig Kaplan and colleagues, confirmed this breakthrough through mathematical proof. This “aperiodic monotile” solves a 50-year-old mathematical mystery, representing a significant discovery that sits between order and chaos, potentially inspiring further exploration into complex geometric tiling.
The discovery of einstein tiles could influence architectural design and materials science by offering new ways to create non-repeating, aesthetically pleasing patterns for tiling and decoration. Here are some practical and scientific uses for the einstein tile.
- Photonic Crystals: Physicists construct nanoscale crystals out of Einstein tiles to control light movement. Shining a laser through these structures creates swirling, pinwheel-shaped scattering patterns that respond to the direction of the light’s spin.
- Advanced Material Design: By extruding the two-dimensional tile patterns into 2.5-dimensional forms, engineers create honeycombs with a zero Poisson’s ratio across a wide range of densities.
- Tougher Composites: Using the hat tiling to form continuous ductile networks within hard matrices provides increased stiffness and toughness compared to traditional hexagon-based networks.
- Continuously Varying Properties: Families of tiles based on the hat feature continuously varying geometry, which can give rise to materials with customized, variable mechanical behaviors like shifting elasticity.
- Molecular Quasicrystals: Researchers discovered chemical and molecular analogs that spontaneously self-assemble into chiral, non-periodic surfaces, paving the way for unique electronic and optical nano-devices.
- Art, Design, and Puzzles: Crafters, 3D printing enthusiasts, and artists use the monotile to design interlocking tiles, decorative mosaics, textiles, and tiling games that feature order without predictable repetition.
What Is the Difference Between Tessellations and Tiles?
While both tiles and tessellations can be made up of polygons without leaving any gaps or overlaps, there’s a significant difference between them. As we’ve said earlier, tessellations must have discernable repeating patterns. Tiles, on the other hand, may or may not include them.
In example #1, we see hexagons and rhombuses of equal size that fit together to cover the plane and form a pattern that is repeatable and predictable. So, example #1 shows a tessellation:
In example #2, we see rectangles filling up the plane without leaving any gaps or overlaps but also without a repeatable pattern. So, example 2 shows a tiling, not a tessellation:
Why
See Theoretical Knowledge Vs Practical Application.
How
Whether a shape can tessellate (i.e., cover a flat surface with no gaps or overlaps) depends entirely on interior angles. For a regular shape (where all sides and angles are equal) to tessellate, its interior angles must divide exactly into 360 degrees. This is because the shapes must fit together perfectly around a single point (vertex) without leaving a gap or overlapping.
The Rule of 360°
When the corners of shapes meet at a single point, they form a full circle. Therefore, the formula for a perfect tessellation at any vertex is:
If the number of shapes required is not a whole number, that shape cannot tessellate by itself.
Finding the Interior Angle
To test any regular shape, you first find its interior angle using this geometry formula, where n is the number of sides:
The Math in Action: What Works vs. What Fails
Only three regular shapes pass this mathematical test: triangles, squares, and hexagons.
| Shape | Number of Sides n | Interior Angle Formula | Interior Angle | Does it divide into $360°? | Tessellates? |
|---|---|---|---|---|---|
| Triangle | 3 | ((3 – 2) × 180°)/3 | 60° | Yes (360 ÷ 60 = 6) | Yes (6 meet at a point) |
| Square | 4 | ((4 – 2) × 180°)/4 | 90° | Yes (360 ÷ 90 = 4) | Yes (4 meet at a point) |
| Pentagon | 5 | ((5 – 2) × 180°)/5 | 108° | No (360 ÷ 108 = 3.33) | No (leaves a 360° – 324° = 36° gap) |
| Hexagon | 6 | ((6 – 2) × 180°)/6 | 120° | Yes (360 ÷ 120 = 3) | Yes (3 meet at a point) |
| Heptagon | 7 | ((7 – 2) × 180°)/7 | 128.57° | No (360 ÷ 128.57 = 2.8) | No (shapes will overlap) |
As the number of sides increases beyond 6, the interior angles keep getting larger than 120° but remain smaller than 180°. Because no whole number multiplied by these angles can equal exactly 360°, no other regular shapes can tessellate on their own.
Many of the References and Additional Reading websites and Videos will assist you with understanding and applying tessellations.
As some professors say: “It is intuitively obvious to even the most casual observer.“
References
Sikorski, Melissa. “What Is a Tessellation in Math? Definition, Types & Real-World Examples” Mathnasium, July 25, 2024. https://www.mathnasium.com/blog/what-is-tessellation-in-math.
[ ℰ ] Kirk, Donna. “10.5 Tessellations – Contemporary Mathematics.” OpenStax, March 22, 2023. https://openstax.org/books/contemporary-mathematics/pages/10-5-tessellations.
Gerbis, Nicholas. “How Tessellations Work.” HowStuffWorks, January 1, 1970. https://science.howstuffworks.com/math-concepts/tessellations.htm.
F. Champagne : Art and Design. “Tessellation Artists around the Globe.” February 25, 2017. https://champagnedesign.com/2017/02/24/tessellation-artists-around-the-globe/.
Lucinda. “Fun With Tessellations.” Navigating By Joy, May 15, 2013. https://navigatingbyjoy.com/2013/05/15/fun-with-tessellations/.
“Real Life Applications of Tessellations.” Grand Valley State University. Accessed October 2, 2026.
https://www2.gvsu.edu/oxfordj/angie.html.
“Einstein Problem.” Wikipedia, September 23, 2026. https://en.wikipedia.org/wiki/Einstein_problem.
Shields, Jesslyn. “The ‘Einstein’ Tile: Mathematicians Find a Shape That Never Repeats.” HowStuffWorks, January 1, 1970. https://science.howstuffworks.com/math-concepts/einstein-tile.htm.
Capella, Larissa G. “Crystal Made from 13-Sided ‘einstein’ Shape Bends Light in Ways Nobody Imagined.” Live Science, August 6, 2026. https://www.livescience.com/physics-mathematics/crystal-made-from-13-sided-einstein-shape-bends-light-in-ways-nobody-imagined.
Socolar, Joshua E. S. “Quasicrystalline Structure of the Hat Monotile Tilings.” Physical Review B 108, no. 22 (2023). https://doi.org/10.1103/PhysRevB.108.224109.
Additional Reading
Cottier, Cody. “At Long Last, Hobbyist Discovers ‘Einstein’ Tile.” Discover Magazine, April 27, 2023. https://www.discovermagazine.com/at-long-last-hobbyist-discovers-einstein-tile-44867.
Many aperiodic tiles run roughshod over an orderly background, “but this one,” Kaplan says, “just kind of sits there very nicely. It’s almost ridiculously well-behaved.”
Seymour, Dale, and Jill Britton. 1989. Introduction to Tessellations. Dale Seymour Publication.
This book explores a class of geometric patterns called tessellations, a pattern made up of one or more shapes which completely cover a surface without any gaps or overlaps. Focus in on one of the more elementary types of tessellations and investigation of them through concepts basic to art, geometry, and design. The seven chapters look at which shapes will tessellate, why certain shapes tessellate and others do not, how many different tessellating patterns can be created using two or more regular polygons; considerations of symmetry, and what techniques can be used to generate these intricate designs. Step-by-step instructions and suggested activities are provided for student production of both copied and original tessellations. An appendix, “Algebraic Analysis of Regular Polygons Around a Point,” a glossary, and a bibliography conclude the book.
Klarreich, Erica. “Hobbyist Finds Math’s Elusive ‘Einstein’ Tile.” Quanta Magazine, April 4, 2023. https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/.
Taggart, Emma. “The History of Tessellations: The Mathematic Art of Repeating Patterns.” My Modern Met, September 18, 2021. https://mymodernmet.com/tessellation-art/.
Tessellations are geometric patterns of repeating shapes that cover surfaces without gaps or overlaps. Dating back to 4000 BCE, this mathematical art form appears across history, most notably in Islamic zillij tilework found at the Alhambra. While regular and semi-regular tilings repeat, Roger Penrose discovered unique aperiodic patterns. Furthermore, artists like M.C. Escher and Koloman Moser famously integrated these precise mathematical structures into their own creative, decorative works.
Kunstler, Julianna. “TESSELLATIONS – Julianna Kunstler Art Lessons.” Julianna Kunstler. Accessed October 1, 2026. https://juliannakunstler.com/tessellations.
Art with Kunstler explores the history and geometric principles behind tessellations, from ancient mosaics to M.C. Escher’s illusions. Learn which basic shapes seamlessly cover a surface and how to construct custom tiles using translation, rotation, and reflection techniques.
Levin, Janna. “What Can Tiling Patterns Teach Us?” Quanta Magazine, July 3, 2024. https://www.quantamagazine.org/what-can-tiling-patterns-teach-us-20240703/.
If you cover a surface with tiles, repetitive patterns always emerge — or do they? Mathematician Natalie Priebe Frank joins host Janna Levin to discuss the significance of aperiodic tilings—shapes that cover a plane without repeating. Following decades of mathematical search, including the famous Penrose tilings, the 2023 discovery of an “aperiodic monotile” marked a major breakthrough. These complex geometric patterns, similar to quasicrystals, offer unique structural insights, bridging the gap between theoretical mathematics, computational logic, and physical
phenomena in the natural world.
“A Tip of the Hat: Celebrating the Aperiodic Monotile Discovery.” University of Cambridge. Accessed October 3, 2026. https://www.maths.cam.ac.uk/features/tip-hat-celebrating-aperiodic-monotile-discovery.
In 2023, mathematical hobbyist David Smith discovered the “hat,” an aperiodic monotile—or “einstein”—capable of tiling a plane without repeating patterns. Collaborating with Joseph Myers, Chaim Goodman-Strauss, and Craig Kaplan, Smith provided a formal proof for this breakthrough. Beyond its theoretical significance in understanding quasicrystals, the hat has inspired global artistic projects, including the Einstein Mad Hat Awards, highlighting the unique cultural impact of this mathematical discovery.
NovaPrism. “What Will Happen If a Non-Academic Mathematician Proves Such a Big Conjecture in Math, like Odd Perfect Numbers or Something?” Quora, October 11, 2026. https://qr.ae/pFmoE9.
Videos
What is Tessellation? | by M. C. Escher inspired Tessellation Art
What is Tessellation? An educational video animation, which explains the basic principles behind tessellating shapes and patterns. It shows a simple visual demonstration of tessellating triangles, squares and hexagons. It also explains how they can be transformed using translation, rotation and glide reflection to create shapes like fish.
Tessellations: Part1
An introduction to tessellations. This video explains WHY there are only three regular polygons that tessellate on their own. It demonstrates visually and mathematically that only triangles, squares, and hexagons will be the only regular polygons to tessellate. It also demonstrates translation, rotation, and glide-reflection tessellations.
12.1 Tessellations of Regular and Irregular Polygons
Matt Bresciani reviews the definitions of polygons and tessellations from a previous grade. This lesson demonstrates how to determine if specific shapes can tile a plane based on whether their interior angles sum to 360 degrees.
Tessellation – Ian Bailey-Mortimer
Notes
Archimedean tessellation
An Archimedean tessellation, also known as a semiregular tessellation, is a tiling that uses only regular polygons arranged so that two or more different polygons are around each vertex and each vertex involves the same pattern of polygons. There are eight such tessellations, two involving triangles and squares, two involving triangles and hexagons, and one each involving squares and octagons; triangles and dodecagons; squares, hexagons and dodecagons; and triangles, squares and hexagons.
Darling, David. “Archimedean Tessellation.” David Darling. Accessed October 1, 2026. https://www.daviddarling.info/encyclopedia/A/Archimedean_tessellation.html.
einstein
The term “einstein” comes from the German “ein stein,” or “one stone” — more loosely, “one tile” or “one shape.”
The Einstein Tile (Not THAT Einstein)
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