Tessellation: Exploring the Mathematics of Repeating Patterns

Tessellation, also known as tiling, is the covering of a surface with one or more geometric shapes, called tiles, with no overlaps and no gaps. Mathematically, it's a fascinating area connecting geometry, art, and even physics. This article provides a comprehensive exploration of tessellations, covering their mathematical underpinnings, historical context, artistic applications, and real-world examples.

What is a Tessellation?

At its core, a tessellation is a pattern formed by repeating a shape or set of shapes to cover a plane. The key characteristics are:

No Gaps: The tiles must fit together perfectly, leaving no empty spaces between them.
No Overlaps: The tiles cannot overlap each other.
Complete Coverage: The tiles must cover the entire surface.

Tessellations can be classified based on the types of shapes used and the way they are arranged. Simple tessellations involve a single shape, while complex tessellations utilize multiple shapes.

Types of Tessellations

Tessellations can be broadly classified into the following categories:

Regular Tessellations

A regular tessellation is made up of only one type of regular polygon (a polygon with all sides and angles equal). There are only three regular polygons that can tessellate the plane:

Equilateral Triangles: These form a very common and stable tessellation. Think of triangular support structures in bridges or the arrangement of atoms in some crystal lattices.
Squares: Perhaps the most ubiquitous tessellation, seen in floor tiles, graph paper, and city grids around the world. The perfectly orthogonal nature of squares makes them ideal for practical applications.
Regular Hexagons: Found in beehives and some molecular structures, hexagons provide efficient space utilization and structural integrity. Their six-fold symmetry offers unique properties.

These three are the only possible regular tessellations because the interior angle of the polygon must be a factor of 360 degrees to meet at a vertex. For example, an equilateral triangle has angles of 60 degrees, and six triangles can meet at a point (6 * 60 = 360). A square has angles of 90 degrees, and four can meet at a point. A hexagon has angles of 120 degrees, and three can meet at a point. A regular pentagon, with angles of 108 degrees, cannot tessellate because 360 is not evenly divisible by 108.

Semi-Regular Tessellations

Semi-regular tessellations (also called Archimedean tessellations) use two or more different regular polygons. The arrangement of polygons at each vertex must be the same. There are eight possible semi-regular tessellations:

Triangle-square-square (3.4.4.6)
Triangle-square-hexagon (3.6.3.6)
Triangle-triangle-square-square (3.3.4.3.4)
Triangle-triangle-triangle-square (3.3.3.4.4)
Triangle-triangle-triangle-triangle-hexagon (3.3.3.3.6)
Square-square-square (4.8.8)
Triangle-dodecagon-dodecagon (4.6.12)
Triangle-square-dodecagon (3.12.12)

The notation in parentheses represents the order of the polygons around a vertex, going in a clockwise or counter-clockwise direction.

Irregular Tessellations

Irregular tessellations are formed by irregular polygons (polygons where sides and angles are not equal). Any triangle or quadrilateral (convex or concave) can tessellate the plane. This flexibility allows for a wide range of artistic and practical applications.

Aperiodic Tessellations

Aperiodic tessellations are tilings that use a specific set of tiles that can only tile the plane non-periodically. This means the pattern never repeats itself exactly. The most famous example is the Penrose tiling, discovered by Roger Penrose in the 1970s. Penrose tilings are aperiodic using two different rhombuses. These tilings have interesting mathematical properties and have been found in surprising places, like the patterns on some ancient Islamic buildings.

Mathematical Principles of Tessellations

Understanding the mathematics behind tessellations involves concepts from geometry, including angles, polygons, and symmetry. The key principle is that the angles around a vertex must add up to 360 degrees.

Angle Sum Property

As mentioned earlier, the sum of angles at each vertex must equal 360 degrees. This principle dictates which polygons can form tessellations. Regular polygons must have interior angles that are factors of 360.

Symmetry

Symmetry plays a crucial role in tessellations. There are several types of symmetry that can be present in a tessellation:

Translation: The pattern can be shifted (translated) along a line and still look the same.
Rotation: The pattern can be rotated around a point and still look the same.
Reflection: The pattern can be reflected across a line and still look the same.
Glide Reflection: A combination of reflection and translation.

These symmetries are described by what are known as wallpaper groups. There are 17 wallpaper groups, each representing a unique combination of symmetries that can exist in a 2D repeating pattern. Understanding wallpaper groups allows mathematicians and artists to classify and generate different types of tessellations systematically.

Euclidean and Non-Euclidean Geometry

Traditionally, tessellations are studied within the framework of Euclidean geometry, which deals with flat surfaces. However, tessellations can also be explored in non-Euclidean geometries, such as hyperbolic geometry. In hyperbolic geometry, parallel lines diverge, and the sum of angles in a triangle is less than 180 degrees. This allows for the creation of tessellations with polygons that would not be possible in Euclidean space. M.C. Escher famously explored hyperbolic tessellations in his later works, aided by the mathematical insights of H.S.M. Coxeter.

Historical and Cultural Significance

The use of tessellations dates back to ancient civilizations and can be found in various forms of art, architecture, and decorative patterns across the globe.

Ancient Civilizations

Ancient Rome: Roman mosaics often feature intricate tessellations using small colored tiles (tesserae) to create decorative patterns and depictions of scenes. These mosaics have been found throughout the Roman Empire, from Italy to North Africa and Britain.
Ancient Greece: Greek architecture and pottery often incorporate geometric patterns and tessellations. Meander patterns, for example, are a form of tessellation that appears frequently in Greek art.
Islamic Art: Islamic art is renowned for its complex geometric patterns and tessellations. The use of tessellations in Islamic art is rooted in religious beliefs that emphasize the infinite and the unity of all things. Mosques and palaces around the Islamic world showcase stunning examples of tessellations using various geometric shapes. The Alhambra palace in Granada, Spain, is a prime example, featuring intricate mosaics and tilework with various tessellated patterns.

Modern Applications

Tessellations continue to be relevant in modern times, finding applications in diverse fields:

Architecture: Tessellated surfaces are used in building facades, roofs, and interior designs to create visually appealing and structurally sound structures. Examples include the Eden Project in Cornwall, UK, with its geodesic domes composed of hexagonal panels.
Computer Graphics: Tessellation is a technique used in computer graphics to increase the detail of 3D models by subdividing polygons into smaller ones. This allows for smoother surfaces and more realistic renderings.
Textile Design: Tessellations are used in textile design to create repeating patterns on fabrics. These patterns can range from simple geometric designs to complex and intricate motifs.
Packaging: Tessellations can be used to efficiently pack products, minimizing waste and maximizing space utilization.
Science: Tessellating shapes are found in nature, such as the hexagonal cells of a honeycomb or the scales of certain fish. Understanding tessellations can help scientists model and understand these natural phenomena.

Examples of Tessellations in Art and Nature

Tessellations are not just mathematical concepts; they are also found in art and nature, providing inspiration and practical applications.

M.C. Escher

Maurits Cornelis Escher (1898-1972) was a Dutch graphic artist known for his mathematically inspired woodcuts, lithographs, and mezzotints. Escher's work often features tessellations, impossible constructions, and explorations of infinity. He was fascinated by the concept of tessellation and used it extensively in his art to create visually stunning and intellectually stimulating pieces. His works like "Reptiles", "Sky and Water", and "Circle Limit III" are famous examples of tessellations transforming into different forms and exploring the boundaries of perception. His work bridged the gap between mathematics and art, making mathematical concepts accessible and engaging to a wider audience.

Honeycomb

The honeycomb is a classic example of a natural tessellation. Bees construct their honeycombs using hexagonal cells, which fit together perfectly to create a strong and efficient structure. The hexagonal shape maximizes the amount of honey that can be stored while minimizing the amount of wax needed to build the comb. This efficient use of resources is a testament to the evolutionary advantages of tessellated structures.

Giraffe Spots

The spots on a giraffe, while not perfect tessellations, exhibit a pattern that resembles a tessellation. The irregular shapes of the spots fit together in a way that covers the giraffe's body efficiently. This pattern provides camouflage, helping the giraffe blend in with its environment. Although the spots vary in size and shape, their arrangement showcases a naturally occurring tessellation-like pattern.

Fractal Tessellations

Fractal tessellations combine the principles of fractals and tessellations to create complex and self-similar patterns. Fractals are geometric shapes that exhibit self-similarity at different scales. When fractals are used as tiles in a tessellation, the resulting pattern can be infinitely complex and visually stunning. These types of tessellations can be found in mathematical visualizations and computer-generated art. Examples of fractal tessellations include those based on the Sierpinski triangle or the Koch snowflake.

How to Create Your Own Tessellations

Creating tessellations can be a fun and educational activity. Here are some simple techniques you can use to create your own tessellations:

Basic Translation Method

Start with a Square: Begin with a square piece of paper or cardboard.
Cut and Translate: Cut a shape from one side of the square. Then, translate (slide) that shape to the opposite side and attach it.
Repeat: Repeat the process on the other two sides of the square.
Tessellate: You now have a tile that can be tessellated. Trace the tile repeatedly on a piece of paper to create a tessellated pattern.

Rotation Method

Start with a Shape: Begin with a regular polygon like a square or an equilateral triangle.
Cut and Rotate: Cut a shape from one side of the polygon. Then, rotate that shape around a vertex and attach it to another side.
Repeat: Repeat the process as needed.
Tessellate: Trace the tile repeatedly to create a tessellated pattern.

Using Software

There are various software programs and online tools available that can help you create tessellations. These tools allow you to experiment with different shapes, colors, and symmetries to create intricate and visually appealing patterns. Some popular software options include:

TesselManiac!
Adobe Illustrator
Geogebra

The Future of Tessellations

Tessellations continue to be an area of active research and exploration. New types of tessellations are being discovered, and new applications are being found in various fields. Some potential future developments include:

New Materials: The development of new materials with unique properties could lead to new types of tessellated structures with enhanced strength, flexibility, or functionality.
Robotics: Tessellated robots could be designed to adapt to different environments and perform various tasks. These robots could be composed of modular tiles that can rearrange themselves to change the robot's shape and function.
Nanotechnology: Tessellations could be used in nanotechnology to create self-assembling structures with specific properties. These structures could be used in applications such as drug delivery, energy storage, and sensing.

Conclusion

Tessellation is a rich and fascinating area of mathematics that connects geometry, art, and science. From the simple patterns of floor tiles to the complex designs of Islamic mosaics and the innovative art of M.C. Escher, tessellations have captivated and inspired people for centuries. By understanding the mathematical principles behind tessellations, we can appreciate their beauty and functionality and explore their potential applications in various fields. Whether you are a mathematician, an artist, or simply curious about the world around you, tessellations offer a unique and rewarding subject to explore.

So, next time you see a repeating pattern, take a moment to appreciate the mathematical elegance and cultural significance of tessellations!