Tessellation in Mathematics
The Building Blocks of Tessellations
At its heart, a tessellation is like a perfectly laid tile floor that extends forever in all directions. The most basic rule is simple: the shapes must fit together with no empty spaces (gaps) and no parts lying on top of each other (overlaps). The point where the corners of several tiles meet is called a vertex (plural: vertices). The arrangement of shapes around a vertex is a key to understanding the pattern.
The simplest tessellations are made from a single, repeated shape. Let's start with regular polygons[1]. A regular polygon has all sides the same length and all interior angles the same size. Not all regular polygons can tessellate by themselves. For a single regular polygon to tessellate, its interior angles must divide evenly into 360°.
Let's test this with the three regular polygons that work: equilateral triangles, squares, and regular hexagons.
| Regular Polygon | Sides (n) | Interior Angle | Angles Around a Vertex | Does it Tessellate? |
|---|---|---|---|---|
| Equilateral Triangle | 3 | 60° | 6 x 60° = 360° | Yes |
| Square | 4 | 90° | 4 x 90° = 360° | Yes |
| Regular Pentagon | 5 | 108° | 3 x 108° = 324° (Gap!) | No |
| Regular Hexagon | 6 | 120° | 3 x 120° = 360° | Yes |
As the table shows, only the triangle, square, and hexagon can create a regular tessellation. A regular pentagon leaves a gap, and any polygon with more than six sides would have interior angles larger than 120°, meaning you can't even fit three around a point without overlapping.
Beyond Regular Polygons: Semi-Regular and Irregular Tessellations
The world of tessellations becomes much richer when we move beyond a single shape. A semi-regular tessellation (or Archimedean tiling) uses two or more types of regular polygons, and the same arrangement of polygons appears in the same order at every vertex. There are only 8 possible semi-regular tessellations. A classic example is the pattern of squares and regular octagons, often seen in tiled floors.
But we don't have to be limited to regular polygons! Irregular tessellations use shapes that are not regular. Any triangle or any quadrilateral (four-sided shape) can tessellate on its own. You can test this yourself by drawing a random quadrilateral on a piece of paper. If you trace it, rotate it 180° around the midpoint of each side, the shapes will fit together perfectly. This is because the interior angles of any quadrilateral still sum to 360°, allowing them to fit around a point.
Furthermore, artists like M.C. Escher[2] famously used irregular shapes that resembled animals, birds, and fish to create mesmerizing tessellations. He started with a simple geometric grid (like squares or hexagons) and then modified the sides by cutting a shape from one side and adding it to the opposite side. This process, called a slide transformation, ensures the piece still fits together perfectly.
Tessellations in Nature and Human Design
Tessellations are not just a human invention; they are everywhere in the natural world. The most famous example is the honeycomb built by bees. The regular hexagonal pattern is a marvel of efficiency: it uses the least amount of wax to create the largest possible storage space for honey. This is a perfect example of mathematics in nature solving an optimization problem.
Other natural tessellations include the pattern of scales on a snake or fish, the cracking of dried mud, the surface of a pineapple, and the arrangement of cells in a leaf. In each case, the tessellating pattern provides strength, flexibility, or efficient use of materials.
Humans have adopted these efficient patterns for our own use. Think about a brick wall: the rectangular bricks are laid in a staggered pattern to create a strong, stable structure. Tiled floors and ceilings in buildings, paving stones on a path, and the geometric patterns in Islamic art are all examples of human-made tessellations. In modern technology, tessellating patterns are used in computer graphics to create realistic 3D models and in engineering to design strong, lightweight structures for everything from airplanes to stadiums.
Create Your Own Simple Tessellation
Creating your own tessellation is a fun and easy way to understand the concept. Let's start with a square, one of the three regular tessellations.
- Draw a square on a piece of paper or cardstock.
- On the top side, draw a random curved or zig-zag line from the left corner to the right corner. Cut along this line.
- Without rotating it, slide this cut-out piece directly to the bottom side of the square and tape it there.
- Now, on the left side, draw another random line from the top corner to the bottom corner. Cut along this line.
- Without rotating it, slide this piece directly to the right side and tape it there.
You now have a strange, amoeba-like shape. This is your tile. If you trace this shape repeatedly, fitting the traced copies together like a puzzle, you will see that they cover the page with no gaps or overlaps. You have created an irregular tessellation! You can now decorate your tile to look like a creature or an abstract object.
Common Mistakes and Important Questions
Q: Can a circle tessellate a plane?
A: No, a circle cannot tessellate. No matter how you arrange circles, there will always be gaps between them. A tessellation requires the shapes to cover the plane completely with no gaps.
Q: Is every pattern a tessellation?
A: No. A pattern like polka dots on a shirt is not a tessellation because the dots do not fit together to cover the entire surface; there is background fabric visible between the dots. A true tessellation leaves no uncovered space.
Q: Why are there only three regular tessellations?
A: This is due to the mathematical constraint that the interior angles of the polygons meeting at a vertex must add up to 360°. As shown in the table, only the equilateral triangle, square, and regular hexagon have interior angles that are factors of 360°.
Footnote
[1] Regular Polygon: A polygon that is both equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Examples include the equilateral triangle and the square.
[2] M.C. Escher: Maurits Cornelis Escher (1898-1972) was a Dutch graphic artist famous for his mathematically inspired woodcuts, lithographs, and mezzotints, which often featured impossible constructions, explorations of infinity, and complex tessellations.