Tessellation — Definition, Formula & Examples
A tessellation is a repeating pattern of one or more shapes that completely covers a flat surface (plane) without any gaps or overlaps. Common examples include the hexagonal pattern in honeycombs and the square tiles on a bathroom floor.
A tessellation (or tiling) of the plane is a collection of closed shapes that covers the Euclidean plane such that every point in the plane belongs to at least one shape, and the interiors of any two distinct shapes do not intersect. A regular tessellation uses congruent copies of a single regular polygon; only equilateral triangles, squares, and regular hexagons can form regular tessellations.
Key Formula
Where:
- = Number of sides of the regular polygon
How It Works
To check whether a regular polygon can tessellate by itself, look at its interior angle. The copies of the polygon meet at each vertex, so the interior angle must divide evenly into . For example, a square has a interior angle, and , so four squares fit perfectly around every vertex. You can also build semi-regular tessellations by combining two or more types of regular polygons, as long as the angles at each vertex still sum to exactly . Tessellations rely on geometric transformations — translations, rotations, and reflections — to repeat the basic motif across the plane.
Worked Example
Problem: Determine whether a regular octagon (8 sides) can tessellate the plane by itself.
Step 1: Calculate the interior angle of a regular octagon.
Step 2: Check whether 135° divides evenly into 360°.
Step 3: Since the result is not a whole number, you cannot fit a whole number of regular octagons around a single vertex without gaps or overlaps.
Answer: A regular octagon cannot tessellate the plane by itself because its 135° interior angle does not divide evenly into 360°.
Another Example
Problem: Show that equilateral triangles can tessellate the plane.
Step 1: Find the interior angle of an equilateral triangle.
Step 2: Divide 360° by 60° to see how many triangles meet at one vertex.
Step 3: Six equilateral triangles fit perfectly around each vertex, so the pattern extends across the entire plane with no gaps.
Answer: Equilateral triangles tessellate the plane, with six triangles meeting at every vertex.
Why It Matters
Tessellations appear throughout geometry courses when studying transformations, symmetry, and angle relationships. Architects and graphic designers use tessellations to create efficient and visually appealing patterns in flooring, wallpaper, and digital art. The concept also connects to crystallography, where scientists analyze how atoms arrange themselves in repeating lattice structures.
Common Mistakes
Mistake: Assuming any regular polygon can tessellate the plane.
Correction: Only equilateral triangles, squares, and regular hexagons work as regular tessellations. Always check that the interior angle divides evenly into 360°.
Mistake: Forgetting that irregular or non-convex shapes can also tessellate.
Correction: Every triangle and every quadrilateral (even non-convex ones) can tessellate the plane through appropriate rotations and translations. The "only three" rule applies strictly to regular polygons.