Tessellations — Mathematics Knowledge System

Overview

A tessellation (or tiling) is a covering of the plane using one or more repeated shapes, called tiles, with no gaps and no overlaps. A tessellation is regular if it uses only one type of regular polygon, meeting edge-to-edge; only three regular polygons tessellate the plane this way: equilateral triangles, squares, and regular hexagons. A tessellation is semi-regular if it mixes two or more types of regular polygons, arranged identically at every vertex. The key numerical requirement behind every tessellation is that the interior angles meeting at any single vertex must sum to exactly 360°, since the tiles must fit together with no gap and no overlap all the way around that point.

Intuition

Think of tiling a bathroom floor: whatever shape of tile you pick, the angles of the tiles meeting at any corner point must add up to a full 360° turn, or you'll either leave a gap or force tiles to overlap. A regular hexagon's interior angle is 120°, and exactly three of them (3 × 120° = 360°) fit perfectly around a point — which is why honeycomb cells, floor tiles, and many crystal structures favor hexagons. A regular pentagon's interior angle is 108°, and no whole number of 108° angles sums to exactly 360° (three give 324°, four give 432°), which is the precise reason regular pentagons alone can never tile the plane.

Interactive Graph

Formal Definition

Definition

For a regular polygon with n sides, its interior angle is given below; a set of regular polygons tessellates edge-to-edge at a vertex exactly when their interior angles sum to 360°:

\theta_n = \frac{(n-2)\cdot 180^\circ}{n}

Interior angle of a regular n-gon

\sum_{\text{tiles at a vertex}} \theta = 360^\circ

Vertex condition for a tessellation with no gaps or overlaps

n \in \{3, 4, 6\}

Only these regular polygons tile the plane by themselves (triangle, square, hexagon)

Notation

Notation	Meaning
$\theta_n$	Interior angle of a regular polygon with n sides
$\text{vertex configuration}$	Notation listing, in cyclic order, the number of sides of each polygon meeting at a vertex (e.g. 3.3.3.3.3.3 = six triangles at a point)

Properties

Regular tessellations (exactly three exist)

\text{Equilateral triangles (6 per vertex), squares (4 per vertex), and regular hexagons (3 per vertex) are the only regular polygons that tile the plane alone.}

Condition: Because their interior angles (60°, 90°, 120°) are the only ones among regular polygons that divide evenly into 360°.

Example: Hexagon: 120° × 3 = 360°. Triangle: 60° × 6 = 360°. Square: 90° × 4 = 360°.

Vertex-angle sum condition

\sum \theta_i = 360^\circ

Condition: Necessary for any edge-to-edge tessellation: the interior angles of all tiles meeting at a single vertex must sum to exactly a full turn.

Semi-regular (Archimedean) tessellations

\text{Eight combinations of two or more regular polygon types can tile the plane with the same vertex configuration repeated at every vertex.}

Example: 3.3.3.3.6 (four triangles and one hexagon) satisfies 4(60°) + 120° = 360°.

Every triangle and every quadrilateral tessellates

\text{Any triangle (not just equilateral) and any quadrilateral (not just a square) can tile the plane, generally without requiring regularity.}

Condition: Two copies of any triangle, rotated 180°, form a parallelogram, which tiles the plane by translation; a similar trick works for any quadrilateral.

Applications

Floor and wall tiling design must respect the 360° vertex condition to avoid gaps or forced cuts at joints, directly using tessellation geometry.

Worked Examples

Find the interior angle of a regular hexagon (n = 6).
$\theta_6 = \frac{(6-2)\times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ$
Check how many 120° angles sum to 360°.
$120^\circ \times 3 = 360^\circ$

Answer: Yes — three 120° angles sum to exactly 360°, so regular hexagons tessellate with 3 meeting at each vertex.

Practice Problems

Difficulty 2/10

Find the interior angle of a regular octagon (n = 8), and determine how many would be needed to exactly surround a point if regular octagons alone could tessellate.

Difficulty 3/10

Verify the semi-regular vertex configuration 3.3.3.4.4 (three triangles and two squares meeting at a vertex) sums to 360°.

Difficulty 5/10

A designer wants to tile a floor using only regular 12-gons (dodecagons, interior angle 150°) and equilateral triangles (60°) at every vertex. If 2 dodecagons meet at a vertex, how many triangles are needed to complete the 360° with no gap, and does this use a whole number of triangles?

Common Mistakes

Common Mistake

Assuming any regular polygon can tessellate the plane on its own if enough copies are used.

Only equilateral triangles, squares, and regular hexagons tile the plane alone — their interior angles (60°, 90°, 120°) are the only ones among regular polygons that divide evenly into 360°.

Common Mistake

Thinking tessellations require regular polygons.

Any triangle or any quadrilateral (regular or not) tiles the plane — two copies of an arbitrary triangle rotated 180° form a parallelogram, which tiles by simple translation.

Quiz

Summary

A tessellation covers the plane with tiles, no gaps or overlaps; the angles of tiles meeting at any vertex must sum to exactly 360°.
Only three regular polygons tile the plane alone: equilateral triangles (60°, 6 per vertex), squares (90°, 4 per vertex), and regular hexagons (120°, 3 per vertex).
Semi-regular (Archimedean) tessellations mix regular polygon types with the same vertex configuration repeated everywhere, e.g. 3.3.3.3.6.
Any triangle or quadrilateral, regular or not, can tile the plane — regularity is not required in general.
Regular pentagons and most other regular polygons cannot tessellate alone because their interior angle does not divide evenly into 360°.

References

WebsiteWikipedia — Tessellation