Geometry › Tessellations

Tessellations

Tiles fitting together with no gaps and no overlaps. Honeycombs, Escher prints, bathroom floors.

Regular tessellations

Only three regular polygons tile the plane all by themselves.

Tile

Triangles

Six equilateral triangles meet at every vertex (6 × 60° = 360°).

The rule

For a regular polygon to tile alone, its interior angle must divide 360° evenly. That gives only triangles, squares and hexagons.

What counts as a tessellation

A tessellation covers a flat surface using one or more shapes, with no gaps and no overlaps, repeating forever.

The three regular tessellations

Only three regular polygons tile the plane all by themselves:

Triangle — six meet at every vertex (6 × 60° = 360°).
Square — four meet at every vertex (4 × 90° = 360°).
Hexagon — three meet at every vertex (3 × 120° = 360°).

The 360° rule

For any vertex, the angles must sum to exactly 360°. That eliminates pentagons (108° doesn't divide 360 evenly) and any regular polygon with 7+ sides.

Semi-regular tessellations

Mix two or more regular polygons and you get eight more "Archimedean" tilings — like 4.8.8 (one square + two octagons at every vertex).

In the wild

Honeycomb — bees use hexagons because they minimize wax for a given area.
Cracked mud, basalt columns — natural hexagonal tilings.
M.C. Escher prints — birds, fish, lizards interlocking with shape symmetry.
Penrose tilings — non-repeating patterns from two rhombi.