Tessellate

Tessellate

To cover a plane with identically shaped pieces which do not overlap or leave blank spaces. The pieces do not have to be oriented identically. A tessellation may use tiles of one, two, three, or any finite number of shapes.

Tessellation of identical arrow-shaped pieces arranged in interlocking rows, covering a plane with no gaps or overlaps.

Example

Problem: Can a regular hexagon tessellate a plane by itself?

Step 1: Find the interior angle of a regular hexagon. A regular polygon with

n

sides has interior angles given by:

\frac{(n-2) \times 180°}{n} = \frac{(6-2) \times 180°}{6} = 120°

Step 2: For shapes to tessellate, the angles meeting at each vertex must sum to exactly

360°

. Check whether

120°

divides evenly into

360°

\frac{360°}{120°} = 3

Step 3: Since exactly 3 hexagons meet at each vertex with no gap and no overlap, the plane is completely covered.

Answer: Yes. Three regular hexagons meet at each vertex (

3 \times 120° = 360°

), so regular hexagons tessellate the plane perfectly.

Why It Matters

Tessellations appear throughout architecture, art, and nature—from honeycomb structures to tiled floors. Understanding which shapes tessellate helps in design, manufacturing, and material science where covering a surface without waste is essential. The concept also connects geometry to symmetry and transformations, forming a bridge to more advanced topics like group theory.

Common Mistakes

Mistake: Assuming only regular polygons can tessellate.

Correction: Any triangle and any quadrilateral (even irregular ones) can tessellate the plane. Regularity is not required—what matters is that the angles at each vertex sum to

360°

Related Terms

Plane — The flat surface that a tessellation covers
Polygon — Common shapes used as tessellation tiles
Regular Polygon — Only three regular polygons tessellate alone
Symmetry — Tessellations often exhibit translational symmetry
Interior Angle — Key to determining if a shape tessellates