Question

Explain why not all regular polygons will tessellate.


WRITER

Answer 1

A tessellation by a single regular polygon means copies of that polygon meet edge-to-edge with no gaps or overlaps. Look at what happens at any vertex where several polygons meet: the interior angles that meet there must add up exactly to 360°.

For a regular n-gon the interior angle is
180° − 360°/n.
If m such polygons meet at a vertex we need
m(180° − 360°/n) = 360°.

Solve for m:
m = 2n/(n − 2).

For the tessellation to work m must be a positive integer. The only n ≥ 3 that make 2n/(n − 2) an integer are n = 3, 4, and 6:
- n = 3 (equilateral triangle): interior angle 60°, 360/60 = 6 fit at a point.
- n = 4 (square): interior angle 90°, 360/90 = 4 fit.
- n = 6 (regular hexagon): interior angle 120°, 360/120 = 3 fit.

No other regular n-gon has an interior angle that divides 360°, so no other single-type regular polygon can tessellate the plane. (Mixed tessellations or nonregular polygons allow many more possibilities.)