Make: Volume 87

114 makezine.com

SKILL BUILDER: Tiling Trigonometry

circle perfectly. Square tiles have vertex angles of

90°, so four of them come together at each vertex

(Figure

D

). Six triangles come together at each

vertex in a triangular tiling (Figure

E

).

IMPOSSIBLE TILINGS

But what happens when we get to five sides (a

pentagon)? The interior angle is 360° / (2 * 5), or

36°. That means that half the vertex angle is 180°

– 90° – 36° = 54°, and the vertex angle is twice

that, or 108°. That means that 3.3333 pentagons

would fit around a vertex — so it does not work.

There is a good discussion of this problem in

the 2017 Quanta magazine article by