Mathematics
Tessellations and the Classification of Regular Tilings
Quick fact
Only three regular polygons can tile the entire plane without gaps or overlaps: equilateral triangles, squares, and regular hexagons. This surprising limitation arises directly from the requirement that the interior angles around a vertex must sum to exactly 360°.
Why this is interesting
You've seen hexagonal tiles in bathrooms and square grids on graph paper, but have you ever wondered why you never see a regular pentagon tiling a floor? What's so special about the shapes that can tile a plane perfectly?
Read the full explanation
Understanding Tessellations and the Classification of Regular Tilings
Imagine you are covering a flat floor with tiles, each tile being a regular polygon—all sides and angles equal. To avoid gaps or overlaps, the corners of the tiles must meet at a point. At that point, the sum of the angles of all tiles meeting must be exactly 360 degrees, because they make a complete turn around the point. For a regular n-gon, each interior angle measures (n-2)×180°/n. To fit exactly, 360° divided by this angle must be a whole number. Let's check: for n=3 (triangle), angle = 60°, so 360/60 = 6 tiles fit; for n=4 (square), angle = 90°, so 4 tiles fit; for n=5 (pentagon), angle = 108°, and 360/108 ≈ 3.33, not an integer, so pentagons cannot tile alone. For n=6 (hexagon), angle = 120°, so 3 tiles fit. For n=7, angle = 128.57°, 360/128.57 ≈ 2.8, not an integer, and as n increases beyond 6, the angle increases and the ratio becomes less than 3 and greater than 2, never an integer. Thus only n=3,4,6 work. This is why you see honeycombs and Spanish tiling patterns with hexagons.
A deeper explanation
The reason only these three regular tilings exist is rooted in the angle sum condition at a vertex, a consequence of the plane being flat (zero curvature). For a tiling to be regular, it must use congruent regular polygons and have identical vertex configurations—meaning the same arrangement of polygons around every vertex. The vertex configuration notation (e.g., 4.4.4 for squares) records the number of sides of each polygon meeting at a vertex. The three regular tilings are (3.3.3.3.3.3), (4.4.4.4), and (6.6.6). This classification is a special case of the broader theory of uniform tilings, which allow different types of regular polygons but still require the same vertex arrangement; there are 11 such Archimedean tilings. The classification also depends on the curvature of the surface: on a sphere, you get regular tilings like the Platonic solids, and on hyperbolic planes, infinitely many regular tilings exist. Because the Euclidean plane has zero curvature, the interior-angle sum constraint is constant and restrictive.