a regular tiling

Of all the ways to tile a flat floor, the most perfectly uniform use a single regular polygon repeated edge-to-edge, with every tile identical and every meeting point looking the same. These are the regular tilings. The striking fact is how few there are: out of all the regular polygons, only three can do it — the equilateral triangle, the square, and the regular hexagon. Just three patterns achieve this highest grade of tiling perfection.

The reason is once again the 360-degree vertex condition, applied with the extra demand that the polygons be regular and identical. For copies of a single regular n-gon to surround a vertex with no gap, its interior angle must divide 360 evenly. The interior angle of a regular n-gon is (n - 2) x 180 / n degrees. Equilateral triangle: 60 degrees, and 360 / 60 = 6 fit. Square: 90 degrees, and 360 / 90 = 4 fit. Regular hexagon: 120 degrees, and 360 / 120 = 3 fit. For a regular pentagon the angle is 108, which does not divide 360; for a regular heptagon and beyond the angle is too large (more than 120 but less than 180), so only two would fit with a leftover gap, never closing. Hence exactly three.

These three tilings are often labelled by the vertex configuration — the list of polygons around each vertex: 3.3.3.3.3.3 (six triangles), 4.4.4.4 (four squares), and 6.6.6 (three hexagons). They are the foundation on which the semiregular (Archimedean) tilings build, mixing different regular polygons while still keeping every vertex alike. Nature favours the hexagonal one: bees build honeycomb in regular hexagons because, among the three, hexagons enclose the most area for the least total wall — a real efficiency result, not just an aesthetic preference.