Platonic and Archimedean tilings with interactive vertex figure explorer
A tessellation (or tiling) is a covering of the plane by shapes with no gaps or overlaps. When we restrict to regular polygons and require the same arrangement at every vertex, the results are remarkably constrained: there are exactly 3 regular and 8 semi-regular (Archimedean) tilings. This classification, known since antiquity, is a consequence of the simple fact that the angles at each vertex must sum to exactly 360°.
Beyond these 11 vertex-transitive tilings lie infinitely many others, but these form the foundation of tiling theory and appear throughout art, architecture, and nature.
Only three regular polygons can tile the plane by themselves: equilateral triangles, squares, and regular hexagons. These are the Platonic tilings, named by analogy with the Platonic solids. The key constraint is that the interior angle must divide 360° evenly: 60° (triangles, 6 per vertex), 90° (squares, 4 per vertex), or 120° (hexagons, 3 per vertex).
The only three ways to tile the plane with a single regular polygon. Hover over a tile to highlight its vertex figure.
Key insight: Why not regular pentagons? The interior angle of a regular pentagon is 108°, and 360°/108° = 3.33... Since you can't fit a non-integer number of pentagons at a vertex, regular pentagons cannot tile the plane periodically. This observation foreshadows the deep theory of aperiodic tilings.
If we allow different regular polygons but still require the same arrangement at every vertex, we get the 8 Archimedean tilings. Each is described by its vertex configuration -- the sequence of polygon sizes around each vertex. For example, (3.4.6.4) means each vertex is surrounded by a triangle, square, hexagon, and square in that order.
The 8 semi-regular tilings use two or more types of regular polygons, with the same arrangement at every vertex.
Key insight: The vertex configuration determines the tiling (almost) uniquely. The constraint is purely arithmetic: the interior angles at each vertex must sum to 360°. For a regular n-gon, the interior angle is (n-2)·180°/n. Finding all integer sequences whose corresponding angles sum to 360° yields exactly the 3 + 8 = 11 vertex-transitive tilings.
Build your own vertex figure by adding regular polygons around a central point. The angle sum must reach exactly 360° for a valid tiling. Can you discover all 11 vertex-transitive tilings? Watch the angle budget decrease as you add each polygon.
Build a vertex figure by adding regular polygons. The angles must sum to exactly 360° to tile the plane.
Key insight: Not every arrangement that sums to 360° can actually extend to tile the full plane. The vertex configuration (3.3.3.3.6) can tile the plane in two non-isomorphic ways, while some valid-looking configurations are impossible to extend beyond a single vertex. The global extension problem is subtler than the local angle constraint.