Tessellations
Which regular polygons tile the plane? Only three: triangles, squares, and hexagons. Combine different regular polygons at each vertex and you get exactly eight semi-regular (Archimedean) tilings. Explore them all — pan, zoom, recolor, and see the symmetry groups that underlie crystallography and decorative art.
Why only three regular tessellations?
At each vertex of a tessellation, the interior angles of the meeting polygons must sum to exactly 360°. An equilateral triangle has 60° interior angles: six fit (6 × 60 = 360). A square has 90°: four fit (4 × 90 = 360). A regular hexagon has 120°: three fit (3 × 120 = 360). A regular pentagon has 108°, and 360/108 ≈ 3.33 — not an integer, so pentagons cannot tile. For heptagons and beyond, the interior angle exceeds 120° and even three copies overshoot 360°. The constraint is arithmetic, and it is absolute.
Semi-regular tilings
If we allow different regular polygons at a vertex (but require the same arrangement at every vertex), we get the semi-regular or Archimedean tilings. There are exactly eight, plus two mirror-image pairs. The vertex configuration notation — like 3.6.3.6 for the trihexagonal tiling — lists the polygons around each vertex in order. These tilings appear throughout Islamic geometric art, Roman mosaics, and the crystal structures of real materials.
Symmetry groups and crystallography
Every tessellation has a wallpaper group — one of exactly 17 distinct symmetry groups that describe all possible periodic patterns in two dimensions. This classification, proved complete by Fedorov in 1891, constrains what crystals can look like when viewed down a symmetry axis. The hexagonal tiling has p6m symmetry (six-fold rotation plus reflection), the same group that governs graphene and snowflake formation. The snub square tiling has p4g symmetry, with four-fold rotations and glide reflections.
Beyond periodicity
Periodic tilings repeat by translation. But some sets of tiles — like the Penrose rhombs or the recently discovered hat monotile — can tile the plane only aperiodically, never settling into a repeating pattern. Aperiodic tilings sit at the boundary between order and disorder, their diffraction patterns showing sharp Bragg peaks with forbidden rotational symmetries. The discovery of quasicrystals with these exact patterns won Dan Shechtman the 2011 Nobel Prize in Chemistry.