a tessellation

A tessellation is a way of covering a flat surface completely with shapes that fit together leaving no gaps and no overlaps — the way bathroom tiles, brick walls, honeycomb, and pavements do. Every spot on the plane is inside exactly one tile (or on a shared edge), and the pattern can in principle go on forever in all directions. The word comes from the Latin tessella, a small square tile used in Roman mosaics.

The decisive condition happens at the vertices — the points where tile corners meet. For the tiling to close up with no gap and no overlap, the interior angles of the tiles meeting at each vertex must sum to exactly 360 degrees, a full turn. That single arithmetic fact controls which shapes can tile and how. For example, six equilateral triangles (60 degrees each, 6 x 60 = 360) fit around a point; four squares (4 x 90 = 360) fit; three regular hexagons (3 x 120 = 360) fit. Tessellations are classified by how regular they are: a regular tiling uses one type of regular polygon (only triangles, squares, or hexagons work); a semiregular or Archimedean tiling uses several regular polygons but the same arrangement at every vertex (eight of these exist); demiregular tilings allow more than one kind of vertex arrangement.

Tessellations are where symmetry becomes infinite and periodic: the symmetry group of a tessellation that repeats by translation is one of the seventeen wallpaper groups. Not every shape needs to be a regular polygon — any triangle and any quadrilateral whatsoever can tile the plane, certain pentagons can, and famous artists like M. C. Escher tiled with interlocking birds and fish. But there is also a deep frontier: some shapes, like the Penrose tiles, cover the plane only in non-repeating (aperiodic) ways, showing that 'tiling the plane' and 'tiling it periodically' are genuinely different questions.