tessella

The Greek word tessares means "four"—referring to the four sides of a square. Latin diminished it to tessella, a small square stone used in mosaics. Roman craftsmen covered floors, walls, and baths with tessellae: tiny cubes of marble, glass, or ceramic fitted together to form images. The finest examples survive in Pompeii, Antioch, and Ravenna.

Mathematicians adopted the term in the 19th century to describe the problem of covering a flat surface with repeated shapes, leaving no gaps or overlaps. Regular tessellations use a single regular polygon: only equilateral triangles, squares, and regular hexagons tile the plane perfectly. Euclid knew this. The proof is simple: the interior angles must sum to exactly 360 degrees at each vertex.

In 1891, the Russian crystallographer Evgraf Fedorov proved there are exactly 17 distinct wallpaper groups—17 ways to tile the plane with a repeating pattern. Islamic architects at the Alhambra in Granada had discovered nearly all 17 by the 14th century, centuries before the mathematical proof existed. Art preceded theorem.

M.C. Escher visited the Alhambra in 1922 and 1936. The tessellated walls transformed his art. He spent decades creating tessellations where the tiles were recognizable figures—birds, fish, lizards—interlocking with mathematical precision. In 2023, a retired printing technician named David Smith discovered an "einstein" tile: a single shape that tiles the plane but never repeats. The 2,000-year-old mosaic problem still yields surprises.