What Is a Tessellation? Covering a Surface With No Gaps

At any vertex of a tiling, the angles of the shapes meeting there must sum to exactly 360 degrees, or there is a gap or an overlap. The interior angle of an equilateral triangle is 60, so six meet cleanly. A square is 90, so four meet. A regular hexagon is 120, so three meet. A regular pentagon is 108, which divides into 360 three and a third times, so it cannot work, and every regular polygon above six has an angle larger than 120 and smaller than 180, so only two could meet and two is not enough to close a corner. That is the entire proof, and it explains the hexagonal honeycomb, since a hexagon encloses the most area for the least perimeter among shapes that tile, meaning bees use the least wax for the most storage, a result proved formally as the honeycomb conjecture only in 1999.

Relaxing the conditions opens the subject up considerably:

• •Semiregular tilings use more than one regular polygon with the same arrangement at every vertex, and there are exactly eight of them, including the octagon and square pattern found on countless bathroom floors
• •Any triangle tiles the plane, and so does any quadrilateral whatever, including non-convex ones, which surprises most people
• •Fifteen types of convex pentagon tile the plane, a list completed only in 2017 after a computer-assisted proof showed no more exist; five of them were found by Marjorie Rice, an amateur with no mathematical training beyond school, working from a magazine column in the 1970s
• •The plane admits exactly seventeen distinct symmetry groups, the wallpaper groups, which classify every possible repeating pattern by its combination of translations, rotations, reflections and glide reflections, and all seventeen appear in the tilework of the Alhambra
• •In three dimensions the equivalent classification gives 230 space groups, which is the foundation of crystallography
• •On a curved surface the rules change, which is why a football uses pentagons and hexagons together and a flat surface cannot

The artist most associated with the subject arrived at it by looking at a building. M. C. Escher visited the Alhambra in Granada in 1922 and again in 1936, copied the tilework in detail, and was struck by the mathematical rigour of patterns that, following Islamic convention, contained no living creatures. His own contribution was to keep the underlying symmetry and fill the shapes with figures, so that interlocking birds, fish, reptiles and horsemen tile the plane with no gaps, each animal's outline serving as the outline of its neighbours. He worked out the systematic possibilities himself, corresponded with the mathematician George Polya and later with Roger Penrose and the crystallographer Caroline MacGillavry, and his notebooks contain a personal classification system that reinvents much of the mathematics of wallpaper groups without the vocabulary. His prints are now used to teach the subject, which is a reversal of the usual direction between art and mathematics.

A tiling can cover the plane without ever repeating, which was not obvious and was proved in stages. In 1961 Hao Wang conjectured that any set of tiles capable of tiling the plane could do so periodically, and his student Robert Berger disproved it in 1964 with an aperiodic set of 20,426 tiles, later reduced to 104. Roger Penrose found sets of two tiles in the 1970s, the kite and dart, which tile the plane in patterns with fivefold symmetry that never repeat exactly. That was regarded as a mathematical curiosity until 1982, when Dan Shechtman observed a diffraction pattern with fivefold symmetry in a metal alloy, which crystallography said was impossible; he was told to go and read a textbook and left his research group, and received the Nobel Prize in Chemistry in 2011 for the discovery of quasicrystals, which are three-dimensional aperiodic structures. The long-standing open question was whether a single tile could do it, and in March 2023 a team including a retired printing technician named David Smith found one, a thirteen-sided shape called the hat, followed within months by a version requiring no reflections, called the spectre.

Only equilateral triangles, squares and regular hexagons tile a flat surface alone, because the angles at every vertex must sum to exactly 360 degrees, which also explains why bees build hexagons. Eight semiregular tilings exist, any triangle or quadrilateral tiles, exactly fifteen convex pentagons do, and every repeating flat pattern falls into one of seventeen symmetry groups, all present in the Alhambra. Escher filled those symmetries with animals, Penrose found tile sets that never repeat, and a single aperiodic tile was discovered in 2023.