Tessellation
A temple mosaic from the ancient Sumerian city of Uruk IV dates to between 3400 and 3100 BC. This artifact displays a tessellation pattern in colored tiles, marking one of humanity's earliest uses of geometric covering. The Sumerians built wall decorations using patterns of clay tiles around 4000 BC. Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity. These blocks sometimes displayed geometric patterns that covered floors and walls without gaps or overlaps. Roman builders continued this tradition with rhombille mosaics found across their empire. In 1619, Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his work on geometry. Kepler was possibly the first to explore and explain the hexagonal structures of honeycomb and snowflakes. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations.
A tessellation covers a surface often a plane using one or more geometric shapes called tiles. No overlaps and no gaps define the core rule of these arrangements. Regular tessellations use identical regular polygonal tiles all of the same shape. Only three shapes can form such regular tessellations: equilateral triangles squares and regular hexagons. Semi-regular tessellations use more than one type of regular polygon but maintain the same arrangement at every corner. Eight types of semi-regular tessellations exist if mirror-image pairs count as separate entities. Irregular tessellations can be made from other shapes like pentagons polyominoes and almost any kind of geometric shape. The Conway criterion provides a sufficient set of rules for deciding whether a given shape tiles the plane periodically without reflections. No general rule has been found for determining whether a given shape can tile the plane or not. This means there are many unsolved problems concerning tessellations in mathematics. A normal tiling requires every tile to be topologically equivalent to a disk with uniform bounds on size.
Penrose tilings use two different quadrilateral prototiles to create non-periodic patterns that never repeat exactly. These belong to a general class of aperiodic tilings which use tiles that cannot tessellate periodically. The recursive process of substitution tiling generates aperiodic patterns through methods like rep-tiles. Pinwheel tilings appear in infinitely many orientations while lacking translational symmetry. An einstein tile is a single shape that forces aperiodic tiling. The first such tile dubbed a hat was discovered in 2023 by David Smith. Smith worked alongside Joseph Samuel Myers Craig S. Kaplan and Chaim Goodman-Strauss on this discovery. Their paper appeared as arXiv:2303.10798 in March 2023. Wang tiles are squares colored on each edge placed so abutting edges have matching colors. A suitable set of Wang dominoes can tile the plane but only aperiodically. Since the halting problem is undecidable the problem of deciding whether a Wang domino set can tile the plane is also undecidable. Truchet tiles split into two triangles of contrasting colors were used by Sébastien Truchet in 1704.
M. C. Escher visited Spain in 1936 and became inspired by Moorish wall tilings at the Alhambra palace. He made four Circle Limit drawings using hyperbolic geometry for artistic effect. His woodcut Circle Limit IV from 1960 required a pencil and ink study showing the necessary geometry. Escher explained that no single component of all the series ever reaches the boundary line when viewed from infinitely far away. Tessellated designs appear on textiles woven stitched or printed with interlocking motifs of patch shapes. Quilts often use tessellation patterns to design these repeating geometric arrangements. Origami paper folding uses pleats to connect molecules like twist folds together in a repeating fashion. Manufacturing industries apply tessellation principles to reduce material wastage during cutting operations. Sheet metal cutouts for car doors or drink cans benefit from efficient tiling strategies. Mudcrack-like cracking of thin films shows self-organization using micro and nanotechnologies.
Honeycombs display hexagonal cells as a well-known example of tessellation in nature. Botany describes checkered patterns on flower petals tree bark or fruit as tessellate. Flowers including fritillary and some species of Colchicum are characteristically tessellate. Basaltic lava flows often display columnar jointing resulting from contraction forces causing cracks as the lava cools. The Giant's Causeway in Northern Ireland provides an array of columns formed by this process. Tessellated pavement found at Eaglehawk Neck on the Tasman Peninsula of Tasmania fractures into rectangular blocks. Foams pack according to Plateau's laws requiring minimal surfaces to separate cells of equal volume. Lord Kelvin proposed a packing using only one solid called bitruncated cubic honeycomb with slightly curved faces in 1887. Denis Weaire and Robert Phelan proposed the Weaire, Phelan structure in 1993 which uses less surface area than Kelvin's foam.
Tessellation extends to three dimensions where certain polyhedra stack in regular crystal patterns to fill space exactly. The cube remains the only Platonic polyhedron capable of tiling three-dimensional space regularly. Rhombic dodecahedra truncated octahedrons and triangular quadrilateral hexagonal prisms also tile space naturally. Naturally occurring rhombic dodecahedra appear as crystals of andradite garnet and fluorite. Uniform honeycombs can be constructed using the Wythoff construction method. A Schwarz triangle tiles a sphere while tessellations exist in non-Euclidean geometries like hyperbolic geometry. Nine Coxeter group families generate compact convex uniform honeycombs in three-dimensional hyperbolic space. These are represented by permutations of rings within Coxeter diagrams for each family. The Schmitt-Conway biprism is a convex polyhedron that tiles space only aperiodically. Ludwig Schläfli pioneered this extension by defining polyschemes now called polytopes in spaces with more dimensions.
Common questions
When did the Sumerians create the earliest known tessellation patterns in Uruk IV?
The Sumerian city of Uruk IV created a temple mosaic with tessellation patterns between 3400 and 3100 BC. This artifact marks one of humanity's earliest uses of geometric covering using colored tiles.
Who proved that every periodic tiling of the plane features one of seventeen different groups of isometries?
Russian crystallographer Yevgraf Fyodorov made this proof in 1891. His work marked the unofficial beginning of the mathematical study of tessellations.
What shapes can form regular tessellations without gaps or overlaps?
Only three shapes can form such regular tessellations: equilateral triangles, squares, and regular hexagons. These identical regular polygonal tiles all share the same shape to cover a surface completely.
Which tile was discovered in March 2023 as the first einstein tile for aperiodic tiling?
David Smith discovered the hat tile in March 2023 alongside Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. Their paper appeared as arXiv:2303.10798 on the 2nd of May 1536.