A tessellation is a covering of a surface—most commonly the Euclidean plane—by shapes (tiles) that meet edge‑to‑edge without gaps or overlaps; the term derives from Latin tessella, the diminutive of tessera, a small square used in mosaics (Merriam‑Webster; Merriam‑Webster: tessella). (merriam-webster.com)

Historical background

Decorative tessellations are documented in classical mosaics and pavements and reached a sophisticated abstract vocabulary in Islamic art, where repeating geometric patterns became a principal ornamental tradition (Britannica: tessellated pavement; The Metropolitan Museum of Art). (britannica.com)

Johannes Kepler gave an early mathematical treatment of plane tilings in 1619, later inspiring systematic classifications of periodic symmetry; in the 20th century, the graphic work of M. C. Escher explored interlocking animal and human figures and hyperbolic tilings after his 1936 studies at the Alhambra (Britannica: M. C. Escher). (britannica.com)

Mathematical classifications and symmetry

In two dimensions, regular tessellations by a single regular polygon exist only for the equilateral triangle, square, and regular hexagon. Semiregular (Archimedean) tessellations use two or more regular polygons with identical vertex configurations; there are eight such types (MathWorld: Regular Tessellation; MathWorld: Semiregular Tessellation; [Grünbaum & Shephard, Tilings and Patterns](book://Branko Grünbaum and G. C. Shephard|Tilings and Patterns|W. H. Freeman|1987)). (mathworld.wolfram.com)

Periodic plane tilings are organized by the 17 wallpaper groups, tabulated in crystallography’s reference work International Tables for Crystallography, Volume A (International Tables, Vol. A home). The International Union of Crystallography provides freely accessible plane‑group tables and diagrams for teaching and reference (IUCr: International Tables Volume A overview). (it.iucr.org)

Beyond regularity, important families include monohedral tilings (one prototile), edge‑to‑edge tilings, and combinatorial classifications. A notable milestone is the complete classification of convex pentagons that tile the plane: in 2017 Michael Rao proved that exactly 15 families exist (Rao 2017). (arxiv.org)

Aperiodic tilings and quasicrystals

Non‑periodic tilings lack translational symmetry; some are forced non‑periodic by their prototiles and matching rules, forming aperiodic tilings. Roger Penrose introduced famous two‑tile aperiodic sets with fivefold symmetry in the 1970s, now a standard gateway to the subject (Britannica: Penrose tiling; Britannica: Roger Penrose). Their mathematical diffraction is closely related to the discovery of Quasicrystal order in solids, which prompted the IUCr to generalize the definition of “crystal” in 1992 and update it in 2021 to emphasize essentially sharp diffraction patterns (IUCr Online Dictionary: Crystal; IUCr: change to the definition of crystal; Britannica: quasicrystal). (dictionary.iucr.org)

In 2023–2024, researchers found and published a strictly chiral aperiodic monotile—an “einstein” that tiles only non‑periodically without reflections—by modifying the “hat” prototile to a family called the “spectres”; the proof appears in Combinatorial Theory (2024) (Combinatorial Theory version; arXiv v2). (escholarship.org)

Higher‑dimensional tessellations and honeycombs

Tessellations generalize to 3‑space as honeycombs (space‑fillings). Examples include the cubic honeycomb and uniform space‑fillings by truncated octahedra. Optimization questions link geometry and physics: William Thomson (Lord Kelvin) conjectured the least‑area equal‑cell foam; in 1994 Weaire and Phelan proposed a structure with lower area than Kelvin’s, now a benchmark in foam and materials studies (Nature 1994). In the plane, the “honeycomb theorem” proved by Thomas C. Hales shows that the regular hexagonal tiling minimizes perimeter among equal‑area partitions (Hales 2001, Discrete & Computational Geometry; arXiv preprint). (nature.com)

Computational geometry and natural models

Voronoi partitions, also called Dirichlet or Voronoi tessellations, decompose a space into regions closest to a set of sites; they model diverse phenomena from cellular patterns to facility location and underpin Delaunay triangulations. A standard reference is Okabe, Boots, Sugihara, and Chiu’s monograph ([Spatial Tessellations](book://Atsuyuki Okabe; Barry Boots; Kokichi Sugihara; Sung Nok Chiu|Spatial Tessellations: Concepts and Applications of Voronoi Diagrams|Wiley|2000)). For symmetry‑driven periodic tilings and their classification, see Conway, Burgiel, and Goodman‑Strauss ([The Symmetries of Things](book://John H. Conway; Heidi Burgiel; Chaim Goodman‑Strauss|The Symmetries of Things|A K Peters|2008)). (barnesandnoble.com)

Tessellation in computer graphics

Modern GPUs implement hardware tessellation to adapt geometric detail at render time. In OpenGL 4.x, tessellation includes programmable control and evaluation shaders bracketing a fixed‑function tessellator (Khronos OpenGL wiki; Khronos: Tessellation Evaluation Shader). Microsoft’s Direct3D 11 pipeline offers analogous hull, tessellator, and domain stages for adaptive subdivision (Microsoft Learn: D3D11 features; Microsoft Learn: Tessellation stages). (khronos.org)

Art, architecture, and culture

Escher’s series Regular Division of the Plane and his circle‑limit prints popularized mathematically precise interlocking figures; museum collections in The Hague house major holdings (Escher in Het Paleis). Islamic art’s geometric ornament demonstrates repeating modules based on circles, stars, and polygons, extended with sophisticated symmetries and color organization (The Metropolitan Museum of Art). (escherinhetpaleis.nl)

Terminology and related concepts

Key notions include prototile (the generating tile shape), matching rules, vertex‑configuration notation, and symmetry groups (wallpaper and frieze). In crystallography, two‑dimensional periodic symmetries are codified by the 17 plane groups in the International Tables; quasicrystalline order connects aperiodic tilings to long‑range order characterized by sharp diffraction (International Tables, Vol. A; IUCr Online Dictionary: Quasicrystal). (it.iucr.org)

Internal links: Roger Penrose, Quasicrystal, M. C. Escher, International Union of Crystallography, Wallpaper group, Voronoi diagram, OpenGL, Direct3D.