+- MKLab (https://mklab.gr)
+-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1)
+--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3)
+---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13)
+----- Forum: GEOMETRY (https://mklab.gr/forumdisplay.php?fid=150)
+----- Thread: Truchet Tilings and their Generalisations (/showthread.php?tid=1813)
Truchet Tilings and their Generalisations - mklabgr - 09-03-2026
Truchet Tilings and their Generalisations
Authors: E. A. Lord & S. Ranganathan
Journal:Resonance, Vol. 11, No. 6, June 2006, pp. 42–50
The article explores Truchet tilings, originating with the French mathematician and cleric Sébastien Truchet in 1704. A basic Truchet tile is a square divided diagonally into two contrasting regions; by rotating the same tile into four possible orientations and arranging many copies, one can generate an enormous variety of geometric patterns. The authors emphasize that this is essentially an early form of combinatorial encoding: a tiling can be represented by a sequence of symbols specifying the orientation of each tile. Periodic Truchet patterns can realize 12 of the 17 wallpaper symmetry groups, while modifications of the basic tile can produce random, aperiodic and quasiperiodic structures. A particularly interesting extension replaces the square with a $60^\circ$ rhombus, allowing threefold and sixfold symmetries and inflation rules reminiscent of Penrose tilings.
Lord and Ranganathan then extend the concept from the plane into three dimensions. Instead of decorating squares, they decorate cubic cells with curved surface patches that join continuously when cubes are assembled. Appropriate arrangements generate complicated periodic surfaces related to triply periodic minimal surfaces, including Schoen's "batwing" surface. The same combinatorial philosophy can also describe three-dimensional weaving: a cubic unit containing three mutually perpendicular threads can be repeated according to translation, reflection, inversion or screw operations. Under suitable constraints, the possible weaves reduce to several fundamental classes such as OOO, OOI, OII and III.
The broader message is that Truchet's seemingly simple idea anticipated a very modern way of thinking about geometry and materials: complex structures can be represented by compact symbolic codes and reconstructed algorithmically. The authors describe such a code metaphorically as an "inorganic gene". With computers, this approach becomes particularly powerful for exploring enormous families of two- and three-dimensional structures, including possible applications to composite materials, crystallography and materials science.
Key takeaways
- A single simple tile, combined through different orientations, can generate extraordinarily complex patterns.
- Truchet tilings connect geometry, symmetry and combinatorics through symbolic encoding.
- The idea naturally generalises from ordinary 2D tilings to aperiodic/quasiperiodic patterns, minimal surfaces and 3D weaving.
- The paper's deeper insight is computational: a geometrical structure can be treated like information—a short symbolic "gene" capable of generating an entire pattern.
ARTICLE