Geometric art uses mathematics to produce images. Many of the images presented here have been created and made available by the Korean geometric artist Ghee Beom Kim now living in Sydney, Australia. As he says, “Most of the stuff I created here are the result of trial on simple initial ideas, which might lead to some complex structure”. Perhaps, just perhaps, the plethora of patterns around us are the product of such a process… natural selection operating on mutation “trials”?
A tessellation is a collection of figures that fills space with no overlaps and no gaps. Tessellation can be Euclidean or non-Euclidean like hyper-tiling. In particular, hyperbolic tessellation on a circle reveal fractality and self-similarity. Tessellations frequently appeared in the art of M. C. Escher and, apart from their appearance in ancient architecture and mosaics they occur in nature’s Basalt columns, honeycombs and various shells. In Latin, “tessella” (from the Greek word “tessera” meaning 4) is a small cubical piece of clay, stone or glass used to make mosaics. An important tesselation is theVoronoi Diagram and its sister Delauney Triangulation whose irregularity and complexity has as its basis simple mathematical rules.
Thue-Morse Knot Tessellation
The Thue–Morse sequence was first studied by Eugène Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property to show to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.
Thanks go to Mark Dow for sharing his fabulous images:
There are an infinite number of uniform tilings on the hyperbolic plane where 1/p + 1/q + 1/r < 1 with p,q,r being the orders of reflection symmetry at three points of the fundamental domain triangle.
Penrose tilings have remarkable properties:1) they are nonperiodic so that a shifted copy will never match the original exactly, 2) they contain infinite copies of a finite region of tiling, and 3) they are quasicrystal so that when implemented as a physical structure they produce Bragg diffraction reflecting the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called “deflation” or “inflation hence giving them cosmological importance.
In mathematics, a Voronoi Diagram is a decomposition of a metric space determined by distances to a specified discrete set of objects: Georgy Voronoi (1907) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal für die Reine und Angewandte Mathematik, 133:97-178, 1907.
Tessellations in Nature