Tilings/Tessellations

Mark Thompson • Tessellation comes from the Latin word Tessella meaning tiny square.

• Earliest recorded use was in the ancient Sumerian city of Uruk IV. Regular Tessellations

• Tessellations which are comprised of only one shape. • Must be regular polygons, i.e. Equiangular and equilateral. Semiregular Tessellations

• Similar to Regular Tessellations in most ways, but without the restriction that all polygons be congruent to one another. Truchet Tiles

www.interactiveshaderformat.com/sketches/1351

• The patterns are not rotationally symmetric. • First described by Sébastien Truchet in 1704. Rep-Tiles Tiling with Convex polygons

• The Euclidean Tessellation Theorem: For any Euclidean tessellation by reasonably round tiles, considering configurations C(r)of tiles meeting larger and larger disks of radius r, the average imbalance of the tiles in C(r) must approach zero. [sic]

• Corollary: No convex n-gon, n>=7, admits a Euclidean tessellation.[sic] Tessellations in Hyperbolic Geometry Spiral Tessellations

• Given by the equation r = eΦ x, x determines how fast the spiral uncoils, and Φ is the azimuth (i.e. the angle from the positive x axis to the positive y axis). Applications/Occurrence in Nature • Voronoi tessellations, which subdivides a plane into regions based off of distance to a certain point in the subdivision. These are used in Hydrology to measure rainfall, in Biology to study cells and bone microarchitecture, among other things.

• Gilbert Tessellation are used in crystallogy, or in modeling mudcracks.

• Used in manufacturing for yield losses(material wasting).

• Colchicum flower whose leaves have a tessellation like formation.

Exam Question

• There are 8 semi-regular tessellations, find 4 of them, Denote using vertex notation or drawings and descriptions the any vertex.

N.B. Every vertex must have the same formation of polygons around it. nrich.maths.org/content/id/4832/polygons.swf References

Baragar, A. (2001) A survey of classical and modern geometries: with computer activities. Upper Saddle River, NJ: Prentice-Hall, Inc.

Goodman-Strauss, Chaim (2010). Tessellations. La Matematica vol 3. Retrieved November 28, 2016, from arXiv:1606.04459v1 [math.HO]

Fulton, Chandler. “Tessellations.” The American Mathematical Monthly, vol. 99, no. 5, 1992, pp. 442–445. www.jstor.org/stable/2325088

Dutch, Steven (July 21, 1999) “Logarithmic Spiral Tessellations”. Retrieved November 28, 2016 http://www.uwgb.edu/dutchs/symmetry/log-spir.htm