9/23/11&1&TESSELLATIONS Tessellation: Tiling a plane Filling a plane with a shape or image – no gaps From Latin “tessella” - a small cubical piece of clay, stone or glass used to make mosaics. The word "tessella" means "small square” … from "tessera” - square … which in its turn is from the Greek word for "four” Wikipedia…9/23/11&2&Regular Tessellations Only three “regular” tilings – all the same shape Variations on these regular shapes work too… Semi-Regular Tessellations Two or more regular polygons – 8 possibilities9/23/11&3&Demiregular Tessellations Harder to define… Tile a plane with images - Escher9/23/11&4&M. C. Escher Dutch graphic artist – (1898 – 1972) known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations. Wikipedia… Tile a plane with images - Escher9/23/11&5&Tile a plane with images - Escher VLSI Version?9/23/11&6&How to do it… Exploit symmetries Translation Rotation Reflection Start with a regular tessellation square, triangle, hexagon Then apply transformations and see what they end up looking like Simple example: Translation Start with a square http://euler.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures9/23/11&7&Simple example: Translation Start with a square in this example It works for any of the regular tessellations The Tessellation Project Simple example: “Glide Reflection” Start with a triangle… The Tessellation Project9/23/11&8&Simple example: Reflection Start with a triangle… Tessellations.org Simple example: Reflection Start with a triangle… Tessellations.org9/23/11&9&Escher Translation Translation http://euler.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures9/23/11&10&Other Symmetries Translation, Rotation, Reflection Using squares, triangles, and hexagons http://www.dustinjpurdy.com/Math/PDF/Tessellation.pdf The Nine Heesch Types Tessellation classification code developed by German mathematician Heinrich Heesch http://www.dustinjpurdy.com/Math/PDF/Tessellation.pdf9/23/11&11&For your Tesselations… Start with a regular tessellation grid vertices only can be a good way to go too Then start tweaking the lines according to the transformations and see what it looks like Some Web Sources – good info… http://euler.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures http://www.dustinjpurdy.com/Math/PDF/Tessellation.pdf http://library.thinkquest.org/16661/ http://tessellations.org/ http://www.mcescher.com/9/23/11&12&One more – Escher
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