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__FORCETOC__ Tiling can be viewed as a generative process as well as a mathematical pattern. Tiling is a sort of algorithm, or set of operations, which can be used to cover an arbitrary surface with a design. Tiling can also be used to divide a plane into zones for computational purposes. We will later study an example of this...cellular automata. Tiling also has deep ties to notions about symmetry, and can be extended into 3 dimensional space or even multi-dimensional hyperspace. == Tiling using regular polygons == There are only 3 ways regular polygons can be used to uniformly tile a plane. The corresponding tiles are the triangle, square, and hexagon. {{SingleImage|imageWidthPlusTen=610|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/01.jpg|caption=}} == Tiling using 2 or more regular polygons == There are only 8 ways 2 or more regular polygons can be used to uniformly tile a plane. {{SingleImage|imageWidthPlusTen=720|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/02.jpg|caption=}} Note this presentation showing 2 forms of the tiling to the upper right. {{SingleImage|imageWidthPlusTen=650|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/03.jpg|caption=}} If the constraint of uniformity is released some of the above can be altered to create an infinite number of variations. {{SingleImage|imageWidthPlusTen=672|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/04.jpg|caption=}} If irregular polygons are allowed other families of uniform tiling can be generated. Using only irregular hexigons 3 types of tilings are possible...Interestingly in clusters of 2, 3, and 4 tiles. {{SingleImage|imageWidthPlusTen=396|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/05.jpg|caption=}} There are no uniform or periodic tiling schemes with 5 fold symmetry. Penrose tiling is created via a generative procedure which places limitations as to which tile edges are allowed to meet. Penrose tiling exhibits only local 5 fold symmetry..."pulling the camera back" will reveal symmetry flaws. {{SingleImage|imageWidthPlusTen=638|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/06.jpg|caption=}} Many modern mathematical notions about tiling and symmetry were discovered by crystalographers seeking to create systematic classification systems based on the underlying lattice structures. Crystals are, in a sense, self-organizing structures where local "rules" result in an emergent form. Penrose tiling was discovered before any corresponding material was known. Later quasicrystals were shown to form according to the aperiodic tiling rules akin to Penrose tiling. {{SingleImage|imageWidthPlusTen=766|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/07.jpg|caption=}} Artisans in the Islamic world were constrained by religious laws not allowing the depiction of God or his creations in the natural world. Following an artistic impulse to experiment in form and color, they became masters of tiling. {{SingleImage|imageWidthPlusTen=540|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/08.jpg|caption=}} {{SingleImage|imageWidthPlusTen=651|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/tile/09.jpg|caption=}}
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