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Tilings and Patterns: Second Edition

The single tile, depending on its detailed shape, can appear in the same orientation throughout the whole tiling or more usually in a number of different rotated or reflected orientations which fit together. The translation symmetry demands that any aspect of the tiling, be it a single tile in a particular orientation or a group of tiles, repeats itself periodically over the whole tiling. Each of the above sections of infinite tilings consists of a single tile shape. In each case it is clear how the pattern, based on a single tile shape, will repeat itself periodically over a complete tiling.


Tilings and Patterns

Here is one way to visualize the vertices around the origin of the E8 lattice: The 2-dimensional Penrose Tiling in the above image was generated by Quasitiler as a section of a 5-dimensional cubic lattice based on the 5-dimensional HyperCube shown in the center above the Penrose Tiling plane. The E8 root vectors form a Witting Polytope. To see what some structures related to the cell and cell, and many other interesting things, look like, see a Kyoto WWW page Professor Koji Miyazaki at Kyoto has done much work on golden isozonahedra and related geometric structures. It is the associative 4-dim spacetime of the D4-D5-E6 model.

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