Wednesday, May 21, 2008

Tessellations Upon a Theme by Archimedes



The nine vectors which determine the Archimedean Truncated Cuboctahedron may be used to form tilings, using the Generalized Dual Method. Here two different tilings, each arising from 3*9=27 planes, are gradually built up, from an axis of 3-fold symmetry, outward. The first has a central region tiled by Truncated Cuboctahedra, Cubes, and Truncated Octahedra. The second has a central region tiled by Truncated Cuboctahedra and Octagonal Prisms.

Friday, May 9, 2008

Interpolation


Using Mathematica, one can read in a photo and then manipulate the color data.

Here I used interpolation functions to assign RGB colors to the tiles, according to tile centers.

Thursday, May 8, 2008

Penrose Tilings


The fascinating tiling discovered by Sir Rogen Penrose in the middle 1970s may be constructed using the Generalized Dual Method, using an arrangement of lines. Let a set of equally-spaced parallel lines, one unit apart, be called a grid. Let a set of n non-zero, independent vectors be called a star. Let the star be the five unit vectors from center to vertices of a regular pentagon (of unit circumradious). Now create a grid perpendicular to each vector of the star, each grid centered upon the origin and containing one line passing through the origin. Now offset all five grids by 2/5, measured in the direction of each grid's star vector.

Above is an example of the resulting tiling. This strategy leads to a Penrose tiling with 5-fold symmetry governing the entire tiling. If the grids have inifnitely many lines, the entire plane is tiled. If the grids have an equal and finite number of lines, the Penrose tiling arises in a central region of a regular decagon tiled by 36- and 72-degree rhombs.

If the edges of the rhombs are suppressed, and the 36- and 72-degree rhombs colored differently, one obtains the effect seen above.