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2011-02-11 Bette Burgoyne
2011-02-03 Emma McNally
2011-02-01 Truchet and Mondrian
2011-01-22 Your beautiful eyes
2011-01-21 Pascal Lemoine
2011-01-20 Extra views
2011-01-18 Clint Fulkerson
2010-11-06 Gerry Judah
2010-11-01 Constance Jacobson
2010-10-31 Long range
February 1st 2011
Truchet and Mondrian
Recently I wrote a variation on the Truchet pattern. Recall that the Truchet pattern involves a tiling as well as a set of decorated tiles. The decorated tiles are distributed randomly on the tiling, creating a pattern.
In the variation, instead of having a fixed set of decoration, a decoration for each (square) tiles is constructed as follows. Through each edge of the tile, there will be a segment of line perpendicular to it. The actual location of the line segment is chosen randomly. Inside the tile, the line segment coming from all four edges are somehow combined to complete the decoration. When they are assembled, an interesting pattern appears.
A typical tile decoration. The endpoints on the edges of the outgoing segments are chosen randomly.
Nine tiles. The edges of the tiles are drawn in light gray. The pattern is drawn in black.
The pattern, without the underlying tiling.
Now we can color the middle regions of each tile and get something reminiscent of Mondrian's famous pattern. The second image is simply a zoom, to get the size of the pattern compared to the size of the image in a ratio closer to Mondrian's paintings.
Of course it is not very interesting to draw Mondrian-like pictures, and in order to do this, one could simply draw random vertical lines, horizontal lines and color some of the resulting regions. What is interesting about this way of drawing Mondrian-like pattern is that it is local. The algorithm needs to know only what is going on in the tile the pixel belongs to. Such "pixel by pixel" drawing algorithms can be scaled easily and are well-suited for pattern piling. Here are two pictures resulting from piling Mondrian-like patterns. Click on the pictures for zoomable images. The two pictures differ only by their "magnification step", namely the scaling factor between two piled patterns. It is equal to 2 in the first one and to 6 in the second one.
[Update : A picture of the fractal Mondrian pattern is available as open edition prints here.]