M.C. Escher Interlocking Tiling

Escher said that he felt irresistible joy with putting multiple copies of an image on a drawing and making them fit together in a structure. Over a period of 40 years, he gradually learn about the mathematical rules and about techniques used by other cultures, and the science of packing molecules. Within each interlocking shape, half of the boundary is a clone of the other half. These identical sections of the boundary make it possible for neighboring shapes to mate together without gaps. This pre-condition often prevents an arbitrary shape from mating together, or it makes a good fitting shape look meaningless (absurd silhouette). Here is another rule, the boundary of a silhouette cannot cross itself. The vertices of Escher's lizard drawing form a regular hexagon. Three of the vertices are rotation points for the 3 pairs of line segments. When you replace the straight edges of the hexagon with these curved lines, the lines wind in and out of the hexagon. It is difficult to let the lines come to close proximity (to form the legs of the lizard) and yet never intersect another boundary line. Benoit Mandelbrot stated this as "self avoiding curves" in his discussion of Koch Island. Through 40 year of practice, Escher found a way to mediate the need for the eyeballs to find meaning and beauty, and the need for the mathematical rules to be followed. This is a tension which haunts every student who tries to use these computer programs to draw interlocking shapes. You lack the freedom to draw the clone half of the silhouette. Escher has a wonderful description of this thinking process in his book "Exploring the Infinite". Here is a direct quote: "A contour line between two interlocking figures has a double function, and the act of tracing such a line therefore presents a special difficulty. On either side of it, a figure takes shape simultaneously. But, as the human mind can't be busy with two things at the same moment, there must be a quick and continuous jumping from one side to the other. The desire to overcome this fascinating difficulty is perhaps the very reason for my continuing activity in this field".

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