Here is a tessellation paradox I recall from school were we got to cut out the four pieces from a paper, rearrange them and try to find out why the area didn't match.
The upper figure is a square 13 units on a side, with an area of 169. The lower figure, made by rearranging the pieces, measures 8 by 21 units, for an area of 168. Where did the missing unit of area go?
Take a look at this DWF where you can zoom in an find the issue. Here is the DWG I created for this purpose.
The solution to the puzzle is the slanting line. If we were to draw the figure accurately on graph paper, the slanting line would not pass exactly through the ends of the adjoining pieces. The "missing" unit of area is actually spread out along the slanting line as a very narrow gap or overlap too small to be easily noticed.
Via http://www.uwgb.edu/DutchS/symmetry/gentess1.htm
I found the above website when I read about Penrose tiles on Swedish DN.se on Peter J. Lu research on Decagonal and Quasicrystalline Tilings in Medieval Islamic Architecture. Very interesting reading.
The image to the right shows how the tilings are made up of pattern using girih tiles.
"We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons (girih tiles) decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West."
Original research paper in Science [ pdf ] [ supporting online material ]
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