Extreme Constructions with Hexaboloes



It is impossible to construct rectangles with all hexabolos . Using only 106 hexaboloes you get an area of 106*6/2=318 s-square units or 159 h-square units. Rectangles with this area are 106s x 3s, 53s x 6s and 53h x 3h. The first one is impossible to build, because the number of available s-edges is to small to cover the circumference.
It took me some time until the computer program was fast enough to get the solution for the other cases. The main idea was to sum the maximum number of s-edges or h-edges respectively, which a used hexabolo can contribute to the circumference of the rectangle. If a bound which grows with the number of used pieces was exceeded, backtracking started.
With an additional piece you can construct a 18s x 18s square as shown by Andrew Clarke . With the 107 hexaboloes it is possible to build one trapezium and one symmetric pentagon, which are shown here.
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