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.