Bilenses


Polyforms with lenses
By combining lenses B and C in pairs we obtain eleven distinct pieces or bilenses. Reflections and rotations (ten) are not considered for the counting. They cover a total area of 12B + 10C The relation C/B = 0.8 is far from 0.618 (golden mean) which seems guarantees two distinct pieces tile the plane with pentagonal symmetry for large areas. Figure at right with perimeter 28 contains the eleven bilenses marked with their codes.
A covered area with minimum perimeter should be a better solution. Is possible to fit the bilenses into the symmetric area with perimeter 26 showed at rigth? Is there "parity issues" which impedes to obtain a solution? What is the minimum perimeter of a totally covered shape?
Farms
A farm is a figure composed of polyforms where each polyform touches others at least by one side while the whole set encloses an empty area. The largest the area enclosed, the better and difficult to find farm. The farm in the figure at right encloses a total area equals to 24B + 11C What is the maximum area enclosed by a bilenses farm?
Not every farm is perfect as the one showed at the right. In fact is not trivial to work with polyforms. The tiling with lenses must be carry out with care in order to avoid dead-ends.
A Lenses Editor Applet
The figures in this page where worked out first using lenses forms cut out from cardboard. Later a Lenses Editor Applet was written for playing with the pieces. From the screen the images were captured for finally append text with a graphic editor. Go to the Lenses Editor Applet

J0R6E M1RELE5 2002/09/02

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