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Perfect Rectangles

Jaime Rangel-Mondragón
Organization: Universidad Autonoma de Querétaro
Department: Facultad de Informatica
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The task of perfectly squaring a given rectangle involves finding out a (possibly non-existent) finite set of squares of different sizes covering it without overlapping or gaps. Even when this set is given, finding the actual position of the squares inside the given rectangle can be computationally arduous. In this work we assume that none of the rectangle or its perfect squaring are given and endeavor towards the computational search of the perfect squarings of those rectangles admitting them, following the work of Brooks, Smith and Stone (a list of references is included). We begin with an examination of a straightforward approach using backtrack, one which the author has found pedagogically useful, and continue with the implementation of theoretical tools involving the recursive generation of 3-connected simple planar graphs. These graphs provide the graph-theoretical background to reduce our search to that of solving a suitable network of resistors using Kirchhoff laws. Finally, our results are filtered in order to select non-isomorphic instances toward setting up a catalog of all perfectly squared rectangles of order at most 14. This notebook was developed with Mathematica version 4.

*Mathematics > Geometry > Plane Geometry
*Mathematics > Geometry > Tiling
*Mathematics > Recreational Mathematics

Graphs, Networks, Tessellations, backtrack
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PerfectRectangles.nb (897.1 KB) - Mathematica Notebook

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