Tilings, or tessellations, refers to a branch of discrete geometry that involves covering the plane (or space) with shapes, and without overlaps. The discipline finds application in a variety of areas, including crystallography, self-assembly, art and design, materials science, biology, and computer graphics, to name a few. Recreational mathematicians enjoy the topic as well, due perhaps to tilings’ connection to art and games. Despite the potential application, the field abounds with fundamental open questions reflecting the complexity of the real world that tilings model.
Consider the challenge of understanding which convex polygons give rise to monohedral tilings of the plane (monohedral tilings are those in which all tiles of the tiling are congruent to one another). Quick verification indicates that any triangle or quadrilateral, convex or not, admits tilings of the plane. Skipping the question of pentagons momentarily, the problem of classifying the convex hexagons that tile the plane is solved , though the solution is nontrivial. Convex polygons with seven or more sides do not admit any tilings of the plane [1, 4]; this result is also nontrivial and relates to Euler’s famous formula for planar graphs. Thus remains the problem of convex pentagons, which may be stated as the following: In terms of the measures of the angles and sides, classify all convex pentagons that admit monohedral tilings of the plane.
It seems surprising that this simply-stated problem has not been solved, despite a rich history of effort. Attempts date back to Hilbert’s famous 23 problems, and include the spotlight of Martin Gardner’s Scientific American column and notable contributions from amateur mathematician Marjorie Rice . Our team recently made progress on the problem, which we will outline here.
Pentagons admit the most complex monohedral tilings among convex polygons; there are existing types that admit no tilings in which every tile is in the same transitivity class, with respect to the symmetry group of the tiling. This was first demonstrated in 1968 . Theoretically, our work focused on showing that if a pentagon admits tilings with \(i\) transitivity classes, there is a maximum number of ways that the tiles of such a tiling can meet one another. This led to the development of a computer algorithm that can, for each positive integer \(i\), exhaustively list all such pentagons .
Figure 1. A hypothetical 3-block with a choice of labeling and a demon-
stration of how a centrally placed 3-block would be surrounded in a choice
of how it might admit a tile-transitive tiling.
For example, if a hypothetical pentagon admits a tiling of the plane having three transitivity classes, then inside the tiling the pentagon must form into clusters of three pentagons (see Figure 1) so that the cluster of three pentagons tiles the plane in a tile-transitive manner. For such a hypothetical pentagon, we can program a computer to list all labelings of the pentagons comprising this cluster, as well as all the ways such a cluster of three pentagons can tile the plane in a tile-transitive manner. For example, labeling the pentagons in the cluster of three in Figure 1 and requiring it to tile the plane in a specific tile-transitive manner yields a patch of tiles surrounding a centrally-placed cluster of three pentagons, from which we can comprehend relationships among the angles and sides.
From the resulting system of equations, one must then determine if an actual convex pentagon can satisfy this set of equations, and if so, if such a pentagon is among the types already known. In our example, the answer is yes, the equations can form a pentagon, and no, the pentagon is not among the previously-known types. This particular pentagon was the first new type found since 1985 (see Figure 2).
Figure 2. Pentagon type 15.
Moving forward, there is still work to be done on the problem. A classification still eludes us, and more new pentagon types may yet be found. Whether any of these patterns will have direct applications in science is uncertain. However, it seems that furthering our knowledge of how the most basic shapes fit together will be important in understanding the complexity of the real world.
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 Mann, C., McLoud-Mann, J., & Von Derau, D. (2015). Convex pentagons that admit i-block transitive tilings. Cornell University Library. E-print, arXiv:1510.01186.
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