Tiling of the Sphere by Congruent Equilateral Pentagons

Min Yan
Hong Kong University of Science and Technology



Content: The classification of the tiling of the sphere by congruent triangles was started in 1922 and completed 80 years later in 2002. We studied the similar problem for the pentagons and obtained the classification for the minimal case of the dodecahedron and some cases where there is enough variety of edge lengths. In contrast, the classification for the tilings of the sphere by congruent equilateral pentagons (i.e., no variety in edge length) calls for completely different technique, that includes the following: 1. The distribution of angles at vertices of degree 3 for general tilings of surfaces, and the distribution for spherical pentagonal tilings in particular. 2. The topological structure of the tiling in case there are only few vertices of degree > 3. 3. Massive amount of numerical calculation that yields rigorous conclusion on the possible pentagons suitable for tiling. At the end, we conclude that there are total of 8 tilings of the sphere by congruent equilateral pentagons. We also got a glimpse on the tilings by congruent almost equilateral (four edges having equal length) pentagons, which is the only remaining challenging case for the complete classification.

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