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<p>Can the sphere be tiled by an odd number of congruent polygons in an edge-to-edge arrangement? The answer is negative under a natural setting where polygons have at least 3 edges and at least 3 polygons meet at each vertex. The parity phenomenon has come to light after a voluminous classification of spherical tilings. We offer a short proof of it and draw a parallel to a result in polyhedra by Branko Grünbaum.</p>

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A parity phenomenon of spherical tilings

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月27日

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  • Hoi Ping Luk

摘要

Can the sphere be tiled by an odd number of congruent polygons in an edge-to-edge arrangement? The answer is negative under a natural setting where polygons have at least 3 edges and at least 3 polygons meet at each vertex. The parity phenomenon has come to light after a voluminous classification of spherical tilings. We offer a short proof of it and draw a parallel to a result in polyhedra by Branko Grünbaum.