A parity phenomenon of spherical tilings

Abstract 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.

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Publication Details

Journal
Archiv der Mathematik
Published
2026-03-27
DOI
https://doi.org/10.1007/s00013-026-02233-2
Primary Topic
Quasicrystal Structures and Properties
Type
article
Field-Weighted Citation Impact
0.00
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article

A parity phenomenon of spherical tilings

Hoi Ping Luk
Archiv der Mathematik
Quasicrystal Structures and Properties
article

A parity phenomenon of spherical tilings

Hoi Ping Luk
article en

Abstract

Abstract 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.

Archiv der Mathematik
University of West Bohemia in Pilsen (CZ)
Openalex Percentile: Top 7%
Quasicrystal Structures and Properties
DOIOpen Access PDFOpenAlex
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A parity phenomenon of spherical tilings — Hoi Ping Luk · Archiv der Mathematik (2026) | TGRS Research Map | TGRS