THESIS
2023
1 online resource (xlix, 1114 pages) : illustrations
Abstract
In this thesis, we classify the edge-to-edge tilings of the sphere by congruent
polygons. For simplicity, we call them the tilings. By the Euler formula, these
polygons are triangles, quadrilaterals, and pentagons. Duncan Sommerville’s
1923 effort commenced the quest for classification of the tilings by congruent
triangles. The endeavour eventually came to fruition in 2002 following the work
of Yukaku Ueno and Yoshio Agaoka. Various mathematicians have since made
significant progress on the other aspects of the subject matter. Notably, Min
Yan and his collaborators have largely settled the classification of the tilings by
congruent pentagons. Yohji Akama et al. have built preliminaries for the quadrilateral
counterpart. We call a polygon almost equilateral if all but one of its edges
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In this thesis, we classify the edge-to-edge tilings of the sphere by congruent
polygons. For simplicity, we call them the tilings. By the Euler formula, these
polygons are triangles, quadrilaterals, and pentagons. Duncan Sommerville’s
1923 effort commenced the quest for classification of the tilings by congruent
triangles. The endeavour eventually came to fruition in 2002 following the work
of Yukaku Ueno and Yoshio Agaoka. Various mathematicians have since made
significant progress on the other aspects of the subject matter. Notably, Min
Yan and his collaborators have largely settled the classification of the tilings by
congruent pentagons. Yohji Akama et al. have built preliminaries for the quadrilateral
counterpart. We call a polygon almost equilateral if all but one of its edges
have the same length. Namely, an almost equilateral quadrilateral has exactly
three edges of equal length and an almost equilateral pentagon has exactly four
edges of equal length. The outstanding problems are on the tilings by congruent
almost equilateral quadrilaterals, by congruent quadrilaterals with exactly three
distinct edges, and by congruent almost equilateral pentagons respectively. They
are challenges at a different level and the tools from previous works regrettably
fall short. Modern concepts and techniques are developed in this thesis. The
advancement enables us to not only resolve the outstanding problems but also simplify the classification of the tilings by congruent triangles. In particular, we
attain the tilings by congruent almost equilateral quadrilaterals via two different
strategies. These results conclude the classification for the edge-to-edge tilings
of the sphere by congruent polygons under a unified framework.
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