Combinatorial Tilings of the Sphere by Pentagons

Abstract

A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent pentagons. We show that the tiling cannot have only one vertex of degree > 3. Moreover, we construct earth map tilings, which give classifications under the condition that vertices of degree > 3 are at least of distance 4 apart, or under the condition that there are exactly two vertices of degree > 3. * Research was supported by Hong Kong RGC General Research Fund 605610 and 606311.

Tilings of the sphere by congruent quadrilaterals I: edge combination $a^2bc$

“…The earth map tiling is introduced in [12] as the edge-to-edge pentagonal tilings of the sphere with exactly two vertices of degree > 3. We use this name for general tilings with symmetric north/south pole regions and time zones.…”

Tilings of the Sphere by Congruent Quadrilaterals with Exactly Two Equal Edges

“…Further efforts have been put into study of tilings by quadrilaterals by Ueno and Agaoka in [15], by Akama et al in [1], [2], [3], [4], [6], which include quadrilaterals of the types that are equilateral or can be divided into two triangles. On the other hand, Yan et al are on course to give a complete classification of tilings by pentagons in [11], [5], [16], [17], [7], [12], [18], [19], [20]. The problems remain open are the tilings by quadrilaterals with exactly two equal edges and those with exactly three equal edges.…”

Pentagonal Subdivision

“…This problem is motivated by the fact that most vertices in a pentagonal tiling of the sphere have degree 3. In [4], we studied the extreme case of few high degree vertices, which means high concentration of high degree. Specifically, we proved that the tiling cannot have only one high degree vertex.…”