Geometric and combinatorial structure of a class of spherical folding tessellations - I

Abstract

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Tilings of the Sphere by Congruent Quadrilaterals or Triangles

“…We remark that the counting in the lemma (as well as the subsequent Lemmas 5,6,7,8,9,10,11) relies on the criterion for distinguishing angles. The criterion can be angle values.…”

“…π and the parity lemma, the possible vertices are α 2 γ 4 , αβγ 5 , β 2 γ 6 , γ 8 , γ 10 . The angle sum of one of these vertices, and the angle sums of α 4 , β 6 , and the angle sum for triangle imply the following cases. We also update the AVC (5.3).…”

“…Moreover, if αγ 3 δ is a vertex, then adding the angle sums of αδ 2 , αβ 3 , αγ 3 δ together gives 3(α + β + γ + δ) = 6π, a contradiction. It remains to consider the combinations of αβ 3 with one of β 2 γ 2 , γ 4 , γ 3 δ, βγ 4 , γ 6 .…”

“…Therefore αγ 3 δ, γ 5 δ are not vertices. It remains to consider the combinations of β 5 with one of αβγ 2 , β 2 γ 2 , γ 4 , γ 3 δ, βγ 4 , γ 6 .…”

“…The classification of pentagonal tilings was completed in a number of cases [4,5,20,28,29], and is expected to be completed soon [22]. Other notable works on spherical tilings include non-edge-to-edge triangular tilings by Dawson and Doyal [15,16,17,18,19] and f -tilings by Avelino, Breda d'Azevedo, Ribeiro, Santos [6,7,8,9,10]. This paper completely classifies edge-to-edge tilings of the sphere by congruent quadrilaterals.…”