Tilings of the sphere by congruent quadrilaterals II: edge combination a³ b with rational angles

2022/05/30 by Yixi Liao, Liao, Yixi, Erxiao Wang +1 · 2 citations
Materials Science · Mathematics · #05B45 #52C20 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Metric Geometry (math.MG) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2205.14936

openalex publication_date 2022/05/30 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28

Abstract

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of a³b-quadrilaterals with all angles being rational degrees. There are 12 sporadic and 3 infinite sequences of quadrilaterals admitting the 2-layer earth map tilings together with their modifications, and 3 sporadic quadrilaterals admitting 4 exceptional tilings. Among them only 3 quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

Cited by

• Cyclotomic points on varieties and all rational a³b-monotiles
• Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

Related

• Tilings of the sphere by congruent quadrilaterals III: edge combination a³b with general angles
• Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals
• Tilings of the sphere by congruent quadrilaterals I: edge combination a²bc
• Tilings of the sphere by congruent pentagons IV: Edge combination a⁴b with general angles
• Tilings of the sphere by congruent pentagons V: Edge combination a⁴b with rational angles