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TILINGS OF THE SPHERE BY CONGRUENT QUADRILATERALS II: EDGE COMBINATION $a^3b$ WITH RATIONAL ANGLES

Part of: Designs and configurations Discrete geometry

Published online by Cambridge University Press:  07 September 2023

YIXI LIAO Open the ORCID record for YIXI LIAO [Opens in a new window]  and
ERXIAO WANG Open the ORCID record for ERXIAO WANG [Opens in a new window]
YIXI LIAO
Affiliation:
School of Mathematical Sciences Zhejiang Normal University 688 Yingbin Road Jinhua, Zhejiang Province, 321004 China yixiliao@zjnu.edu.cn
ERXIAO WANG*
Affiliation:
School of Mathematical Sciences Zhejiang Normal University 688 Yingbin Road Jinhua, Zhejiang Province, 321004 China
*
wang.eric@zjnu.edu.cn
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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 $a^3b$-quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

MSC classification

Primary: 52C20: Tilings in $2$ dimensions 05B45: Tessellation and tiling problems
Type
Article
Information
Nagoya Mathematical Journal , Volume 253 , March 2024 , pp. 128 - 163
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

This research was supported by key projects of Zhejiang Natural Science Foundation No. LZ22A010003 and ZJNU Shuang-Long Distinguished Professorship Fund No. YS304319159.

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