2024/12/11 by Shu, Junjie, Yixi Liao, Liao, Yixi +2 · 2 citations
Materials Science · Computer Science · Engineering · #Quasicrystal Structures and Properties #Cellular Automata and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2412.08492
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination a4b and with any irrational angle in degree: they are three 1-parameter families of pentagonal subdivisions of the Platonic solids, with 12, 24 and 60 tiles; and a sequence of 1-parameter families of pentagons admitting non-symmetric 3-layer earth map tilings together with their various rearrangements under extra conditions. Their parameter moduli and geometric data are all computed in both exact and numerical form. The total numbers of different tilings for any fixed such pentagon are counted explicitly. As a byproduct, the degenerate pentagons produce naturally many new non-edge-to-edge quadrilateral tilings. A sequel of this paper will handle a4b-pentagons with all angles being rational in degree by solving some trigonometric Diophantine equations, to complete our full classification of edge-to-edge tilings of the sphere by congruent pentagons.