2025/07/09 by Jinjin Liang, Yixi Liao, Liang, Jinjin +5 · 1 citation
Materials Science · Mathematics · #05B45 #11R18 #11Y50 #14Q25 #52C20 #Algebraic and Geometric Analysis #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2507.07038
openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination a4b and with rational angles in degree: they are a one-parameter family of symmetric a4b-pentagonal subdivisions of the tetrahedron with 12 tiles; a sequence of unique symmetric a4b-pentagons admitting a symmetric 3-layer earth map tiling by 4m tiles for any m≥4, among which each odd m case admits two standard flip modifications; and a unique non-symmetric and degenerate a4b-pentagon admitting a non-symmetric 3-layer earth map tiling and its standard flip modification with 20 tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures.