2025/05/22 by Wen Chen, Chen, Wen, Jinjin Liang +3
Materials Science · Mathematics · #05B45 #52C20 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematics and Applications #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2505.16629
openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with 2n(n≥3) tiles, together with rotational modifications for even n; a 1-parameter family of triangles each admitting a unique tiling with 8 tiles; and a sporadic triangle admitting a unique tiling with 16 tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.