Tilings of the Sphere by Congruent Quadrilaterals I: Edge Combination a²bc

Abstract

Edge-to-edge tilings of the sphere by congruent a²bc-quadrilaterals are classified as 3 classes: (1) A 1-parameter family of quadrilateral subdivisions of the octahedron with 24 tiles, and a flip modification for one special parameter; (2) a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n ≥ 3; (3) a 3-layer earth map tiling with 8n tiles for each n ≥ 2, and two flip modifications for each odd n. The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings.

Keywords

Spherical tiling / Quadrilateral / Classification / Earth map tiling / Subdivision / 52C20 / 05B45

References

[1] Akama Y, van Cleemput N. Spherical tilings by congruent quadrangles: Forbidden cases and substructures. Ars Math. Contemp.. 2015, 8: 297-318

[2] Akama Y, Wang E, Yan M. Tilings of sphere by congruent pentagons III: Edge combination a⁵. Adv. Math.. 2022, 394: 107881

[3] Cheung, H. M., Luk, H. P. and Yan, M., Tilings of the Sphere by Congruent Quadrilaterals or Triangles, arXiv: 2204.02736, 2022.

[4] Davies H. Packings of spherical triangles and tetrahedra. Proc. Colloquium on Convexity (Copenhagen, 1965). 1967, Copenhagen, Kobenhavns Univ. Mat. Inst.: 42-51

[5] Liao Y, Wang E. Tilings of the sphere by congruent quadrilaterals II: Edge combination a³b with rational angles. Nagoya Math. J.. 2024, 253: 128-163

[6] Liao, Y., Wang, E., Qian, P. and Xu, Y., Tilings of the sphere by congruent quadrilaterals III: Edge combination a³b with general angles, Forum Math., 2024, https://doi.org/10.1515/forum-2023-0209.

[7] Rao, M., Exhaustive search of convex pentagons which tile the plane, arXiv:1708.00274, 2017.

[8] Sakano Y, Akama Y. Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi. Hiroshima Mathematical Journal. 2015, 45(3): 309-339

[9] Sommerville D M Y. Division of space by congruent triangles and tetrahedra. Proc. Royal Soc. Edinburgh. 1924, 43: 85-116

[10] Ueno Y, Agaoka Y. Examples of spherical tilings by congruent quadrilaterals. Math. Inform. Sci., Fac. Integrated Arts and Sci., Hiroshima Univ., Ser. IV. 2001, 27: 135-144

[11] Ueno Y, Agaoka Y. Classification of tilings of the 2-dimensional sphere by congruent triangles. Hiroshima Math. J.. 2002, 32(3): 463-540

[12] Wang E, Yan M. Tilings of the sphere by congruent pentagons I: Edge combinations a²b²c and a³bc. Adv. Math.. 2022, 394: 107866

[13] Wang E, Yan M. Tilings of the sphere by congruent pentagons II: Edge combination a³b². Adv. Math.. 2022, 394: 107867

[14] Wang, E. and Yan, M., Moduli of pentagonal subdivision tiling, arXiv: 1907.08776, 2019.

[15] Zong C M. Can you pave the plane with identical tiles?. Notice AMS. 2020, 67(5): 635-646