Tilings of the sphere by congruent pentagons. I: Edge combinations \(a^2b^2c\) and \(a^3 bc \). (English) Zbl 1484.52016
The authors study edge-to-edge tilings of the two-dimensional sphere by congruent images of some spherical pentagon \(P\). It is shown that the lengths of the consecutive edges of such \(P\) must be one of \(a-a-b-b-c\) (called edge combination \(a^2b^2c\)), \(a-a-a-b-c\) (called \(a^3bc\)), \(a-a-a-b-b\) (called \(a^3b^2\)), \(a-a-a-a-b\) (called \(a^4b\)) or \(a-a-a-a-a\) (called \(a^5\)) with distinct \(a,b,c\). The tilings with edge combinations \(a^2b^2c\) (three families with \(12\), \(24\) or \(60\) tiles) and \(a^3bc\) (two unique tilings with \(48\) or \(120\) tiles) are completely classified.
For Part II, see [the authors, Adv. Math. 394, Article ID 107867, 68 p. (2022; Zbl 1484.52017)], for Part III, see [Y. Akama and the authors, ibid. 384, Article ID 107881, 41 p. (2022; Zbl 1484.52014)].
For Part II, see [the authors, Adv. Math. 394, Article ID 107867, 68 p. (2022; Zbl 1484.52017)], for Part III, see [Y. Akama and the authors, ibid. 384, Article ID 107881, 41 p. (2022; Zbl 1484.52014)].
Reviewer: Christian Richter (Jena)
MSC:
| 52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |
| 05B45 | Combinatorial aspects of tessellation and tiling problems |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
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\textit{E. Wang} and \textit{M. Yan}, Adv. Math. 394, Article ID 107866, 36 p. (2022; Zbl 1484.52016)
References:
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