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Title: Geometric and combinatorial structure of a class of spherical folding tessellations – I (English)
Author: Avelino, Catarina P.
Author: Santos, Altino F.
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 67
Issue: 4
Year: 2017
Pages: 891-918
Summary lang: English
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Category: math
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Summary: A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed. (English)
Keyword: dihedral f-tiling
Keyword: combinatorial propertie
Keyword: spherical trigonometry
Keyword: symmetry group
MSC: 20B35
MSC: 52B05
MSC: 52C20
idZBL: Zbl 06819562
idMR: MR3736008
DOI: 10.21136/CMJ.2017.0610-15
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Date available: 2017-11-20T14:52:01Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/146956
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Reference: [1] Avelino, C. P., Santos, A. F.: Spherical and planar folding tessellations by kites and equilateral triangles.Australas. J. Comb. 53 (2012), 109-125. Zbl 1255.05043, MR 2961976
Reference: [2] Avelino, C. P., Santos, A. F.: Spherical folding tesselations by kites and isosceles triangles II.Int. J. Pure Appl. Math. 85 (2013), 45-67. MR 3240748, 10.12732/ijpam.v85i1.5
Reference: [3] Avelino, C. P., Santos, A. F.: Spherical folding tessellations by kites and isosceles triangles: a case of adjacency.Math. Commun. 19 (2014), 1-28. Zbl 1298.52025, MR 3240748
Reference: [4] Avelino, C. P., Santos, A. F.: Spherical folding tessellations by kites and isosceles triangles IV.Ars Math. Contemp. 11 (2016), 59-78. Zbl 1354.52022, MR 3546649, 10.26493/1855-3974.703.05c
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Reference: [7] Breda, A. M., Ribeiro, P. S.: Spherical $f$-tilings by two non congruent classes of isosceles triangles-I.Math. Commun. 17 (2012), 127-149. Zbl 1267.52016, MR 2946138
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Reference: [9] Breda, A. M., Santos, A. F.: Dihedral $f$-tilings of the sphere by rhombi and triangles.Discrete Math. Theor. Comput. Sci. (electronic only) 7 (2005), 123-141. Zbl 1138.52307, MR 2164062
Reference: [10] Dawson, R. J. M.: Tilings of the sphere with isosceles triangles.Discrete Comput. Geom. 30 (2003), 467-487. Zbl 1053.52027, MR 2002969, 10.1007/s00454-003-2846-4
Reference: [11] Dawson, R. J. M., Doyle, B.: Tilings of the sphere with right triangles. I: The asymptotically right families.Electron. J. Comb. 13 (2006), Research paper R48, 31 pages. Zbl 1096.05015, MR 2223523
Reference: [12] Dawson, R. J. M., Doyle, B.: Tilings of the sphere with right triangles. II: The $(1,3,2)$, $(0,2,n)$ subfamily.Electron. J. Comb. 13 (2006), Research paper R49, 22 pages. Zbl 1096.05016, MR 2223524
Reference: [13] Robertson, S. A.: Isometric folding of Riemannian manifolds.Proc. R. Soc. Edinb., Sect. A 79 (1977), 275-284. Zbl 0418.53016, MR 0487893, 10.1017/S0308210500019788
Reference: [14] Ueno, Y., Agaoka, Y.: Classification of tilings of the 2-dimensional sphere by congruent triangles.Hiroshima Math. J. 32 (2002), 463-540. Zbl 1029.52004, MR 1954054, 10.32917/hmj/1151007492
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