| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2017-11-20T14:52:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146956 |
| . |
| 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 |
| Reference:
|
[5] Breda, A. M.: A class of tilings of $S^{2}$.Geom. Dedicata 44 (1992), 241-253. Zbl 0770.52010, MR 1193117, 10.1007/BF00181393 |
| Reference:
|
[6] Breda, A. M., Dawson, R., Ribeiro, P. S.: Spherical $f$-tilings by two noncongruent classes of isosceles triangles-II.Acta Math. Sin., Engl. Ser. 30 (2014), 1435-1464. Zbl 06342513, MR 3229152, 10.1007/s10114-014-3302-5 |
| 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 |
| Reference:
|
[8] Breda, A. M., Ribeiro, P. S., Santos, A. F.: A class of spherical dihedral $f$-tilings.Eur. J. Comb. 30 (2009), 119-132. Zbl 1161.52014, MR 2460222, 10.1016/j.ejc.2008.02.010 |
| 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 |
| . |
