Geometric and combinatorial structure of a class of spherical folding tessellations – I

Catarina P. Avelino; Altino F. Santos

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 4, page 891-918
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Avelino, Catarina P., and Santos, Altino F.. "Geometric and combinatorial structure of a class of spherical folding tessellations – I." Czechoslovak Mathematical Journal 67.4 (2017): 891-918. <http://eudml.org/doc/294407>.

@article{Avelino2017,
abstract = {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.},
author = {Avelino, Catarina P., Santos, Altino F.},
journal = {Czechoslovak Mathematical Journal},
keywords = {dihedral f-tiling; combinatorial propertie; spherical trigonometry; symmetry group},
language = {eng},
number = {4},
pages = {891-918},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Geometric and combinatorial structure of a class of spherical folding tessellations – I},
url = {http://eudml.org/doc/294407},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Avelino, Catarina P.
AU - Santos, Altino F.
TI - Geometric and combinatorial structure of a class of spherical folding tessellations – I
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 891
EP - 918
AB - 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.
LA - eng
KW - dihedral f-tiling; combinatorial propertie; spherical trigonometry; symmetry group
UR - http://eudml.org/doc/294407
ER -

References

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  1. Avelino, C. P., Santos, A. F., Spherical and planar folding tessellations by kites and equilateral triangles, Australas. J. Comb. 53 (2012), 109-125. (2012) Zbl1255.05043MR2961976
  2. Avelino, C. P., Santos, A. F., 10.12732/ijpam.v85i1.5, Int. J. Pure Appl. Math. 85 (2013), 45-67. (2013) MR3240748DOI10.12732/ijpam.v85i1.5
  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. (2014) Zbl1298.52025MR3240748
  4. Avelino, C. P., Santos, A. F., 10.26493/1855-3974.703.05c, Ars Math. Contemp. 11 (2016), 59-78. (2016) Zbl1354.52022MR3546649DOI10.26493/1855-3974.703.05c
  5. Breda, A. M., 10.1007/BF00181393, Geom. Dedicata 44 (1992), 241-253. (1992) Zbl0770.52010MR1193117DOI10.1007/BF00181393
  6. Breda, A. M., Dawson, R., Ribeiro, P. S., 10.1007/s10114-014-3302-5, Acta Math. Sin., Engl. Ser. 30 (2014), 1435-1464. (2014) Zbl06342513MR3229152DOI10.1007/s10114-014-3302-5
  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. (2012) Zbl1267.52016MR2946138
  8. Breda, A. M., Ribeiro, P. S., Santos, A. F., 10.1016/j.ejc.2008.02.010, Eur. J. Comb. 30 (2009), 119-132. (2009) Zbl1161.52014MR2460222DOI10.1016/j.ejc.2008.02.010
  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. (2005) Zbl1138.52307MR2164062
  10. Dawson, R. J. M., 10.1007/s00454-003-2846-4, Discrete Comput. Geom. 30 (2003), 467-487. (2003) Zbl1053.52027MR2002969DOI10.1007/s00454-003-2846-4
  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. (2006) Zbl1096.05015MR2223523
  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. (2006) Zbl1096.05016MR2223524
  13. Robertson, S. A., 10.1017/S0308210500019788, Proc. R. Soc. Edinb., Sect. A 79 (1977), 275-284. (1977) Zbl0418.53016MR0487893DOI10.1017/S0308210500019788
  14. Ueno, Y., Agaoka, Y., 10.32917/hmj/1151007492, Hiroshima Math. J. 32 (2002), 463-540. (2002) Zbl1029.52004MR1954054DOI10.32917/hmj/1151007492

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