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Spherical quadrangles with three equal sides and rational angles

Kris Coolsaet (UGent)
(2017) ARS MATHEMATICA CONTEMPORANEA. 12(2). p.415-424
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Abstract
When the condition of having three equal sides is imposed upon a (convex) spherical quadrangle, the four angles of that quadrangle cannot longer be freely chosen but must satisfy an identity. We derive two simple identities of this kind, one involving ratios of sines, and one involving ratios of tangents, and improve upon an earlier identity by Ueno and Agaoka. The simple form of these identities enable us to further investigate the case in which all of the angles are rational multiples of pi and produce a full classification, consisting of 7 infinite classes and 29 sporadic examples. Apart from being interesting in its own right, these quadrangles play an important role in the study of spherical tilings by congruent quadrangles.
Keywords
Spherical quadrangle, rational angle, spherical tiling

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Citation

Please use this url to cite or link to this publication:

Chicago
Coolsaet, Kris. 2017. “Spherical Quadrangles with Three Equal Sides and Rational Angles.” Ars Mathematica Contemporanea 12 (2): 415–424.
APA
Coolsaet, K. (2017). Spherical quadrangles with three equal sides and rational angles. ARS MATHEMATICA CONTEMPORANEA, 12(2), 415–424.
Vancouver
1.
Coolsaet K. Spherical quadrangles with three equal sides and rational angles. ARS MATHEMATICA CONTEMPORANEA. 2017;12(2):415–24.
MLA
Coolsaet, Kris. “Spherical Quadrangles with Three Equal Sides and Rational Angles.” ARS MATHEMATICA CONTEMPORANEA 12.2 (2017): 415–424. Print.
@article{8547885,
  abstract     = {When the condition of having three equal sides is imposed upon a (convex) spherical quadrangle, the four angles of that quadrangle cannot longer be freely chosen but must satisfy an identity. We derive two simple identities of this kind, one involving ratios of sines, and one involving ratios of tangents, and improve upon an earlier identity by Ueno and Agaoka. 
The simple form of these identities enable us to further investigate the case in which all of the angles are rational multiples of pi and produce a full classification, consisting of 7 infinite classes and 29 sporadic examples. Apart from being interesting in its own right, these quadrangles play an important role in the study of spherical tilings by congruent quadrangles.},
  author       = {Coolsaet, Kris},
  issn         = {1855-3966},
  journal      = {ARS MATHEMATICA CONTEMPORANEA},
  language     = {eng},
  number       = {2},
  pages        = {415--424},
  title        = {Spherical quadrangles with three equal sides and rational angles},
  volume       = {12},
  year         = {2017},
}

Web of Science
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