On right-angled polygons in hyperbolic space

  • Edoardo Dotti
  • Simon T. Drewitz
Original Paper


We study oriented right-angled polygons in hyperbolic spaces of arbitrary dimensions, that is, finite sequences \(( S_0,S_1,\ldots ,S_{p-1})\) of oriented geodesics in the hyperbolic space \(\varvec{H}^{n+2}\) such that consecutive sides are orthogonal. It was previously shown by Delgove and Retailleau (Ann Fac Sci Toulouse Math 23(5):1049–1061, 2014. that three quaternionic parameters define a right-angled hexagon in the 5-dimensional hyperbolic space. We generalise this method to right-angled polygons with an arbitrary number of sides \(p\ge 5\) in a hyperbolic space of arbitrary dimension.


Hyperbolic space Clifford matrix Cross ratio Right-angled polygon Golden ratio 

Mathematics Subject Classification

51M10 15A66 51M20 



Both authors would like to thank their Ph.D. supervisor Prof. Ruth Kellerhals for her encouragement and valuable advice throughout the work on this paper and for her constant support.


  1. 1.
    Ahlfors, L.V.: Möbius transformations and Clifford numbers. In: Chavel, I., Farkas, H.M. (eds.) Differential geometry and complex analysis, pp. 65–73. Springer, Berlin (1985)CrossRefGoogle Scholar
  2. 2.
    Ahlfors, L.V.: Möbius transformations in \({\mathbb{R}}^n\) expressed through \(2\times 2\) matrices of Clifford numbers. Complex Var. Elliptic Equ. 5(2–4), 215–224 (1986)zbMATHGoogle Scholar
  3. 3.
    Beardon, A.F.: The Geometry of Discrete Groups, vol. 91. Springer, Berlin (2012)zbMATHGoogle Scholar
  4. 4.
    Cao, C., Waterman, P.L.: Conjugacy invariants of Möbius groups. In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds.) Quasiconformal mappings and analysis, pp. 109–139. Springer, New York (1998)CrossRefGoogle Scholar
  5. 5.
    Costa, A.F., Martínez, E.: On hyperbolic right-angled polygons. Geom. Dedicata 58(3), 313–326 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dekster, B.V., Wilker, J.B.: Simplexes in spaces of constant curvature. Geom. Dedicata 38(1), 1–12 (1991). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Delgove, F., Retailleau, N.: Sur la classification des hexagones hyperboliques à angles droits en dimension 5. Ann. Fac. Sci. Toulouse Math. 23(5), 1049–1061 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces: Séminaire Orsay. Astérisque 66–67 (1979)Google Scholar
  9. 9.
    Fenchel, W.: Elementary Geometry in Hyperbolic Space, De Gruyter Studies in Mathematics, vol. 11. Walter de Gruyter & Co., Berlin (1989). CrossRefGoogle Scholar
  10. 10.
    Parizet, J.: Quaternions et géométrie. (2006). Accessed 11 July 2017
  11. 11.
    Parker, J.: Hyperbolic spaces. (2007). Accessed: 11 July 2017
  12. 12.
    Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149, 2nd edn. Springer, New York (2006). zbMATHGoogle Scholar
  13. 13.
    Vahlen, K.T.: Über bewegungen und complexe zahlen. Math. Ann. 55(4), 585–593 (1902)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Waterman, P.L.: Möbius transformations in several dimensions. Adv. Math. 101(1), 87–113 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wilker, J.B.: Inversive geometry. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds.) The geometric vein, pp. 379–442. Springer, Berlin (1981)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité de FribourgFribourgSwitzerland

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