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On right-angled polygons in hyperbolic space

  • Edoardo Dotti
  • Simon T. Drewitz
Original Paper
  • 85 Downloads

Abstract

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.  https://doi.org/10.5802/afst.1435) 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.

Keywords

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

Mathematics Subject Classification

51M10 15A66 51M20 

Notes

Acknowledgements

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.

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Copyright information

© 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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