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Symmetries in Graphs, Maps, and Polytopes Workshop

SIGMAP 2014: Symmetries in Graphs, Maps, and Polytopes pp 287–300Cite as

Even-Integer Continued Fractions and the Farey Tree

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 159))

Abstract

Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this tessellation comprises the edges of an infinite tree whose vertices belong to the ideal boundary. Here we show how this tree can be used to give a beautiful geometric representation of even-integer continued fractions. We use this representation to prove some of the fundamental theorems on even-integer continued fractions that are already known, and we also prove some new theorems with this technique, which have familiar counterparts in the theory of regular continued fractions.

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK

    Ian Short

  2. School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, UK

    Mairi Walker

Authors
  1. Ian Short
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  2. Mairi Walker
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Corresponding author

Correspondence to Ian Short .

Editor information

Editors and Affiliations

  1. The Open University, Milton Keynes, United Kingdom

    Jozef Širáň

  2. Comenius University, Bratislava, Slovakia

    Robert Jajcay

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© 2016 Springer International Publishing Switzerland

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Short, I., Walker, M. (2016). Even-Integer Continued Fractions and the Farey Tree. In: Širáň, J., Jajcay, R. (eds) Symmetries in Graphs, Maps, and Polytopes. SIGMAP 2014. Springer Proceedings in Mathematics & Statistics, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-30451-9_15

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