The Mathematical Intelligencer

, Volume 7, Issue 3, pp 20–29 | Cite as

The geometry of markoff numbers

  • Caroline Series


Hyperbolic Plane Continue Fraction Expansion Fundamental Region Binary Quadratic Form Simple Closed Geodesic 
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  1. 1.
    E. B. Christoffel,Observatio Arithmetica, Annali di Mathe- matica, 2nd series, 6(1875), 148–152.Google Scholar
  2. 2.
    H. Cohn, Approach to Markoff’s minimal forms through modular functions.Ann. Math. 61(1955), 1–12.CrossRefMATHGoogle Scholar
  3. 3.
    H. Cohn, Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus.Ada Arithmetica XVIII(1971), 125–136.Google Scholar
  4. 4.
    L. E. Dickson, Studies in the theory of numbers. Chicago: 1930.Google Scholar
  5. 5.
    D. Fowler, Anthyphairetic ratio and Eudoxan proportion.Archive for History of Exact Sciences 24(1981), 69–72.CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Haas, Diophantine approximation on hyperbolic Rie-mann surfaces,Bull. A.M.S. 12(1984), 359–362.Google Scholar
  7. 7.
    J. Lehner, M. Scheingorn, Simple closed geodesics on H+/r(3) arise from the Markoff spectrum, preprint.Google Scholar
  8. 8.
    A. A. Markoff, Sur les formes binaires indefinies, I,Math. Ann. 15(1879), 281–309; II, 17(1880), 379–400.CrossRefGoogle Scholar
  9. 9.
    C. Series, The modular surface and continued fractions. J. London Math. Soc. (1984).Google Scholar
  10. 10.
    A. L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups I.J. Reine Angew. Math. 286/7 (1976), 341–368.Google Scholar
  11. 11.
    H. J. S. Smith, Note on continued fractions.Messenger of Mathematics, 2nd series, 6(1876), 1–14.Google Scholar
  12. 12.
    E. C. Zeeman, An algorithm for Eudoxan and anthi- phairetic ratios, preprint.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Caroline Series
    • 1
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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