Wallpaper Maps

  • M. Douglas McIlroy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6875)


A wallpaper map is a conformal projection of a spherical earth onto regular polygons with which the plane can be tiled continuously. A complete set of distinct wallpaper maps that satisfy certain natural symmetry conditions is derived and illustrated. Though all of the projections have been published before, some generalize to one-parameter families in which the sphere is pre-transformed by a conformal automorphism.


conformal projection elliptic integral spherical tiling planar tiling 


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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards (1965)Google Scholar
  2. 2.
    Adams, O.A.: Elliptic Functions Applied to Conformal World Maps. Special publication no. 112, U.S. Coast and Geodetic Survey (1925)Google Scholar
  3. 3.
    Adams, O.A.: Conformal Projection of the Sphere within a Square. Special publication no. 153, U.S. Coast and Geodetic Survey (1929)Google Scholar
  4. 4.
    Adams, O.A.: Conformal map of the world in a square. Bulletin Géodèsique, 461–473 (1936)Google Scholar
  5. 5.
    Bieberbach, L.: Conformal Mapping. Chelsea (1953), translated by F. SteinhardtGoogle Scholar
  6. 6.
    Coxeter, H.S.M.: Regular Polytopes. Dover, New York (1973), reprint of Macmillan edition of 1963 Google Scholar
  7. 7.
    Guyou, E.: Sur un nouveau système de projection de la sphère. Comptes Rendus de l’Acad/’emie des Sciences 102, 308–310 (1886)zbMATHGoogle Scholar
  8. 8.
    Lee, L.P.: Conformal Projections based on Elliptic Functions. Cartographica Monograph 16 (1976)Google Scholar
  9. 9.
    McIlroy, M.D.: Power series, power serious. Journal of Functional Programming 9, 325–337 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    McIlroy, M.D.: Wallpaper maps (2011), an extended version of the present paper,
  11. 11.
    Pedoe, D.: Geometry: A Comprehensive Course. Dover, New York (1988)zbMATHGoogle Scholar
  12. 12.
    Peirce, C.S.: A quincuncial projection of the sphere. American Journal of Mathematics 2, 394–396 and one unpaginated plate (1879)Google Scholar
  13. 13.
    Schwarz, H.A.: Über einige abbildungsaufgaben. Journal für die reine un angewandte Mathematik 70, 105–120 (1869)CrossRefGoogle Scholar
  14. 14.
    Snyder, J.P.: Flattening the Earth. University of Chicago Press, Chicago (1997)Google Scholar
  15. 15.
    Snyder, J.P., Voxland, P.M.: An Album of Map Projections. Professional Paper 1453, U.S. Geological Survey (1989)Google Scholar
  16. 16.
    Weyl, H.: Symmetry. Princeton University Press, Princeton (1952)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • M. Douglas McIlroy
    • 1
  1. 1.Dartmouth CollegeHanoverUSA

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