Skip to main content

Inversive Geometry

  • Conference paper
Book cover The Geometric Vein

Abstract

Let me begin by describing one of the gems of classical mathematics which first stirred my own enthusiasm for inversive geometry. It illustrates the elegance of the subject and provides a point of interest which we shall glimpse again in the closing chapters of this account.

I would like to thank H. S. M. Coxeter, J. C. Fisher, E. Honig, W. Israel, B. Salzberg, L. Southwell, J. F. Rigby, and B. Wilker for their encouraging enthusiasm. My interest in inversive geometry goes back more than ten years, and for much of this time I have received financial support from Canadian NRC Grant A8100. The opportunity to write the final version of this paper came while I was enjoying a sabbatical year as Visiting Fellow at the Institute of Advanced Studies, The Australian National University.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexander, H. W., Vectorial inversive and non-Euclidean geometry. Amer. Math. Monthly 74 (1967), 128–140. MR 35, #864.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bers, L. and Kra, I. (eds.), A Crash Course on Kleinian Groups. Springer-Verlag, Berlin-Heidelberg-New York 1974. MR 49, #10878.

    MATH  Google Scholar 

  3. Boyd, D. W., The disc-packing constant. Aequationes Math. 7 (1972), 182–193. MR 46, #2557.

    Article  Google Scholar 

  4. Boyd, D. W., Improved bounds for the disc-packing constant. Aequationes Math. 9 (1973), 99–106. MR 49, #5728.

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyd, D. W., The osculatory packing of a three-dimensional sphere. Canad. J. Math. 25 (1973), 303–322. MR 47, #9430.

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyd, D. W., An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing. Math. Comp. 27 (1973), 369–377. MR 49, #3700.

    Article  MathSciNet  MATH  Google Scholar 

  7. Boyd, D. W., A new class of infinite sphere packings. Pacific J. Math. 50 (1974), 383–398. MR 50, #3118.

    MathSciNet  MATH  Google Scholar 

  8. Caraman, P., n-dimensional Quasiconformal (QCf) Mappings. Abacus Press, Tunbridge Wells 1974. MR 38, #3428 and 50, #10249.

    Google Scholar 

  9. Caratheodory, C., The most general transformations of plane regions which transform circles into circles. Bull. Amer. Math. Soc. 43 (1937), 573–579.

    Article  MathSciNet  Google Scholar 

  10. Caratheodory, C., Theory of Functions of a Complex Variable (Vols. 1,2). Chelsea, New York 1954. MR 15, 612.

    Google Scholar 

  11. Clifford, W. H., On the powers of spheres (1868). In Mathematical Papers. Macmillan, London 1882.

    Google Scholar 

  12. Coolidge, J. L., A Treatise on the Circle and the Sphere. Clarendon Press, Oxford 1916.

    MATH  Google Scholar 

  13. Courant, R. and Hilbert, D., Methods of Mathematical Physics (Vol. 2). Wiley, New York 1962. MR 25, #4216.

    MATH  Google Scholar 

  14. Coxeter, H. S. M., Interlocked rings of spheres. Scripta Math. 18 (1952), 113–121. MR 14, 492.

    MathSciNet  MATH  Google Scholar 

  15. Coxeter, H. S. M., Non-Euclidean Geometry (5th edition). University of Toronto Press, Toronto 1965. MR 19, 445.

    Google Scholar 

  16. Coxeter, H. S. M., Introduction to Geometry (2nd edition), Wiley, New York-London-Sydney-Toronto 1969. MR 23, #A1251.

    MATH  Google Scholar 

  17. Coxeter, H. S. M., Inversive distance. Ann. Mat. Pura Appl. (4) 71 (1966), 73–83. MR 34, #3418.

    Article  MathSciNet  MATH  Google Scholar 

  18. Coxeter, H. S. M., The inversive plane and hyperbolic space. Abh. Math. Sem. Univ. Hamburg 29 (1966), 217–242. MR 33, #7920.

    Article  MathSciNet  MATH  Google Scholar 

  19. Coxeter, H. S. M., The Lorentz group and the group of homographies. In Proc. Internal Conf. Theory of Groups (Canberra 1965). Gordon and Breach, New York 1967. MR 37, #5768.

    Google Scholar 

  20. Coxeter, H. S. M., Mid-circles and loxodromes. Ontario Mathematics Gazette 5 (1967), 4–15.

    Google Scholar 

  21. Coxeter, H. S. M., The problem of Apollonius. Canad. Math. Bull. 11 (1968), 1–17. Reprinted in Amer. Math. Monthly 75 (1968), 5–15. MR 37, #5767.

    Article  MATH  Google Scholar 

  22. Coxeter, H. S. M., Loxodromic sequences of tangent spheres. Aequationes Math. 1 (1968), 104–121. MR 38, #3765.

    Article  MathSciNet  MATH  Google Scholar 

  23. Coxeter, H. S. M. and Greitzer, S. L., Geometry Revisited. Mathematical Association of America, Washington 1977.

    Google Scholar 

  24. Darboux, G., Sur les relations entre les groupes de points de cercles et de spheres dans le plan et dans l’espace. Ann. Ecole Norm. Sup. 1 (1872), 323–392.

    MathSciNet  Google Scholar 

  25. Darboux, G., Théorie Générate des Surfaces (3rd edition). Chelsea, New York 1972. MR 53, #79-82.

    Google Scholar 

  26. Dembowski, P., Finite Géométries. Springer-Verlag, Berlin-New York 1968. MR 38, #1597.

    MATH  Google Scholar 

  27. Dembowski, P. and Hughes, D. R., On finite inversive planes. J. London Math. Soc. 40 (1965), 171–182. MR 30, #2382.

    Article  MathSciNet  MATH  Google Scholar 

  28. Ebner, D. W., A purely geometrical introduction of spinors in special relativity by means of conformal mappings on the celestial sphere. Annalen der Physik (7) 30 (1973), 206–210.

    Article  MathSciNet  Google Scholar 

  29. Ewald, G., Begründung der Geometrie der ebenen Schnitte einer Semiquadrik. Arch. Math. 8 (1957), 203–208. MR 19, 1190.

    Article  MathSciNet  MATH  Google Scholar 

  30. Ewald, G., Geometry: An Introduction. Wadsworth, Belmont, California 1971. MR 51, #1575.

    MATH  Google Scholar 

  31. Fejes Toth, L., Regular Figures. Pergamon Press, London 1964. MR 29, #2705.

    MATH  Google Scholar 

  32. Flanders, H., Liouville’s theorem on conformal mapping. J. Math. Mech. 15 (1966), 157–161. MR 32, #1626.

    MathSciNet  MATH  Google Scholar 

  33. Ford, L. R., Automorphic Functions (2nd edition). Chelsea, New York 1951.

    Google Scholar 

  34. Fricke, R. and Klein, F., Vorlesungen über die Theorie der automorphen Funktionen (erster Band). Teubner, Stuttgart 1897. Johnson Reprint Corporation, New York 1965.

    Google Scholar 

  35. Gehring, F. W., The Liouville theorem in space. Notices Amer. Math. Soc. 7 (1960), 523–524.

    Google Scholar 

  36. Gehring, F. W., Rings and quasiconformal mappings in space. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 98–105. MR 23, #A3261.

    Article  MathSciNet  MATH  Google Scholar 

  37. Gehring, F. W., Symmetrization of rings in space. Trans. Amer. Math. Soc. 101 (1961), 499–519. MR 24, #A2677.

    Article  MathSciNet  MATH  Google Scholar 

  38. Gehring, F. W., Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 25, #3166.

    Article  MathSciNet  MATH  Google Scholar 

  39. Gehring, F. W., Quasiconformal mappings in space. Bull. Amer. Math. Soc. 69 (1963), 146–164. MR 26, #2606.

    Article  MathSciNet  MATH  Google Scholar 

  40. Gehring, F. W., Quasiconformal mappings in ℝn. In Lectures on Quasiconformal Mapping. Department of Mathematics, University of Maryland, College Park 1975. MR 52, #14281.

    Google Scholar 

  41. Gehring, F. W., Quasiconformal mappings. In Complex Analysis and Its Applications (Vol. II). International Atomic Energy Agency, Vienna 1976.

    Google Scholar 

  42. Gerber, L., Sequences of isoclinical spheres. Aequationes Math. 17 (1978), 53–72.

    Article  MathSciNet  MATH  Google Scholar 

  43. Gibbons, J. and Webb, C., Circle-preserving functions of spheres. Trans. Amer. Math. Soc. 248 (1979), 67–83.

    Article  MathSciNet  MATH  Google Scholar 

  44. Gosset, T., The kiss precise. Nature 139 (1937), 62.

    Article  Google Scholar 

  45. Gosset, T., The hexlet. Nature 139 (1937), 251.

    Article  Google Scholar 

  46. Greub, W., Linear Algebra (4th edition). Springer-Verlag, New York 1967. MR 37, #221.

    MATH  Google Scholar 

  47. Guggenheimer, H. W., Differential Geometry. McGraw-Hill, New York-San Francisco-Toronto-London 1963. MR 27, #6194.

    MATH  Google Scholar 

  48. Hartman, P., Systems of total differential equations and Liouville’s theorem on conformal mappings. Amer. J. Math. 69 (1947), 327–332. MR 9, 59.

    Article  MathSciNet  MATH  Google Scholar 

  49. Hartman, P., On isometries and on a theorem of Liouville. Math. Z. 69 (1958), 202–210. MR 21, #7521.

    Article  MathSciNet  MATH  Google Scholar 

  50. Helms, L. L., Introduction to Potential Theory. Wiley, New York-London-Sydney 1969. MR 41, #5638.

    MATH  Google Scholar 

  51. Iwata, S., Generalizations of Ohara-Iwata’s theorem in Wasan to the n-dimensional space. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 4 (1970), 243–250. MR 48, #12234.

    MathSciNet  Google Scholar 

  52. Iwata, S., Generalization of Steiner’s contact circle theorem to the n-dimensional space. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 4 (1970/71), 349–354. MR 46, #4343.

    MathSciNet  Google Scholar 

  53. Iwata, S., On a theorem connected with the contact hyperspheres. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 5 (1972), 5–7. MR 48, #1017.

    MathSciNet  Google Scholar 

  54. Iwata, S., Generalizations of Pappus’s tangent circle theorem to the n-dimensional space, Bull. Gifu Coll. Ed. 1 (1974), 55–58. MR 51, #6551.

    MathSciNet  Google Scholar 

  55. Iwata, S. and Naito, J., The problem of Apollonius in the n-dimensional space. Sci. Rep. Fac. Ed. Gifu univ. Natur. Sci. 4 (1969), 138–148. MR 41, #6038.

    MathSciNet  Google Scholar 

  56. Iwata, S. and Naito, J., A generalization of Wilker’s calculation to the n-dimensional space. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 4 (1970), 251–255. MR 49, #3657.

    MathSciNet  Google Scholar 

  57. Iwata, S. and Naito, J., Relations between the radii of successively tangent hyperspheres touching a hyperellipsoid. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 5 (1973), 121–130. MR 48, #12236.

    MathSciNet  Google Scholar 

  58. Kellogg, O. D., Foundations of Potential Theory. Ungar, New York 1929.

    Google Scholar 

  59. de Kerekjarto, B., Sur le groupe des homographies et des antihomographies d’une variable complex. Comment. Math. Helv. 13 (1940), 68–82. MR 2, 322-323.

    Article  MathSciNet  Google Scholar 

  60. de Kerekjarto, B., Sur le caractere topologique du groupe homographique de la sphere. Acta Math. 74 (1941), 311–341. MR 7, 137.

    Article  MathSciNet  Google Scholar 

  61. de Kerekjarto, B., Sur les groupes transitifs de la droite. Acta. Univ. Szeged. Sect. Sci. Math. 10 (1941), 21–35. MR 2, 322.

    MathSciNet  Google Scholar 

  62. Kra, I., Automorphic forms and Kleinian groups. Benjamin, Reading, Massachusetts 1972. MR 50, #10242.

    MATH  Google Scholar 

  63. Lachlan, R., On systems of circles and spheres. Philos. Trans. Roy. Soc. London A 111 (1886), 481–625.

    Google Scholar 

  64. Lagrange, R., Produits d’Iversions et Métrique Conforme. Gauthier-Villars, Paris 1957. MR 19, 162.

    Google Scholar 

  65. Lehner, J., Discontinuous groups and automorphic functions. American Mathematical Society, Providence 1964. MR 29, #1332.

    MATH  Google Scholar 

  66. Lehner, J., A short course in automorphic functions. Holt Rinehart and Winston, New York-Toronto-London 1966. MR 34, #1519.

    Google Scholar 

  67. Liouville, J., Théorème sur l’equation dx 2 + dy 2 + dz 2 = λ(dα2 + 2 + dγ2). J. Math. Pures Appl. (1) 15 (1850), 103.

    Google Scholar 

  68. Magnus, W., Noneuclidean Tesselations and Their Groups. Academic Press, New York-London 1974. MR 50, #4774.

    MATH  Google Scholar 

  69. Mauldon, J. G., Sets of equally inclined spheres. Canad. J. Math. 14 (1962), 509–516. MR 25, #5425.

    Article  MathSciNet  MATH  Google Scholar 

  70. Mauldon, J. G., Equally inclined spheres. Proc. Camb. Phil. Soc. 58 (1962), 420–421.

    Article  MATH  Google Scholar 

  71. Mauldon, J. G., Bunches of cones. Amer. Math. Monthly 69 (1962), 206–207.

    Article  MathSciNet  MATH  Google Scholar 

  72. Mäurer, H., Ein axiomatischer Aufbau der mindestens 3-dimensionalen Möbius-Geometrie. Math. Z. 103 (1968), 282–305. MR 36, #7016.

    Article  MathSciNet  MATH  Google Scholar 

  73. Melzak, Z. A., Infinite packings of discs. Canad. J. Math. 18 (1966), 838–852. MR 34, #3443.

    Article  MathSciNet  MATH  Google Scholar 

  74. Morley, F., The hexlet. Nature 139 (1937), 72–73.

    Article  Google Scholar 

  75. Morley, F. and Morley, F. V., Inversive Geometry. G. Bell and Sons, London 1933.

    Google Scholar 

  76. Naito, J., Some properties of the contact hyperspheres in the n-dimensional space. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 4 (1970), 256–267. MR 49, #1313.

    MathSciNet  Google Scholar 

  77. Naito, J., A generalization of Malfatti’s problem. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci. 5 (1975), 277–286. MR 52, #15218.

    MathSciNet  Google Scholar 

  78. Nevanlinna, R., On differentiable mappings. In Analytic Functions. Princeton University Press, Princeton 1960. MR 22, #7075.

    Google Scholar 

  79. Pedoe, D., On a theorem in geometry. Amer. Math. Monthly 74 (1967), 627–640. MR 35, #6012.

    Article  MathSciNet  MATH  Google Scholar 

  80. Phillips, R., Liouville’s theorem. Pacific J. Math. 28 (1969), 397–405. MR 42, #4715.

    MathSciNet  MATH  Google Scholar 

  81. Rešetnjak, J. G., On conformal mappings of a space. Dokl. Akad. Nauk. SSSR 130 (1960), 1196–1198. Soviet Math. Dokl. 1 (1960), 153–155. MR 22, #9935.

    Google Scholar 

  82. Rešetnjak, J. G., Liouville’s conformal mapping theorem under minimal regularity hypothesis (Russian). Sibirsk. Mat. Z. 8 (1967), 835–840. MR 36, #1630.

    MathSciNet  Google Scholar 

  83. Scherk, P., Some concepts of conformal geometry. Amer. Math. Monthly 67 (1960), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  84. Schwerdtfeger, H., Geometry of Complex Numbers. University of Toronto Press, Toronto 1962. MR 24, #A2880.

    MATH  Google Scholar 

  85. Siegel, C. L., Topics in Complex Function Theory (3 vols.). Wiley, New York 1969-73. MR 41, #1977.

    MATH  Google Scholar 

  86. Soddy, F., The kiss precise. Nature 137 (1936), 1021.

    Article  Google Scholar 

  87. Soddy, F., The hexlet. Nature 138 (1936), 958.

    Article  Google Scholar 

  88. Soddy, F., The bowl of integers and the hexlet. Nature 139 (1937), 77–79.

    Article  MATH  Google Scholar 

  89. Soddy, F., The hexlet. Nature 139 (1937), 154.

    Article  Google Scholar 

  90. Sommerville, D. M. Y., The Elements of Non-Euclidean Geometry, Bell and Sons, London 1914.

    MATH  Google Scholar 

  91. Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. 3. Publish or Perish, Boston 1975. MR 51, #8962.

    Google Scholar 

  92. Volenec, V., An axiomatic foundation of the circle plane with orthogonality. Glasnik Mat. 8 (No. 28, 1973), 85–92. MR 47, #7574.

    MathSciNet  Google Scholar 

  93. Volenec, V., Axiomatic foundations of the n-dimensional Möbius geometry. Publ. Math. Debrecen 23 (1976) 89–102. MR 54, #11171.

    MathSciNet  MATH  Google Scholar 

  94. Webb, C. and Gibbons, J., The inversive group of S n. Notices Amer. Math. Soc. 24 (1977), A335.

    Google Scholar 

  95. Wilker, J. B., Four proofs of a generalization of the Descartes circle theorem. Amer. Math. Monthly 76 (1969), 278–282. MR 39, #7511.

    Article  MathSciNet  MATH  Google Scholar 

  96. Wilker, J. B., Circular sequences of disks and balls. Notices Amer. Math. Soc. 19 (1972), A193.

    Google Scholar 

  97. Wilker, J. B., The interval of disk packing exponents. Proc. Amer. Math. Soc. 41 (1973), 255–260. MR 50, #3120.

    Article  MathSciNet  MATH  Google Scholar 

  98. Wilker, J. B., Sizing up a solid packing. Period. Math. Hungar. 8 (1977), 117–134. MR 58, #30759.

    Article  MathSciNet  MATH  Google Scholar 

  99. Wise, M. E., On the radii of five packed spheres in mutual contact. Philips Res. Reports 15 (1960), 101. MR 22, #11293.

    MathSciNet  Google Scholar 

  100. Woods, F. S., Higher Geometry. Ginn, Boston 1922. Dover, New York 1961. MR 23, #A2094.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag New York Inc.

About this paper

Cite this paper

Wilker, J.B. (1981). Inversive Geometry. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds) The Geometric Vein. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5648-9_27

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-5648-9_27

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-5650-2

  • Online ISBN: 978-1-4612-5648-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics