Skip to main content
Log in

Nonrational configurations, polytopes, and surfaces

  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

Abstract

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.

Contributions are most welcome.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. M. Aigner and G. M. Ziegler,Proofs from THE BOOK, 3rd ed., Springer-Verlag, Heidelberg, 2004.

    Book  Google Scholar 

  2. L. J. Billera and B. S. Munson,Polarity and inner products in oriented matroids, European J. Combinatorics, 5 (1984), 293–308.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Bokowski, G. Ewald, and P. Klein-schmidt,On combinatorial and affine automorphisms of polytopes, Israel J. Math. 47 (1984) 123–130.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Bokowski and A. Guedes de Oliveira, Simplicial convex A-polytopes do not have the isotopy property, Portugaliae Math. 47 (1990) 309–318.

    MATH  MathSciNet  Google Scholar 

  5. U. Brehm,A universality theorem for realization spaces of maps. Abstracts for the Conferences 20/1997 “Discrete Geometry” and 39/1998 “Geometry,” Mathema-tisches Forschungsinstitut Oberwolfach.

  6. U. Brehm,A universality theorem for realization spaces of polyhedral maps, in preparation, 2007.

  7. H. S. M. Coxeter,Regular Polytopes, Macmillan, New York 1963; second, corrected reprint, Dover, New York, 1973.

    MATH  Google Scholar 

  8. G. Gévay,Kepler hypersolids, in “Intuitive geometry,“ Szeged, 1991, Colloq. Math. Soc. János Bolyai 63, North-Holland, Amsterdam, 1994, pp. 119–129.

  9. P. Gritzmann,Polyhedrische Realisierungen geschlossener 2-dimensionaler Mannigfaltigkeiten im R3, PhD thesis, Univ. Siegen, 1980, 86 pages.

  10. B. Grünbaum,Convex Polytopes, Interscience, London, 1967. Second edition prepared by V. Kaibel, V. Klee, and G. M. Ziegler, Graduate Texts in Mathematics 221, Springer, New York, 2003.

    MATH  Google Scholar 

  11. H. Günzel,On the universal partition theorem for A-polytopes, Discrete Comput. Geometry, 19 (1998) 521–552.

    Article  MATH  Google Scholar 

  12. B. Jaggi, P. Mani-Levitska, B. Sturmfels, and N. White,Uniform oriented matroids without the isotopy property, Discrete Comput. Geometry 4 (1989) 97–100.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Jordan and M. Steiner,Configuration spaces of mechanical linkages, Discrete Comput. Geometry, 22 (1999) 297–315.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Kapovich and J. J. Millson,Universality theorems for configuration spaces of planar linkages, Topology 41 (2002) 1051–1107.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. P. S. Kung, ed.,A Source Book in Matroid Theory, Birkhäuser, 1986.

  16. S. MacLane,Some interpretations of abstract linear algebra in terms of projective geometry, American J. Math. 58 (1936) 236–240.

    Article  MathSciNet  Google Scholar 

  17. N. E. Mnëv,The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in Topology and Geometry—Rohlin Seminar, O. Y. Viro, ed., Lecture Notes in Mathematics 1346, Springer, Berlin 1988, pp. 527–544.

    Chapter  Google Scholar 

  18. N.E. Mnëv,The universality theorem on the oriented matroid stratification of the space of real matrices, in “Discrete and Computational Geometry,” J. E. Goodman, R. Pollack, and W. Steiger, eds., DI-MACS Series in Discrete Math. Theor. Computer Science, vol. 6, Amer. Math. Soc, 1991, pp. 237–243.

  19. S. Onn and B. Sturmfels,A quantitative Steinitz’ theorem, Beiträge zur Algebra und Geometrie 35 (1994) 125–129.

    MATH  MathSciNet  Google Scholar 

  20. A. Paffenholz and G. M. Ziegler,The Etconstruction for lattices, spheres and polytopes, Discrete Comput. Geometry 32 (2004) 601–624.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. A. Perles,At most 2 d+1 neighborly simplices in Ed, Annals of Discrete Math. 20 (1984) 253–254.

    MathSciNet  Google Scholar 

  22. A. Ribo Mor,Realization and Counting Problems for Planar Structures: Trees and Linkages, Polytopes and Polyominoes, PhD thesis, FU Berlin, 2005, 23+167 pages.

  23. J. Richter-Gebert,Realization Spaces of Polytopes, Lecture Notes in Mathematics 1643, Springer, 1996.

  24. S. A. Robertson,Polytopes and Symmetry, London Math. Soc. Lecture Note Series 90, Cambridge University Press, Cambridge, 1984.

    MATH  Google Scholar 

  25. T. Rörig,Personal communication, August 2007.

  26. A. Schwartz and G. M. Ziegler,Construction techniques for cubical complexes, odd cubical A-polytopes, and prescribed dual manifolds, Experimental Math. 13 (2004) 385–413.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. W. Shor,Stretchability of pseudolines is NP-hard, in Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift, P. Gritzmann and B. Sturmfels, eds., DIMACS Series Discrete Math. Theor. Computer Science 4, Amer. Math. Soc, 1991, pp. 531–554.

  28. J. Simutis,Geometric realizations of toroidal maps, PhD thesis, UC Davis, 1977, 85 pages.

  29. E. Steinitz,Polyeder und Raumeinteilungen, in Encyklopädie der mathematischen Wissenschaften, mit Einschluss ihrer Anwendungen, Drifter Band: Geometrie, III.1.2., Heft 9, Kapitel III A B 12, W. F. Meyer and H. Mohrmann, eds., B. G. Teubner, Leipzig, 1922, pp. 1–139.

    Google Scholar 

  30. E. Steinitz and H. Rademacher,Vorlesungen über die Theorie der Polyeder, Springer-Verlag, Berlin, 1934. Reprint, Springer-Verlag, 1976.

    Google Scholar 

  31. G. K. C. von Staudt,Beitrage zur Geometrie der Lage, Verlag von Bauer und Raspe, Nürnberg, 1856–1860.

    Google Scholar 

  32. G. M. Ziegler,Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, New York, 1995. Revised edition, 1998; seventh updated printing, 2007.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Günter M. Ziegler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ziegler, G.M. Nonrational configurations, polytopes, and surfaces. The Mathematical Intelligencer 30, 36–42 (2008). https://doi.org/10.1007/BF02985377

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02985377

Keywords

Navigation