Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Equidecomposability of polyhedra with reference to crystallographic groups

  • 81 Accesses

  • 3 Citations

Abstract

This paper deals with the introduction of theoretical statements which are the results of studying equidecomposability of polyhedra with reference to discrete transformation groups. Lattice polygons and paving polyhedra play the most important role. The equivalence of volume equality and equidecomposability of any two of this polyhedra is shown for special discrete groups.

References

  1. 1.

    U. Betke and M. Kneser, Zerlegungen und Bewertung von Gitteropolytopen,J. Reine und Angew. Math. 358 (1985), 202–208.

  2. 2.

    L. Danzer, B. Grünbaum, and G. C. Shepard, Does every type of polyhedron tile three-space?Structural Topology 8 (1983), 3–14.

  3. 3.

    H. E. Debrunner, Pflasterung des euklidischen Raumes mit kongruenten Simplexen, Bern, 1978, Manuscript.

  4. 4.

    H. E. Debrunner, Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln,Arch. Math. (Basel)35 (1980), 583–587.

  5. 5.

    B. Grünbaum and G. C. Shepard, Tiling with congruent tiles,Bull. Amer. Math. Soc. 3 (1980), 951–973.

  6. 6.

    H. Hadwiger, Über Gitter und Polyeder,Monatsh. Math. 57 (1953), 246–254.

  7. 7.

    H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1957.

  8. 8.

    C. Müller, Zerlegungsgleichheit von Punktmengen bezüglich diskreter Transformationsgruppen, Dissertation, Friedrich-Schiller-Universität, Jena, 1986.

Download references

Author information

Additional information

Communicated by J. Wills

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Müller, C. Equidecomposability of polyhedra with reference to crystallographic groups. Discrete Comput Geom 3, 383–389 (1988). https://doi.org/10.1007/BF02187920

Download citation

Keywords

  • Interior Point
  • Discrete Comput Geom
  • Fundamental Domain
  • Equal Area
  • German Democratic Republic