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

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

  • 282 Accesses

  • 1 Citations

Abstract

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. In two previous papers, D. Bochiş and the second author studied the problem for noncubic groups. This paper deals with “full” cubic groups, while “quarter” cubic groups are left for a subsequent paper. Here, “full” and “quarter” refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston.

This paper’s main result is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.

References

  1. 1.

    Bochiş, D.: Estereoedros de Dirichlet en 2 y 3 dimensiones. Ph.D. Thesis, Universidad de Cantabria (1999)

  2. 2.

    Bochiş, D., Santos, F.: On the number of facets of 3-dimensional Dirichlet stereohedra I: groups with reflections. Discrete Comput. Geom. 25(3), 419–444 (2001)

  3. 3.

    Bochiş, D., Santos, F.: On the number of facets of 3-dimensional Dirichlet stereohedra II: non-cubic groups. Beitrge Algebra Geom. 47(1), 89–120 (2006)

  4. 4.

    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)

  5. 5.

    Conway, J.H., Delgado Friedrichs, O., Huson, D.H., Thurston, W.P.: On three-dimensional space groups. Contributions Algebra Geom. 42(2), 475–507 (2001)

  6. 6.

    Delone, B.N.: A proof of the fundamental theorem of the theory of stereohedra. Dokl. Akad. Nauk SSSR 138, 1270–1272 (1961). (In Russian)

  7. 7.

    Dress, A.W., Huson, D.H., Molnár, E.: The classification of face-transitive periodic three-dimensional tilings. Acta Crystallogr. Sect. A 49(6), 806–817 (1993)

  8. 8.

    Engel, P.: Über Wirkungsbereichsteilungen von kubischer Symmetrie. Z. Kristallogr. 154(3–4), 199–215 (1981)

  9. 9.

    Engel, P.: Über Wirkungsbereichsteilungen von kubischer Symmetrie II. Die Typen von Wirkungsbereichspolyedern in den symmorphen kubischen Raumgruppen. Z. Kristallogr. 157(3–4), 259–275 (1981)

  10. 10.

    Erickson, J.: Nice point sets can have nasty delaunay triangulations. Discrete Comput. Geom. 30(1), 109–132 (2003)

  11. 11.

    Erickson, J., Kim, S.: Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes. In: Bezdek, A. (ed.) Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday, pp. 267–278. Marcel-Dekker, New York (2003)

  12. 12.

    Fischer, W.: Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 138, 129–146 (1973)

  13. 13.

    Fischer, W.: Normal homogeneous partitions of three-dimensional Euclidean space which are not partitions into fundamental regions of a space group. Commun. Math. Chem. 9, 101 (1980)

  14. 14.

    Grünbaum, B., Shephard, G.C.: Tilings with congruent tiles. Bull. Am. Math. Soc. 3, 951–973 (1980)

  15. 15.

    Koch, E.: Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 138, 196–215 (1973)

  16. 16.

    Koch, E.: A geometrical classification of cubic point configurations. Z. Kristallogr. 166, 23–52 (1984)

  17. 17.

    Lockwood, E.H., Macmillan, R.H.: Geometric Symmetry. Cambridge University Press, Cambridge (1978)

  18. 18.

    Milnor, J.: Hilbert’s 18 problem: on crystallographic groups, fundamental domains, and on sphere packing. In Proceedings of Symposium in Pure Mathematics of American Mathematical Society, pp. 491–507. Northern Illinois University, Dekalb, Illinois (1974)

  19. 19.

    Sabariego, P., Santos, F.: On the number of facets of 3-dimensional Dirichlet stereohedra IV: quarter groups. Preprint, http://arxiv.org/abs/0708.2114

  20. 20.

    Schattschneider, D., Senechal, M.: Tilings. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 43–63. CRC Press, Boca Raton (1997)

Download references

Author information

Correspondence to Francisco Santos.

Additional information

Research partially supported by the Spanish Ministry of Education and Science, grant number MTM2005-08618-C02-02.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sabariego, P., Santos, F. On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups. Discrete Comput Geom 40, 159–189 (2008). https://doi.org/10.1007/s00454-008-9063-0

Download citation

Keywords

  • Stereohedra
  • Plesiohedra
  • Dirichlet domain
  • Crystallographic group
  • Cubic group