On the number of facets of three-dimensional Dirichlet stereohedra IV: quarter cubic groups

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by F. Santos and D. Bochiş, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 “full” cubic groups cannot have more than 25 facets. Here we study the remaining “quarter” cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.

[1]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[2]  Andras Bezdek.  Arbitrarily large neighborly families of congruent symmetric convex 3- polytopes , 2003 .

[3]  P. Engel,et al.  Über Wirkungsbereichsteilungen von kubischer Symmetrie , 1981 .

[4]  Daniel H. Huson,et al.  On Three-Dimensional Space Groups , 1999 .

[5]  Branko Grünbaum,et al.  Tilings with congruent tiles , 1980 .

[6]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[7]  ELKE KOCH,et al.  Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden , 1973 .

[8]  Francisco Santos,et al.  On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups , 2008, Discret. Comput. Geom..

[9]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[10]  Daciana Bochis.  Estereoedros de Dirichlet en 2 y 3 dimensiones , 1999 .

[11]  E. Koch.  A geometrical classification of cubic point configurations , 1984 .

[12]  W. Fischer.  Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden , 1973 .

[13]  Gesammelte Abhandlungen , 1906, Nature.

[14]  Francisco Santos,et al.  On the number of facets of three-dimensional Dirichlet stereohedra I: Groups with reflections , 2001, Discret. Comput. Geom..

[15]  Jeff Erickson,et al.  Local polyhedra and geometric graphs , 2005, Comput. Geom..

[16]  Jeff Erickson,et al.  Nice Point Sets Can Have Nasty Delaunay Triangulations , 2001, SCG '01.