On Space Groups and Dirichlet-Voronoi Stereohedra

A Dirichlet stereohedron for a crystallographic group G is the Voronoi region of any point with trivial stabilizer in the Voronoi diagram of its orbit.

We prove that Dirichlet stereohedra for three-dimensional crystallographic groups containing reflections in three, two or one independent directions cannot have more than eight, eighteen and fifteen facets, respectively. We show examples where the three bounds are attained.

As a tool for one of the cases, we study how many regions of a Dirichlet tesselation in the plane can be intersected by each region of another Dirichlet tesselation for the same group. We find out that the number is at most seven for all planar crystallographic groups except, perhaps, those of type pgg.

/pdf/on-the-number-of-facets-of-three-dimensional-dirichlet-qnqgnm82pp.pdf

On the number of facets of three-dimensional Dirichlet stereohedra I: Groups with reflections

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. Bochis 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.

https://link.springer.com/content/pdf/10.1007%2Fs00454-008-9063-0.pdf

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

https://upcommons.upc.edu/bitstream/handle/2117/27055/primitive-endm_version_corta.pdf;sequence=1

An algorithm to compute the primitive elements of an embedding dimension three numerical semigroup

V Jornadas de Matemática Discreta y Algorítmica

Triple-loop networks with arbitrarily many minimum distance diagrams

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by F. Santos and D. Bochis, 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.

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 the second author and D. Bochis, 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.

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups

We prove that Dirichlet stereohedra for cubic crystallographic groups cannot have more than 105 facets. This improves a previous bound of 162 [3].

/pdf/on-the-number-of-facets-of-three-dimensional-dirichlet-2g54du8ydd.pdf

Pilar Sabariego

Papers

Triple-loop networks with arbitrarily many minimum distance diagrams

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

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

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups

On the number of facets of three-dimensional Dirichlet stereohedra III: Cubic group.