Engel 38-sided space-filling polyhedron Of the 230 space groups, the symmetries possible in a crystal lattice, the most…
Engel 38-sided space-filling polyhedron
Of the 230 space groups, the symmetries possible in a crystal lattice, the most complicated is group 214, I4_1 32. In the words of Steve Dutch, “This group looks chaotic, but visualizing it is easy. All you do is sit there until little beads of blood form on your forehead.” In 1980, P. Engel used this group to make a 38-sided space-filling polyhedron. In 2016, Moritz Schmitt did a complete study of lattice-based space-filling polyhedra, and the Engel-38 had more faces than anything else.
So, what does Engel-38 look like? Just use the symmetries of I4_! 32 with the following generator point and find the Voronoi cells
via https://community.wolfram.com/groups/-/m/t/2617634
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.