On May 9, 2020, at 4:00 PM, James Propp <jamespropp@gmail.com> wrote:
Bill writes:
Ages ago I mentioned that regular dodex can interlock in an "airtight" sheet
<http://gosper.org/dodex.gif>, analogous to a sheet of cubes ("Martin's Marbles") <http://gosper.org/martinsmarbles.png>.
I'm amazed. (And I still don't see it.) Is this explained anywhere? I'd love to see a physical model that I can walk around so I can understand this.
A MacGyver episode? He’s stuck in the desert and needs to collect condensation to stay hydrated, but all he has to work with are a bunch of D&D dice ...
Isn't this just a plane section perpendicular to (1,1,1) through the 3D endo-dodec checkerboard <http://gosper.org/Endo-dodecahedron_honeycomb_1.png> (absent the endos)?
I don't understand this question. The airtight sheet is a 3-dimensional structure; a plane section of a 3-d honeycomb is a 2-dimensional structure.
Jim, look at a the cut-away dodec. on the left of <http://gosper.org/dodex.gif>. There’s a special plane orthogonal to the 3-fold axis where the cross-sections are regular hexagons. You need to check that when extended above and below this plane the dodec’s don’t interpenetrate, but it seems to me that part of the construction also holds water. -Veit
Anyway, it'd be interesting to classify polyhedra according to whether translates of the given polyhedron can fill a plane. More generally, for any n-dimensional convex polytope K we can ask whether translates of K can fill a k-dimensional subspace. Has anyone looked at this?
Jim Propp