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.
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. 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