From: Bill Gosper <billgosper@gmail.com> Date:Â 5/9/20, 2:46 PM
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>. Isn't this just a plane section perpendicular to (1,1,1) through the 3D endo-dodec checkerboard...
I "see" the cross-section that's a regular hexagonal tiling, I see it as cutting through the "equators" in dodex.gif. I'm pretty convinced the dodecahedrons don't intersect. I see an analogy to martinsmarbles. My brain can't grasp the endo-dodec picture well enough to answer the question. On both the dodex and martinsmarbles, it seems to me that the airtightness is just a 2D slice. There are leaks "of measure zero." Both arrangements have corner points with air on both sides. On dodex it's clearer to me anyway: some of the cutaways show pyramidal air-pockets with their tips touching the "equator". I'm pretty sure the air-pyramids come in pairs with tips (almost?) touching. Is that why you put scare-quotes on "airtight"? On May 9, 2020, at 4:00 PM, James Propp <jamespropp@gmail.com> wrote:
I don't understand this [endo-dodec] question. The airtight sheet is a 3-dimensional structure; a plane section of a 3-d honeycomb is a 2-dimensional structure.
From: Veit Elser <ve10@cornell.edu>
Jim, look at a the cut-away dodec. on the left of <http://gosper.org/dodex.gif>. ThereÂ’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 dodecÂ’s donÂ’t interpenetrate, but it seems to me that part of the construction also holds water.
Heh. --Steve