On Sat, May 9, 2020 at 11:46 AM Bill Gosper <billgosper@gmail.com> wrote:
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 <http://gosper.org/Endo-dodecahedron_honeycomb_1.png> (absent the endos)? I'd like to see that plane sliding along the major diagonal, like the Menger Sponge movie <https://www.youtube.com/watch?v=fWsmq9E4YC0>. —rwg
If you have Mathematica, the Manipulate in gosper.org/rhombosis.nb seems to uphold the conjecture. But it is ugly: Bill Gosper <billgosper@gmail.com> [image: Attachments]8:39 AM (2 hours ago) to Wolfram, In the attached Manipulate, the dodecahedra have been shrunk 3% and have visible air gaps, if you tumble them carefully. So how do I exorcise those groups of one, two, or three rhombi painted on the insides of the cutaway dodecs?? This was made by ClipPlanes of GeometricTransformations of PolyhedronData@"Dodecahedron", which I can provide if you're curious. —Bill