Yes, that helps a lot. Why isn’t this better known? Has anyone built models? (Did Gardner ever write about it? Aside from the cubes case?) Jim On Wed, May 13, 2020 at 5:14 PM Scott Kim <scott@scottkim.com> wrote:
Jim — the key idea is that if you balance a dodecahedron on a vertex, then a horizontal plane halfway between the north and south pole intersects the dodecahedron in a regular hexagon. Hexagons of course tile the plane, and if you grow a dodecahedron out of every hexagon by translating the dodecahedron, the adjacent dodecahedra abut at faces that tilt in or out of vertical at the same angle, so they fit smoothly. This is also true of a cube balancing on a vertex, an octahedron balanced on a face, or an icosahedron balanced on a face — all their equators are regular hexagons, and adjacent faces fit together smoothly (because opposite faces are parallel). The analogous construction for a regular tetrahedron is to balance the tetrahedron on an edge — the equator is then a square, which tiles the plane. to form a sheet of tetrahedra, you have to grow tetrahedra out of the tiling squares in alternating directions, twisting by 90°, to get them to abut properly. In all cases the polyhedra form an "airtight" sheet, as Gosper says. Hope that helps. — Scott
On Wed, May 13, 2020 at 11:20 AM Bill Gosper <billgosper@gmail.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun