When I read your starting article
here I can get the followings
- 2D convex tilings with regular polygons only can use 3-, 4-, 6-, 8-, and 12-gons,
as any other being used regular convex polygon ultimatly results in a non-continuable configuration - morover there is only a single 2D convex tiling with regular polygons only using 8-gons: the uniform 4.8.8-tiling
- occurances of 6-gons variously might be decomposed into 6 3-gons each (6-pyramid)
- occurances of 12-gons variously might be decomposed into 6 3-gons, 6 4-gons, and 1 6-gon each (6-cupola),
in fact within 2 different orientations.
- 3D convex tilings with regular polygons should similarily be restricted to use 3-, 4-, 5-, 6-, 8-, 10-, and 12-gons only,
as any other being used regular convex polygon ultimatly results in a non-continuable configuration - morover usages of 12-gons only can occur within infinite stacks of according 2D tilings.
- sevaral CRF cells can also be excluded for similar reasons:
4-ap, n-ap with n>5, 7-p, 9-p, 10-p, 11-p, n-p with n>12, J52-53, J84-90, J92, snic, snid, grid
About further restrictions on the usage of 10-gons - at least therein - you still seem unfixed.
Within a later post of that very thread you show up a configuration with a tid, but I don't get exactly why it isn't continuable.
Still, that one alone doesn't rule out any other 10-gon usage.
A bit later in that thread you further observe the (non-new) facts of degenerate segmentochora
- tic || cube
- toe || oct
- co || point
- girco || sirco
as well as the obvious decomposition of sirco into the lace tower squacu || op || squacu.
From that you now deduce that
ANY 3D convex tiling with regular polygons should be decomposable into P3 (trip), P4 (cube), P8 (op), T (tet), Y4 (squippy), and Q4 (squacu)
- except for the single case of Weimholt's cube-doe-bilbiro honeycomb.
First of all I don't see the final 10-gon exclusion. But esp. I don't see why that mentioned exception should be the only possible 5-gon usage.
--- rk
