What is a shortlist of first few simplest (say 5~10 simplest) possible shapes of polyhedra/polytopes (with a minimum number of edges shared) to pack the 4-dimensional flat space (say $\mathbb{R}^4$) fully?
By the simplest, I require it to be "with a minimum number of vertex/edge/face/volume shared."
As far as I know,
- The 4-dimensional Tesseract (sort of 4-dimensional "cubic") works:
which requires
0: each 0-vertex V is shared by 16 neighbored cubics.
1: each 1-edge E is shared by 8 neighbored cubics.
2: each 2-face F is shared by 4 neighbored cubics.
3: each 3-volume U is shared by 2 neighbored cubics.
(to make a comparison, the 3-dimensional cubic works to do packing/tessellating in 3D. Where
0: each 0-vertex V is shared by 8 neighbored cubics.
1: each 1-edge E is shared by 4 neighbored cubics.
2: each 2-face F is shared by 2 neighbored cubics.
we can possibly cut out part of 4-dimensional Tesseracts to do fully packing/tessellating in 4D.
- what else polytopes do you know to do packing/tessellating in 4D fully?
The regular tetracombs are
but lots of further tetracombs are known.
--- rk