On 9/18/08, Thomas Colthurst <thomaswc@gmail.com> wrote:
Inspired by the recent thread on Hilbert curves, I decidedto think about cube filling curves that used a face diagonal as their basic move.
For those of you who like to play along at home, here's an elementary formulation of the problem: Given 8 identical cubes, each marked with a line from (0,0,0) to (1,1,0), can you put them together into the [0,2]^3 cube in such a way that the marked lines form a path from (0,0,0) to (2,2,0) ? ... I would be interested if anyone could come up with something more pretty, either in terms of symmetry or in having fewer junction points where the path touches itself.
I haven't made an exhaustive search, but it does look to me as if such a path always revisits some vertex. This doesn't seem to matter very much --- it's cubes that you are visiting here, rather than vertices [just as well, since you must omit all those with odd coordinate sums!] A similar tour based on body-diagonals --- rather than face-diagonals or edges --- is easily seen to be impossible. But exactly why should this be so; and what is the corresponding situation for d-space? WFL