On Thu, Sep 18, 2008 at 2:38 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
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?
Body-diagonal tours are only impossible if you use dyadic subdivision of the cube. For example, if you break up the square into 9 subsquares, you can do a tour like \/\ /\/ \/\ where the subsquares are visited in the order 123 654 789 This can be lifted to a 3d pattern where the 27 subcubes get visited in the order 1 2 3 | 12 11 10 | 25 26 27 6 5 4 | 13 14 15 | 24 23 22 7 8 9 | 16 17 18 | 19 20 21 -Thomas C
WFL
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