From: "Thomas Colthurst" <thomaswc@gmail.com>

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

If \ looks like this up close: --------+ | +-------+ | +-------- (which is what your larger square is doing), and you put little gaps between the \/'s, then that's the Peano curve.

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

Do you mean 1 2 3 | 18 17 16 | 19 20 21 6 5 4 | 13 14 15 | 24 23 22 7 8 9 | 12 11 10 | 25 26 27 ? --Steve