Shaping, continued
Dicing and reshuffling preserve the dimensionality of the grid (N-shapings). However, not every N-shaping can be expressed as a superposition of dicings and reshufflings.
A combination of dicing and reshuffling would produce
1 3 3 4 2 1 3 1 2 4 4 2 4 3
2 4 1 2 4 3 4 2 1 3 3 1 2 1
which is only 8 of the 4!=24 possible combinations.
A more general N-shaping would involve a reordering transformation within DI for some multiindex I.