Posted by Steve Simon on February 18, 2013, 10:14 pm, in reply to "FCC-BCC-equivalency"
It might help to build a model here. I think you are suggesting that the four points in the middle of the north, south, east, and west faces form a square (turned 45 degrees). But this does not become a cube when you consider the next unit cell above, because the height up to the next layer (which is the lattice constant a) is not the same as the edge-length of the square you just described (which is a/sqrt(2)).
Does this help? 
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