It would be really nice if we could see a reason (and hence generic proof) why this works. Even more so if you could say why (or why not) the n-cube's (n-1) net

It would be really nice if we could see a reason (and hence generic proof) why this works. Even more so if you could say why (or why not) the n-cube's (n-1) nets can tile R^(n-1). I'd like to set a notification for when some hero of future-geometry solves this in ten years.