Anything that people do, they find ways to do differently. So for anything human-made we have, we have variations of it. Sometimes those variations are harmless and provide entertaining debate – the designated hitter rule being used in the American League but not in the National League comes to mind – and sometimes they are more obscure.
This picture is from a circa 1905 building in Manhattan with, obviously, tile-arch floors. These can be a little hard to read if you haven’t seen them before, so here are the steel floor beams marked:
This is a side-construction floor, where the hollows (the void spaces) in the terra cotta blocks run parallel to the beams (the light blue lines), so that the sides of the blocks bear against one another. This is the older form of tile-arch floor, and had been partially superseded by the time this building was built by end-construction arches, where the voids run perpendicular to the beams.
The peculiar variation here is that the blocks are staggered sideways. The underside above has a pattern that resembles running bond in brick, with each row of blocks offset from the neighboring rows by half a block length. This raises the question of why? I can’t see a benefit: the blocks aren’t locked together by anything other than the compression inherent in their flat-arch form, so there’s not a great deal of lateral load transfer. More importantly, lateral load transfer would serve little purpose, since ordinary tile-arch floors where the blocks are in line with one another in the spanning direction perform just fine, even when there are small holes cut in them.
It’s possible that someone believed this variation made for a better floor. It’s also possible that this minor variation was meant to be enough to get a patent or to establish a business based on “new and improved” and was ultimately nothing more than a structurally-meaningless business stratagem.