I've written previously about heightfields for mechanical computation: http://lists.canonical.org/pipermail/kragen-tol/2010-June/000919.html. One difficulty mentioned therein is that fabrication technologies capable of producing individual one-off parts like the three-dimensional heightfields called for are rather expensive; drilling a single hole in a hard material can cost tens of cents, and thousands, if not tens of thousands, of such precise holes will be needed for a complete computing device.
Planar fabrication techniques such as acid etching, laser cutting, sawing with a jigsaw or fretsaw or coping saw or piercing saw, waterjet cutting, oxy-acetylene or plasma cutting, or cutting with a hot or abrasive wire are dramatically cheaper, but they can't cut only partway through the material, either with precision or at all. It would be very convenient to be able to achieve the two-dimensional LUT effect using only such planar fabrication techniques.
This is possible by using a tapered probe that measures the width of the hole, rather than its depth; this allows a flat plate containing holes of various widths to be used instead of the cube containing round holes of various depths I proposed before. The holes can be slot-shaped rather than round, since the taper will be cut from a flat plate. The friction and tensile forces resulting from the taper will be the limiting factor on the number of bits in the machine; this can be improved somewhat by making holes shaped like plus signs or rectangles and using two separate tapers in a plus-sign-cross-section- shaped probe, and perhaps by using stepped tapers.
Alternatively, of course, you could construct the three-dimensional heightfield by squeezing together a bunch of parallel plates, like the body of the common stamped-and-riveted laminated Master padlock, perhaps with glue between the plates to improve precision. But that requires some 33 plates for a single 16×16 heightfield rather than one.