Or I'm just dreaming? ;-)
I think that sound/vibration travels much faster in stiffer objects, I suspect that in Arduino time the pulse would arrive at all four corners at the same time - if you were using interrupts to catch it, the internal interrupt priority of the IC and the time it would take to service each one would certainly have a bigger effect than the time the pulse takes to reach all four corners.
I'm thinking on measuring the analog output of each piezo, and compare. The problem is the math, as we don't have constants, since you could do a soft or hard hit...
x position = W * R / (R + L)y position = H * B / (T + B)
Does it have to be transparent?