It may appear at first that the only surfaces in R³ with the GTPP are the grid-cubes themselves, just as in two dimensions the grid-squares are the only closed curves with the GTPP. However it is not too difficult to find an exception: consider a solid 3x3x1 grid-rectangular-prism, with the middle 1x1x1 cube removed. The boundary of this set is a torus with the GTPP (in fact, the solid figure itself also has the GTPP). A demo of this figure can be found here (Surfaces in 3 space). This demo also shows a 3x3x3 cube with three overlapping 1x1x3 rectangular prisms removed. This removal of rectangular prisms can be repeated recursively with each of the 20 1x1x1 cubes remaining, still leaving a surface with the GTPP.