After looking at the intersections of symmetrically placed cylinders and obtaining curved polyhedra, it is tempting to straighten these intersections by looking at intersections of columns instead.
The simplest case is that of three perpendicular columns. The intersection is a cube. Fair enough. But what happens if we rotate all columns by 45 degrees about their axes?
Before we look, let’s make it more interesting. In both cases, we can shift the columns so that their cross sections tile a plane with squares. Surely, every point of space will then be in the intersection of a triplet of perpendicular columns. In other words, the intersection shapes will tile space.
Yes, right, we knew that in the first case. I find the second case infinitely harder to visualize. Fortunately, I have seen enough symmetrical shape to guess what the intersection of the three twisted columns looks like it is a rhombic dodecahedron.
But not all triplets of columns that meet do this in such a simple way, there is a second possibility, in which case the intersection is just a twelfth, namely a pyramid over the face of the rhombic dodecahedron.
Together with the center rhombic dodecahedron they form a stellation of the rhombic dodecahedron, or the Escher Solid, of which you have made a paper model using my slidables.
Above you can see a first few of Escher’s solids busy tiling space.