Well, perhaps this is reading too much in to the language, but you actually draw the circle segments without them crossing, but the gluing to reconstruct the space doesn't permit the segments to be glued back in to the circles they were decomposed from.

So you can draw a decomposition of the shapes in the decomposed space which apparently satisfies the properties, but you run in to trouble when you try to glue those partial solutions back together -- in this case, you can't simultaneously reconstruct the space and the shapes.