Coloring Objects Built From Bricks

We address a question posed by T. Sibley and Stan Wagon. They proved that rhombic Penrose tilings in the plane can be 3-colored, but a key lemma of their proof fails in the natural 3D generalization. In that generalization, an object is built from bricks, each of which is a parellelopiped, and they are glued face-toface. The question is: How many colors are needed to color the bricks of any such object, with no two face-adjacent bricks receiving the same color? We have settled a number of questions on the general problem. For arbitrary parellelopiped bricks, we have proven that zonohedral balls are 4-colorable, and 4 colors are sometimes necessary. For orthogonal bricks, we have several results. First, any object built from such bricks is 4-colorable. Moreover, any genus-zero object (a ball) is 2-colorable. Our most complex result is that any object with no ”dividing” holes (ones that a plane parallel to one of the coordinate planes is divided into two disconnected pieces by the hole), regardless of its genus, is 2-colorable. We have examples, however, that require 3 colors. We prove that all genus-one objects are 3-colorable, as well as object of higher genus subject to certain restrictions, and we conjecture that any object built from orthogonal bricks is 3-colorable.