Coloring Objects Built From Bricks

We address a question posed by Sibley and Wagon. They proved that rhombic Penrose tilings in the plane can be 3colored, 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 parallelopiped, and they are glued face-to-face. 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? For arbitrary parallelopiped bricks, we prove zonohedra are 4-colorable, and 4 colors are sometimes necessary, by establishing two Sibley conjectures for zonohedra. For orthogonal bricks, we narrow the chromatic number to {3, 4}, and have several results. Any genus-0 object (a “ball”) is 2-colorable; any genus-1 object is 3-colorable. For objects of higher genus, we show that if an object’s holes are “nonplanar” in a technical sense, then it is 2-colorable regardless of its genus, and for various special cases of planar holes...

• Type: Conference
• Year: 2003
• Where: CCCG
• Authors: Suzanne Gallagher, Joseph O'Rourke
• Keywords: Arbitrary Parallelopiped Bricks, CCCG 2003, CCCG 2007, Orthogonal Bricks, Rhombic Penrose Tilings