On corners of objects built from parallelepiped bricks

We investigate a question initiated in the work of Sibley and Wagon, who proved that 3 colors suffice to color any collection of 2D parallelograms glued edge-to-edge. Their proof relied on the existence of an “elbow” parallelogram. We explore the existence of analogous “corner” parallelepipeds in 3D objects, which would lead to 4-coloring. Our results are threefold. First, we refine the 2D proof to render information on the number and location of the 2D elbows. Second, we extend the 2D results to 3D for objects satisfying two properties. Third, we exhibit a genus-0 object (a topological ball) that satisfies one but not both of our properties, and fails the 3D extension theorem, establishing that this theorem is, in a sense, tight.