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. Our results are twofold. First, we refine the 2D proof to render information on the number and location of the 2D elbows. Second, we prove that not all of the 2D refinements extend to 3D.