Cube

The cube contains 8 vertices and 12 edges. When tiling this polytope, we were able to use two different tile types (see figures below). The red arms represent bond edge type a and the blue arms represent the complementary bond edge type â. Both the dihedral angles and the angles between two adjacent tile arms equal 90^o.

Part of our research included finding a more accurate way to visually represent rigid tiles, fixed angles, and sticky arm orientation. The diagram below shows a potentially useful representation of rigid tiling for the cube using a lap joint model. This type of model shows not only the two unique tile types with the complementary sticky ends a and â, but also the orientation of the twisted arms and their sticky ends, held in place by the restrictions of rigidity.

In this model the sticky ends are represented by the dark ends of the half-arms. They are held in place respective to the vertex, facing either toward or away from the center of the complex for a and â respectively. This restriction forces the tiles to always bond in a convex manner, preventing the formation of any undesirable incomplete structures during self-assembly.