Cuboctahedron
Theorem A cuboctahedron may be constructed with a minimum of one tile type.
Proof. The cuboctahedron has twelve vertices, twenty-four edges, and the faces are eight equilateral triangles and six squares. It requires twelve tiles with four arms each. We now present a construction that requires only one tile type.
Notice that it is impossible to form any other complete structures from these tiles.
Figure 1: The tile used in the
construction of the cuboctahedron.
Tile Configuration: α1 α2 β1 β2