Tetrahedron
A tetrahedron may be constructed with a minimum of two tile types.
The tetrahedron is made up of four vertices, six edges, and four equilateral triangles for faces. It requires four tiles with three arms each. The tetrahedron requires at least two tile types for its construction because the vertices are of degree three. This follows from the fact that a complete contstruct must have an equal number of hatted and unhatted cohesive ends.
We demonstrate a construction that requires precisely two tile types, and does not permit the residual construction of smaller complete constructs.
Since the tetrahedron is a regular polytope, all vertices will have the same geometric configuration of edges about the vertex.
Tile Type One: consists of three cohesive ends of type b̂ (see Figure 1).
Tile Type Two: consists of one cohesive end of type a, one of type â, and one of type b (see Figure 1).
Figure 1: The two different tile types used in the construction of the tetrahedron. They have the same geometricconfiguration but they have different cohesive ends.