Truncated Tetrahedron

Theorem A truncated tetrahedron may be constructed with a minimum of two tile types.

Proof. The truncated tetrahedron is an Archimedean solid composed of twelve vertices, eighteen edges, and four equilateral triangles and four regular hexagons for faces. It requires twelve tiles with three arms each. We now present a construction that requires precisely two tile types.

Tile Type One: consists of one cohesive end of type b, one of type a, and one of type â such that the a and â cohesive ends are π/3 apart (see Figure 1).

Tile Type Two: consists of one cohesive end of type b̂, one of type a, and one of type â such that the a and â cohesive ends are π/3 apart (see Figure 1).

Remark. Notice that the tiles use more than one cohesive end type. We do this because it restricts the bonding such that there will ultimately be fewer ways for waste structures to form. The truncated tetrahedron naturally has two different types of edges. Edges of the first type form the triangle faces, while edges of the second type do not border a triangular face. So, it makes sense to use two separate cohesion end types because arms that make up the two sorts of edges never intermix. Alternatively, one may look at the tiles themselves. The a and â arms are π/3 radians apart and the third arm of each tile is (2π)/3 radians away from the other two half-edges. The truncated tetrahedron is constructed such that the arms that are (2π)/3 away from the other two half-edges, that is, the b and b̂ arms of any tile, always bond to one another. If the cohesive end types were all a and â, then undesirable bonds could form that would lead to the incidental creation of unwanted complexes.

Six tiles of type one and six tiles of type two assemble as shown in Figure 1. There are twelve similar bonds composed of a and â arms, and these bonds are equivalent because they orient their members in the same way. Furthermore, there are six similar bonds composed of b and b̂ arms which are equivalent because they orient their member tiles in an identical way (see Figure 1). Arm orientation does not present a problem because there are only two types of bonds and they use different cohesive end types.

Notice that it is impossible to form larger or smaller complete structures from these tiles.

We have shown a construction which demonstrates that the absolute minimum we began with is achievable under our present constraints (see Figure 1). We therefore have proven that the minimum number of tiles required to construct the truncated tetrahedron is two.