Octahedron
Theorem An octahedron may be constructed with a minimum of one tile type.Proof: The octahedron is made up of six vertices, twelve edges, and the faces are eight equilateral triangles. It requires six tiles with four arms each. We now present a construction that requires only one tile type.
The Tile Type for the Octahedron: has the configuration α1 α4 β1 β2, two cohesive ends of type â, and two cohesive ends of type a such that the two α-ends have the same cohesive end type and the β-ends have the same cohesive end type. Furthermore, this tile is symmetric with respect to arm orientation so that the two a-arms are equivalent in every way. The same is true for the â-arms. Because there is only one tile type, every tile must bond to another copy of itself. For any arm, there are two different arms that it can bond to on another tile. However, each bond will orient the two involved tiles in exactly the same way because the tile is symmetric. This means that any six tiles of this type will form an octahedron (see Figure 2). Notice that attachment angles make it impossible to form smaller or larger 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 2). We therefore have proven that the minimum number of tiles required to construct the octahedron is one.
Figure 1: The construction of the octahedron. We start
with one four-armed tile consisting of a and â cohesive
ends. Then another tile of the same configuration
is bonded to the first tile. This process of adding tiles
continues until the structure is complete, that is all
cohesive ends have a complementary arm attached.
The octahedron takes eight four-armed tiles to
be a closed and complete structure.
