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Enumerating domino patterns with primitive unit cell area up to area A results in sets of domino patterns for which we can take the convex hull, like those shown in Figure 5.11. For different values ofA, the convex hull is different. But, atA= 8 and greater, the shape of the convex hull stops changing. The boundary of this convex hull constitutes a boundary that cannot be exceeded by any domino pattern (when properly comparing~n-vectors in the way described previously). Consequently, the set of domino patterns with primitive unit cell area up to a value greater than 8 has a convex hull in~n-space of exactly the same shape. This convex hull shape can be seen in Figure 5.11(d), but is shown again in Figure 5.12 with vertex patterns indicated. We will refer to its shape as the limit hull.

The boundary of the limit hull has 7 faces. Each of these faces represents an affine combination of interaction counts that cannot be exceeded. To show this is true, we must make use of all 6 interaction counts (naa, nab, nac, nbb, nbc, ncc). Of

the 7 faces, 3 lie in the planes naa = 0, nbb = 0, and ncc = 0. These 3 faces are

maximal and cannot be exceeded because none of the interaction counts can be negative. Another face lies in the planenac = 2nab. A domino configuration outside

this plane would requirenac>2nab. However, this is not possible because eacha-c

interaction necessitates a nearbya-binteraction. Eacha-binteraction can be placed between twoa-cinteractions so thatnac can be equal to 2nab at most. This is shown

in Figure 5.13. The face of the convex hull lying in the planenab= 2nac is maximal

for the same reason, due to symmetry.

Finally, we have the face lying on the plane 2nbb=N M−nac (whereN×M

is the unit cell the configuration is generated by). In this case, we can see that 2nbb> N M −nac is not possible, since each a-c interaction reduces the number of

possible b-b interactions by 1. The maximum number of b-b interactions is half of N M; therefore, 2nbb=N M−nac is the maximum possible value fornbb for a given

value ofnac. The face lying on the plane 2ncc=N M−nabalso cannot be exceeded,

due to symmetry. The preceding arguments associate all 7 faces of the limit hull with planes that cannot have domino configurations outside of them.

The 6 vertex patterns in Figure 5.12 are unique to their interaction count vectors, even when considering all possible domino configurations. This means that if a set of domino configurations includes one of these special configurations, there must always exist some interaction parameter vector ~ε such that the special con- figuration has the lowest energy. Each of the 6 vertex configurations lies at an intersection of the previously mentioned faces of the boundary of the limit hull. Using the corresponding equations for the bounding planes, it is possible to check that the configurations are unique to their interaction count vector by attempting to build a domino configuration by hand while also keeping to the constraints of the bounding planes. For example, the p4g configuration shown in the upper-left of Figure 5.12 is at the intersection of planes naa = 0,nbb = 0, andncc = 0. Starting

with a single domino and keeping to these constraints, there is only one possible planar domino pattern. A similar process can be done to show each of the 6 vertex configurations are unique to their interaction count vectors.

For a single unit cell to generate a set of configurations with a convex hull that matches the limit hull, it is necessary and sufficient for that unit cell to generate the 6 configurations shown in Figure 5.12. The 6 vertex configurations can be made by the 6×3 unit cell and 4×4 unit cell. Therefore, any unit cell that is a multiple

a

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Figure 5.13: Starting with ana-cinteraction, a corner is created that can be covered in one of two ways. In either case, an a-b interaction is introduced. From there, the a-b interaction can be shared with at most 1 nearby a-c interaction. Because of this effective constraint,nac= 2nab is the maximum possible value for nac, given

of the 12×12 unit cell will generate configurations that have the limit hull as the shape of the convex hull.

The concept of a limit hull is useful because it defines the convex hull over the set of all domino patterns. So, considering the set of all planar domino patterns, there are 6 that can be assigned the unique lowest energy. If the periodicity of the domino patterns is constrained, there is a difference between the shape of the convex hull and the limit hull due to the inability to create the 6 vertex patterns of the limit hull. Thinking more generally, it seems possible that similar limit hulls exist for the other polyomino shapes; however, for polyominoes with a greater number of interaction types, it becomes more difficult to identify the convex hull limiting shape as we have done for the domino patterns.