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By embedding a finite repeat pattern (cluster) from the infinite hexagonal lattice on a torus, we define in fact a mapping of all the clusters forming the lattice into a generic cluster. In other words, the cell layout is wrap-around to form a toroidal surface. In order to be able to perform this mapping, the number of cells in a cluster has to be a rhombic number , defined by two “shifting” parameter i and j as
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A toroidal surface is chosen because it can be easily formed from a rhombus by joining the opposing edges. In SEAMCAT 
, with i=3 and j=2 is used. To illustrate the cyclic nature of the wrap-around cell structure, the cluster of 19 cells is repeated 8 times at rhombus lattice vertices as shown in Figure 188. Note that the original cell cluster remains in the center while the 8 clusters evenly surround this center set. From the figure, it is clear that by first cutting along the blue lines to obtain a rhombus and then joining the opposing edges of the rhombus a toroid can be formed. Furthermore, since the toroid is a continuous surface, there are an infinite number of rhombus lattice vertices but only a few selected have been shown to illustrate the cyclic nature.
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- Distance between (x,y) and (a,b);
- Distance between (x,y) and
Mathinline body (a+3.5*D,b+1.5*\sqrt{3}*D); - Distance between (x,y) and
Mathinline body (a-0.5*D,b+2.5*\sqrt{3}*D); - Distance between (x,y) and
Mathinline body (a-4*D,b+\sqrt{3}*D); - Distance between (x,y) and
Mathinline body (a-3.5*D,b-1.5*\sqrt{3}*D); - Distance between (x,y) and
Mathinline body (a+0.5D,b-2.5*\sqrt{3}*D); - Distance between (x,y) and
,Mathinline body (a+4*D,b-\sqrt{3}*D)
where D is the inter-site distance.
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Figure 185: Wrap-around with ’9’ clusters of 19 cells showing the toroidal nature of the wrap-around surface Anchor F0185 F0185
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