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# There are only two fundamental tilings in N=4.

We have already obtained the two ideal tiling (6), (7) of order N=4 without explanations in previous sections. In order to confirm whether or not the two are all of the possibly independent tiling, we need to investigate the characteristic of pair-symmetrical tiling. Obviously, we can get the four different tiling patterns from the ideal tiling (1) by the rotations R which are , , and (see appendix.) We will describe them shortly by . We adopt transposition T and rotation R as a symmetrical transformation operators in (14). Since , and , it is derived that and . Here we should note the transposition T does not satisfy the association low with rotational operators. E.g., , but . The operators satisfy the following operational table.

According to the table, we can categorize the pairs of tiling patterns into { , , , , , , , } and { , , , }, since

and

Simply, the above categories can be rewritten by

and

These patterns and are not expressed by any symmetrical transformation of each other. The symmetrical transformations of these patterns cover all possible 12 pairs of pair-symmetrical tiling patterns. Therefore, the independent tiling patterns in N=4 are given by

and

based on conjecture 1 (in the case N=4, it is satisfiable theorem). These tiling patterns correspond to , in (11) and (12). Since , are not given by any (N-let-) symmetrical transformation of each other, we call such independent tiling patterns "fundamental tilings".

Next: Relation between fundamental tilings Up: 2D number-tiling Previous: Hierarchical structures of 2D

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