Four or More Layer Repeats

In closest and quasi-closest packings, the only stipulations are

two adjacent layers of hexagonal closest packed planes must be different (A, B or C);

the first and last named layers in the repeat unit must be different (because they are adjacent in the whole pattern);

different letters arranged in the same pattern represent the same lattice.

Thus, while,there are eighteen possible permutations of three letters in four-layers, as illustrated in the cascade diagrams shown here, not all of these patterns are unique. For example, in each cascade one of the three permutations is actually a two-layer (HCP) pattern. Furthermore, many of the remaining twelve patterns are equivalent. For example, pattern (ABAC) is equivalent to pattern (ACAB): (ACAB)
= (ACAB)(ACAB)
= AC)(ABAC)(AB =
(ABAC) After all coincident patterns are eliminated (using, for example, a spreadsheet string-search function), there are aparently three unique four-layer closest packing patterns: (ABAC), (ABCB), and (ACBC). However, (ABCB) and (ACBC) represent the same lattice with different choices of the unit cell. Thus, there are only two unique four-layer packings shown here.

In the same way, it can be shown that of the 30 possible five-letter permutations, four are apparently unique: (ABABC), (ABACB), (ABCAC) and (ABCBC). However, by rechoosing axes and turning the stack end-for-end (reading the stack backwards), it is seen that (ABABC) and (ABCAC) are equivalent. Thus, only three five-layer closest packed patterns are unique: (ABABC), (ABACB), and (ABCBC).

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