  A necklace is an equivalence class of words of length $n$ over an alphabet
under the cyclic shift (rotation) operation. As a classical object, there have
been many algorithmic results for key operations on necklaces, including
counting, generating, ranking, and unranking. This paper generalises the
concept of necklaces to the multidimensional setting. We define
multidimensional necklaces as an equivalence classes over multidimensional
words under the multidimensional cyclic shift operation. Alongside this
definition, we generalise several problems from the one dimensional setting to
the multidimensional setting for multidimensional necklaces with size
$(n_1,n_2,...,n_d)$ over an alphabet of size $q$ including: providing closed
form equations for counting the number of necklaces; an $O(n_1 \cdot n_2 \cdot
... \cdot n_d)$ time algorithm for transforming some necklace $w$ to the next
necklace in the ordering; an $O((n_1 \cdot n_2 \cdot ... \cdot n_d)^5)$ time
algorithm to rank necklaces (determine the number of necklaces smaller than $w$
in the set of necklaces); an $O((n_1\cdot n_2 \cdot ... \cdot n_d)^{6(d + 1)}
\cdot \log^d(q))$ time algorithm to unrank multidimensional necklace (determine
the $i^{th}$ necklace in the set of necklaces). Our results on counting,
ranking, and unranking are further extended to the fixed content setting, where
every necklace has the same Parikh vector, in other words every necklace shares
the same number of occurrences of each symbol. Finally, we study the $k$-centre
problem for necklaces both in the single and multidimensional settings. We
provide strong approximation algorithms for solving this problem in both the
one dimensional and multidimensional settings.
