Round-dance neighbour designs from terraces

In a round-dance neighbour design, an odd number v of objects is arranged successively in (v-1)/2 rings (circular blocks) such that any two of the objects are adjacent to one another in exactly one ring. A round-dance neighbour design is also a Hamiltonian decomposition of the complete graph on v vertices. We show how such designs can be constructed from terraces, which are building blocks for row-complete Latin squares. A round-dance neighbour design is equivalent to a Tuscan square in which the reverse of each row is also a row. Terraces for the cyclic group of order n are used to construct elegantly patterned round-dance neighbour designs for n2^m + 1 objects for any positive integer m.