The PostScript file
Hilbert Order in *d* Dimensions
(161 Kbytes) is a ten-page derivation of the mapping between Hilbert order
and coordinates in any number of dimensions for any order of curve.

Bially's paper, *Space-filling curves: their generation and their application
to bandwidth reduction* (IEEE Transactions on Information Theory, IT-15,
6(Nov. 1969) 658-64) is more general, in that it develops a base hypercube of
size *m*m*..*m* for any *m*, instead of just *m*=2.
Bially expresses his mappings between position on the curve and spatial
coordinates with a series of state diagrams, examples of which he shows in two
and three dimensions, and an abstraction for four. We use a single set of rules
for any number of dimensions as well as for any order.

Curves connecting only nearest neighbours in more than two dimensions are not
unique. We have one, based on a binary reflective Gray code [C. Faloutsos,
*Gray codes for partial match and range queries*, IEEE Trans. on Software
Engineering, 14 10(Oct., 1988) 1381-9] and extended by
reflections. Bially has another. Jagadish (*Linear clustering of objects
with multiple attributes*, Proceedings of the 1990 ACM SIGMOD International
Conference on Management of Data, Atlantic City, NJ, May 23-25, 1990, Hector
Garcia-Molina and H. V. Jagadish, eds., 332-42) shows a third, in 3-D, and
Gilbert shows a fourth 3-D Hilbert curve.