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This paper introduces a new way of generalizing Hilbert's two-dimensionalspace-filling curve to arbitrary dimensions. The new curves, called harmoniousHilbert curves, have the unique property that for any d' < d, the d-dimensionalcurve is compatible with the d'-dimensional curve with respect to the order inwhich the curves visit the points of any d'-dimensional axis-parallel spacethat contains the origin. Similar generalizations to arbitrary dimensions aredescribed for several variants of Peano's curve (the original Peano curve, thecoil curve, the half-coil curve, and the Meurthe curve). The d-dimensionalharmonious Hilbert curves and the Meurthe curves have neutral orientation: ascompared to the curve as a whole, arbitrary pieces of the curve have each of d!possible rotations with equal probability. Thus one could say these curves are`statistically invariant' under rotation---unlike the Peano curves, the coilcurves, the half-coil curves, and the familiar generalization of Hilbert curvesby Butz and Moore.  In addition, prompted by an application in the construction of R-trees, thispaper shows how to construct a 2d-dimensional generalized Hilbert or Peanocurve that traverses the points of a certain d-dimensional diagonally placedsubspace in the order of a given d-dimensional generalized Hilbert or Peanocurve.  Pseudocode is provided for comparison operators based on the curves presentedin this paper.

Harmonious Hilbert curves and other extradimensional space-filling curves