Start by understanding the two-dimensional interleaved-bits curve (Morton curve) and how it interleaves bits to map 1D indices to 2D coordinates. How might this concept extend to three dimensions?

Consider how binary representations of coordinates in 3D space could be interleaved to create a space-filling curve. What pattern would you use to merge the bits of x, y, and z coordinates?

Explore the mathematical formulation of the Morton curve in 2D and generalize it to 3D. How would you define the interleaving function for three coordinates, and what properties must it preserve to ensure a valid space-filling curve?