Governance by those who do the work.

Wednesday, July 6, 2011

Recurrence for Multidimensional Self-Similar Functions

Staring at this diagram of the Peano space-filling curve inspired the recurrence formula, something I have been seeking for a couple of years. It remains to derive the inverse function and generalize the recurrence to produce Hilbert space-filling functions as well.

If the unit cell is a Hamiltonian Path, the resulting function is a space-filling curve. Functions resulting from unit cells which don't connect all their nodes are still self-similar (fractal).

Once the formula was discovered, creating a better space-filling curve, one which is nearly isotropic, is not difficult. This may not be novel in 2 dimensions, but the higher dimensional varieties are.