The core theoretical principle establishes that a Hilbert curve is a continuous function mapping one-dimensional space to two-dimensional Euclidean space by defining itself as the unique limit of a convergent sequence of pseudo-Hilbert approximations. This construction relies on rigorous definitions of continuity, where sufficiently small changes in input ensure arbitrarily small changes in output, thereby ensuring topological preservation despite traversing an infinite area with zero width. The concept bridges finite computational applications and pure mathematics by demonstrating that stability properties observed in discrete grid iterations converge to the formal property of a space-filling curve within continuous domains.
The core theoretical principle establishes that a Hilbert curve is a continuous function mapping one-dimensional space to two-dimensional Euclidean space by defining itself as the unique limit of a c…