Efficiently Filling Space
Accepted for publication at Rocky Mountain Journal of Mathematics
Abstract
In this paper we are concerned with space-filling curves that map a linear perfect set onto a closed domain in Rn. Specifically we prove that there is a space-filling curve, f : [0, 1] → [0, 1]n that is at most n+1-to-1 at every point. We include a rather simple proof that every such space-filling curve is at least n+1-to-1 at infinitely many points. This latter result is corollary to a more general theorem of Hurewicz and is related to, but seemingly independent of the Lebesgue Covering Theorem.
Authors
Dr. Paul D. Humke, Khang V. Huynh, Thong Vo