Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.

Consider a foliation as a collection of continuous nonintersecting curves that start at (0,0) and end at (1,1) and collectively fill the unit square, such as the graphs of functions f_{t}(x) = x^{t} where t >=0. Supposedly there exists a continuous curve G that starts at (1,0), ends at (0,1), fills the unit square, and crosses each f_{t} curve only once.

This initially sounds even more impossible than the Cantor curve. But intuitively a space-filling curve could trace back and forth over the f_{t} curves and only cross at the corners (0,0) and (1,1). Can someone please explain a construction of such a space-filling curve?

`$\{x^a\}_{a \in S(n)}$`

in $[0,1]^2$ where`$S(n) := \{n^{-1},(n-1)^{-1},\dots,1,\dots,n-1,n\}$`

. The union $\gamma_n$ of these curves can be modified within balls of radius $\epsilon$ about the origin and $(1,1)$ to form a single curve $\gamma_{n,\epsilon}$ from the origin to $(1,1)$. I think (not quite sure) that an appropriate limit gives what you're looking for. – Steve Huntsman Aug 27 '10 at 23:14