I'm trying to understand the following proposition about the Hilbert Curves: If (x,y) are the coordinates of a point within the unit square, and d is the distance along the curve when it ...
Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists? Or piecewise differentiable? Must every continuous space-filling curve be nowhere ...
Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x_0,y_0)$ and $(x_1,y_1)$ were represented by $a$ and ...
Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t ...