BOUNDARY VALUE PROBLEMS FOR THE HILBERT CURVE I

together with Margarita Kraus, JGU Mainz

Starting point at $(0,0)$ and generalizations

Arrow schemes

End points of order 1

End points of order 2 right below

End points of order 3 - example

End points of order 4 - example

Summary

Any point of the principal diagonal of the left subsquare is connectable with any point of the secondary diagonal of the whole square.

Proof: The point $(0,0)$ is connectable with all points of the secondary diagonal, therefore, $\left(0,\frac{1}{2}\right)$ is connectable with all points of the principal diagonal of the left subsquare. Thus, connect any point of this part of the principal diagonal with $\left(0,\frac{1}{2}\right)$ and proceed as in the foregoing cases.

Further examples

Starting point $\left(\frac{1}{4},\frac{1}{4}\right),$ end point $\left(\frac{3}{4},\frac{1}{4}\right)$

Starting point $\left(\frac{1}{4},\frac{1}{4}\right),$ end point $\left(\frac{3}{8},\frac{1}{8}\right)$

Closed curve