BOUNDARY VALUE PROBLEMS FOR THE HILBERT CURVE II

together with Margarita Kraus, JGU Mainz

Starting point at \( \left(\frac{1}{2},0\right) \) and generalizations

Starting point approximated in the lower left subsquare

End points of order 1

Found by E.H. Moore: On certain crinkly curves (1900)

End points of order 2 right below

End points of order 3 - example

End points of order 4 - example

Summary

Left image: Any point of the secondary diagonal of the lower left subsquare is connectable with any point of the principal diagonal of the lower right subsquare and with any point of the principal diagonal of the upper left diagonal.

Right image: The target patterns are shown for Hilbert curves which start at a point of the dotted lines in the lower left subsquare. In particular, the intersection point \( \left(\frac{1}{4},\frac{1}{4}\right) \) can be connected with any point of the solid lines in the lower right subsquare and with any point of the solid lines in the upper left subsquare, see the boundary value problems discussed here.

Further examples

The right image shows a closed curve of Hilbert type starting and ending at the center of the square.