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