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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 reaches that point, then points that have nearby d values will also have nearby (x,y) values. The converse can't always be true. There will sometimes be points where the (x,y) coordinates are close but their d values are far apart.

This seems to be a contradiction to me, if the mapping from (x,y) points that are close together doesn't maps to values of d close together, how can values of d close together map to (x,y) points there are close ?

daw
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Tarantula
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    Have you looked at pictures of approximations to the Hilbert curve? The point is that the curve returns to nearly the same place at many far-apart times (close (x,y) but distant d). – Andreas Blass Aug 03 '15 at 16:08

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The statement, an excerpt from Hilbert Curve does seem a bit vague, but this is how I interpret it. Say you have a space filling curve $\gamma$ in the unit square with a length $d$ that ends at $(x_1, y_1)$. If you follow that curve further to a length $d + \epsilon$ you will end up at a nearby point $(x_2, y_2)$ because it is continuous.

On the other hand. If you take two points $(x_1, y_1), (x_2, y_2)$ that are close to each other, the Hilbert Curve may take quite a long excursion starting at $(x_1, y_1)$ before it hits $(x_2, y_2)$. This is a feature of being space filling.

muaddib
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