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I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google search hasn't turned up any comments on this at all. So, is this a general result? Thanks.

bob.sacamento
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  • I also have a question on the topic. What is the Fractal Dimension of a curve (or of the image of the curve) that fills 3-Dimensional space (for example the 3D-Hilbert curve)? I guess it should be between 2 and 3? Regards – Chris Jun 22 '22 at 10:46

1 Answers1

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You seem unclear about the definitions. A curve that fills the unit square is a continuous map $\gamma : [0,1] \rightarrow [0,1]^2$ such that $\gamma([0,1])=[0,1]^2$. For a general curve (not necessarily filling the square), the Hausdorff dimension of that curve is simply the Hausdorff dimension of $\gamma([0,1])$, i.e. of the image of the curve. Since the Hausdorff dimension of the unit square is 2, then yes, by definition the dimension of a curve filling the unit square is 2.

Albert
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