If I have a curve that occupies a finite space (e.g. the unit square) and it has a box-counting dimension > 1, can it still have a finite length?

If not, is there a proof of this?

  • 1
    $\begingroup$ Roughly speaking, a rectifiable curve (i.e. a curve with finite length) can be covered by a number of $\delta$-balls proportional to the ratio of the length and $\delta$. This implies that the box dimension must be 1. $\endgroup$ – Xander Henderson Apr 18 at 20:13

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