Are there other shapes like the Koch snowflake, with infinite perimeter but finite area?

Question

Are there other known paradoxes in which a shape has infinite perimeter but finite area like the Koch snowflake paradox?

Answer 1

(More of a comment, but too long for a comment, and also likely of interest.)

We don't need anything nearly as exotic as the Koch curve or other such fractal (or nowhere differentiable) curves to have such an example. In fact, we can get such an example in which there is ONLY ONE point on the region's boundary curve that isn't "smooth" (by any reasonable usage of the term). Consider the continuous closed curve formed by the union of the graph of $y = \left| x\sin\left(\frac{\pi}{x}\right) \right|$ for $0 < x \leq 1$ and the graph of $y = -\sqrt{\frac{1}{4} - (x - \frac{1}{2})^2}.$ Note that the second function is just the semicircle in the 4th quadrant whose diameter has endpoints $(0,0)$ and $(1,0).$ Here is a graph of the region's boundary curve.

The perimeter of the region certainly has infinite length, since just by considering the sum of the vertical heights of the local maximum values of the top curve (similar to the method used here), we get a series whose divergence can be deduced easily from the divergence of the harmonic series. Also, the region (bounded by the curve defined above) certainly has finite area, since the region lies on or within the rectangle whose vertices are $(0,1)$ and $(1,1)$ and $(1,-1)$ and $(0,-1).$