I am asked to find some curves which don't have length. I'm not sure whether the Koch curve is appropriate as it has infinite length.
Thanks for your help!
You could also look at "space-filling curves". But really they're just other (maybe we can say "worse") examples of curves with infinite length. The term is "non-rectifiable".
Since length is defined as the integral of the magnitude of the derivative, any curve which fails to be differentiable on a set of positive measure won't have a defined length.