Can a fractal have an infinite area if it's bounded by a box?

Question

Can a fractal/or any 2d shape have an infinite area if it's bounded by a box with finite area?

And oppositely, if it can be bounded by a circle/line, how can it's "arc length"/perimeter be infinite?

Any theorems/facts i can take a look at that dives into this topic?

Answer 1

Can a fractal have an infinite area if it's bounded by a box?

Yep, the Menger sponge is pretty much a box (of finite volume) with some pieces "sequentialy cut out of it in the manner of Cantor and Sierpinski" to end up with infinite area

if it can be bounded by a circle/line, how can it's "arc length"/perimeter be infinite?

Just check out Sierpinski's carpet and Koch's snowflake, both bounded (finite area) but having perimeter larger than any real number

Can a fractal/or any 2d shape have an infinite area if it's bounded by a box with finite area?

No, because measures are monotone

Answer 2

I don't know about 2D having infinite area, but a 3D fractal can have infinite surface area. See: Menger sponge.

A 2D fractal can have an infinitely long perimeter. One example is a Koch snowflake. Its perimeter expands by a factor of $\frac{4}{3}$ with each iteration (since the middle subsegment third of each segment is duplicated and then "bent outward" to become the outer two sides of another equilateral triangle) but its area remains within the finite area of the page on which it's printed.