It is simple, if not quite trivial, to find an example of a two-dimensional shape which has an infinite perimeter binding a finite area, e.g. the Koch Snowflake, or the area bounded by the lines x = 1, y = 0, and y = 1 / x (being half the cross-section of the solid known as Gabriel's Horn). This principle can be extended to three dimensions (and possibly more, but my imagination exceeds my training), producing the same Gabriel's Horn, and Eric Haines' "sphereflake", irrespectively.
Given the existence of such counter-intuitive shapes, I feel I must ask: do there exist shapes in two dimensions which have infinite area bounded by finite perimeters (or the n-dimensional analogues)?
"Maybe there are stupid happy people out there... And life isn't fair, and you won't become happier by being jealous of what you can't have... You can never achieve that degree of ignorance... you cannot unknow what you know." -E. Yudkowsky