Bounded infinite shapes

[Aedl Foxe]

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)?

[++$_]

No, because of the isoperimetric inequality.

[Aedl Foxe]

Ah! That was simple. Thank you!

[mfb]

[...] find an example of a two-dimensional shape which has an infinite perimeter binding a finite area, e.g. [...] 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).

I would like to point out that your example has an infinite area (but finite volume if you rotate it around the x-axis).