Infinite Wrapping Paper

There are many examples of shapes with infinite surface area and finite volume- let's find an example. One contender may be the appropriately festive 'Koch Snowflake':

This fractal shape has a finite area, but an infinite perimeter, so could a 3D Koch snowflake, or something similar, be the shape we are after? No, because the question does not specify that the wrapping paper has to precisely follow the edge of the shape, so we could fit such a shape into a sphere of wrapping paper, for instance. Let's try another example, which we will call a 'magic staircase'. It has steps of equal length and width, but the height of each step is half that of the previous step. The first few steps may look something like this:

Recall that $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}...=1$ . Therefore, if the height of the first step (from the ground) is $0.5m$ and the distance between steps is $1m$ , then the area of the front-facing side of the staircase will be $1m^2$ . Let's take the width of the stairs to be $1m$ , making the volume of our magic staircase $1m^3$ . However, it is intuitively clear that such a shape will require an infinite amount of wrapping paper, since the staircase will stretch on forever in the horizontal direction (the shape's surface area is infinite). Therefore the answer is yes, you can buy a present with finite volume requiring an infinite amount of wrapping paper. A more elegant example of such a shape is called Gabriel's Horn .

A sphere is an example of shapes with infinite surface but finite volume e.e