The Melton Ring

Everyone is familiar with a Moebius Strip, a length of paper glued to its own end with a half-twist so that a line drawn down the middle of the paper goes twice around before meeting itself.

But a strip of paper is a real three-dimensional object. Imagine instead of being flat and thin that this paper has a thickness equal to its width. Now the Moebius strip has two edges, one with a line drawn down the middle and one without.

Now imagine that the cross-section of this object were five-sided instead of four-sided before the line was drawn down the middle. Now the line drawn down the middle goes five times around before meeting itself.

What if the object had seven sides? Nineteen? Six hundred and forty-one?

What if it had an infinite number of sides? In other words, a circular cross-section.
And you gave the ends a twist that was irrational to the circumference of the cross-section before gluing the ends together?

Now if you start anywhere on the surface of this object and draw a line perpendicular to the cross-section, that line will be a closed loop infinite in length before it comes back and meets itself. Furthermore, that line will touch every point on the surface.

Do you have a three-dimensional object with a one-dimensional surface? If not, why not?