I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What is the shortest length of paper needed to do this?
I know if the smoothness condition is omitted then any length will suffice (fold paper into an 'accordion' pattern, twist etc as detailed in this post: Constructing a Möbius strip using a square paper? Is it possible?)
I can show that using any rectangle of paper of length to width ratio greater than $$\sqrt{3}:1$$
the construction in the picture can make a smooth non-self-intersecting Möbius Strip (with all the folds slightly rounded). In words you fold the strip along the lines drawn and the ends meet on the perpendicular of the equilateral triangle.I don't know whether this construction is optimal.
Is there a method to find the minimum ratio? Is there a general formula describing the minimum length of paper of width 1 unit needed to make a smooth non-intersecting paper band containing n half twists?