Constructing a Möbius strip using a square paper? Is it possible?

Question

I understand that, from a topological perspective, it is irrelevant whether we choose the quotient of the square $[0,1]\times [0,1]$ (by identifying points $(0,t)$ and $(1,1-t)$) or the quotient of the rectangle $[0,1]\times[0,a]$, $a>1$ (where the obvious points are identified), as a model of the Möbius strip. However, it seems pretty hard, if not impossible, to actually physically construct a Möbius stirp using a square like paper.

I wonder if it is possible to determine the least $a\geq 1$, such that he Möbius strip is physically constructible using a paper congruent to $[0,1]\times[0,a]$. I guess that this answer may depend on factors such as the size and the thickness of the paper that we use. This may be related to the mathematics of paper folding. Is there a general answer to this question, taking relevant qualities of a paper into account? Does anyone know of a source answering this question?

Answer 1

Suppose you have a square $ABCD$, fold point $A$ onto point $C$, giving you an isosceles right triangle. Next fold $D$ onto $B$ giving you a smaller isoceles right triangle. If you did this correctly sides $AB$ and $CD$ should be lined up (as the hypotenuse) if you were to tape them together the resulting figure would be a Mobius strip.

Answer 2

This is really a comment (or at best, a partial answer), but too long and illustrated for a comment.

It suffices if $a > 2$. Here's a proof by picture, for $a = 3$:

Essentially, a "half-turn" of the paper takes one width. Making two half-turns (to reverse the paper) takes two widths, and then returning to the starting point takes one width.

This results in a nice loop (it can be unbent and is flexible.)

You can reduce the length of the last flap, and just bend before you tape. You can do this all the way down to $a = 2$. In this case, in the last step you have

and you just fold the whole thing in half and tape the ends. But with $a = 2$ exactly, you can't unbend it without ripping the paper.

Answer 3

(apologies for the lack of mathematics in this answer, am posting as wiki, please do improve it)

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If you've got any square paper handy, you can fold a (slightly unconventional) Mobius strip in it like this pictured below. It's like a Mobius strip in that:

• Using a twist, the whole of one side connects to the whole of the opposite side inverted ($AB>CD$), the top left of the front connecting to the bottom right of the back, the same as if it was a long strip with one twist.
• You can fold it and unfold it
• It's a ring with a twist in it (not an open-topped "ice cream" cone)
• If you follow a line along the strip with a pencil in an uninterrupted line, when you unfold it, both sides will be marked.

It's different to a standard mobius strip in that it's folded over itself (but all the above still seems to be true). There's more to this answer than just folding a square in quarters then twisting it: as you'll see if you try it, that wouldn't match up the whole of one side against the whole of the other. This does.

I'm not sure if this is strictly a Möbius strip, but it has the main properties and can be folded from a square. Colours are purely as a visual aid. The end shows where on the front side (1) connects to where on the back side (2). If you were to label a conventional 'ribbon' Möbius strip in the same way ($A>D$ down each edge), the connection points would come out the same.