I was thinking that there might be a way to get a hexagon along the lines of the methods of the post Math with scissors. I told my kids about the idea and they were excited to do some math experiments.
I started with three strips of paper.
Taped them together like this:
Then I wanted to tape the three ends up like this,
but I knew there had to be some twisting of the strands involved. In the end we are going to cut each of the strands down the middle.
So we experimented. I knew that at least two of the strands had to have some odd number of half twists in them to have any chance of getting a hexagon (can you see why?). It took us several tries, but my son and I both came up with our own solutions for how to get a hexagon:
Try it and see if you can get it. Here are our solutions:
My son’s solution: Put a single half twist in each of the three strands, twisting two in one direction and the other one in the opposite direction.
My solution: Put a half twist in each of two of the strands, twisting them in opposite directions. Leave the third strand untwisted.
Of course, the experts will want to conjecture and prove necessary and sufficient conditions to get a nice flat hexagon. I also have an idea for making a five-(or more)-pointed-star along similar lines. I’ll let you know what I come up with.