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In addition to the answers above, here are some remarks from my paper. (Sorry for self-advertisement.)

1. An other solution. It is based on idea of Yashenko. This way you can incresae the perimeter just a bit, but it is done by repeating one fold (which is very simple but not "simple" in the sense below).

alt text

2. It is still not known if you can increase the perimeter by a sequence of simple natural folds; i.e., folds like this: alt text I just learned that this problem also appears in Pak's book, Problem 40.16b; it is marked by [$*$] which means that the problem is open.
show/hide this revision's text 2 added 14 characters in body

In addition to the answers above, here are some remarks from my paper. (Sorry for self-advertisement.)

1. An other solution. It is based on idea of Yashenko. This way you can incresae the perimeter just a bit, but it is done by repeating one fold (which is very simple but not "simple" in the above sense)sense below).

alt text

2. It is still not known if you can increase the perimeter by a sequence of simple folds; i.e., folds like this: alt text
show/hide this revision's text 1

In addition to the answers above, here are some remarks from my paper. (Sorry for self-advertisement.)

1. An other solution. It is based on idea of Yashenko. This way you can incresae the perimeter just a bit, but it is done by repeating one fold (which is simple but not in the above sense).

alt text

2. It is still not known if you can increase the perimeter by a sequence of simple folds; i.e., folds like this: alt text