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There is a general version of this question which is known as "the rumpled dollar problem". It was posed by V.I. Arnold at his seminar in 1956. It appears as the very first problem in "Arnold's Problems":

Is it possible to increase the perimeter of a rectangle by a sequence of foldings and unfoldings?

According to the same sourse source (p. 182),

Alexei Tarasov has shown that a rectangle admits a realizable folding with arbitrarily large perimeter. A realizable folding means that it could be realized in such a way as if the rectangle were made of infinitely thin but absolutely nontensile paper. Thus, a folding is a map $f:B\to\mathbb R^2$ which is isometric on every polygon of some subdivision of the rectangle $B$. Moreover, the folding $f$ is realizable as a piecewise isometric homotopy which, in turn, can be approximated by some isotopy of space (which corresponds to the impossibility of self-intersection of a paper sheet during the folding process).

Have a look at

• A. Tarasov, Solution of Arnold’s “folded rouble” problem. (in Russian) Chebyshevskii Sb. 5 (2004), 174–187.

• I. Yashenko, Make your dollar bigger now!!! Math. Intelligencer 20 (1998), no. 2, 38–40.

A history of the problem is also briefly discussed in Tabachnikov's review of "Arnold's Problems":

It is interesting that the problem was solved by origami practitioners way before it was posed (at least, in 1797, in the Japanese origami book “Senbazuru Orikata”).

There is a general version of this question which is known as "the rumpled dollar problem". It was posed by V.I. Arnold at his seminar in 1956(see . It appears as the very first problem in "Arnold's Problems", p.2, problem 1956-1). :

Is it possible to increase the perimeter of a rectangle by a sequence of foldings and unfoldings?

According to the same sourse (p. 182),

Alexei Tarasov has shown that a rectangle admits a realizable folding with arbitrarily large perimeter. A realizable folding means that it could be realized in such a way as if the rectangle were made of infinitely thin but absolutely nontensile paper. Thus, a folding is a map $f:B\to\mathbb R^2$ which is isometric on every polygon of some subdivision of the rectangle $B$. Moreover, the folding $f$ is realizable as a piecewise isometric homotopy which, in turn, can be approximated by some isotopy of space (which corresponds to the impossibility of self-intersection of a paper sheet during the folding process).

Have a look at

• A. Tarasov, Solution of Arnold’s folded rouble problem. (in Russian) Chebyshevskii Sb. 5 (2004), 174–187.

• I. Yashenko, Make your dollar bigger now!!! Math. Intelligencer 20 (1998), no. 2, 38–40.

A history of the problem is also briefly discussed in Tabachnikov's review of "Arnold's Problems":

It is interesting that the problem was solved by origami practitioners way before it was posed (at least, in 1797, in the Japanese origami book Senbazuru Orikata”).

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There is a general version of this question which is also known as "the rumpled dollar problem"problem". It was asked posed by V.I. Arnold at his seminar in 1956 (see "Arnold's Problems) ", p.2, problem 1956-1).

Is it possible to increase the perimeter of a rectangle by a sequence of foldings and unfoldings?

According to the same sourse (p. 182),

Alexei Tarasov has shown that a rectangle admits a realizable folding with arbitrarily large perimeter. A realizable folding means that it could be realized in such a way as if the rectangle were made of infinitely thin but absolutely nontensile paper. Thus, a folding is a map $f:B\to\mathbb R^2$ which is isometric on every polygon of some subdivision of the rectangle $B$. Moreover, the folding $f$ is realizable as a piecewise isometric homotopy which, in turn, can be approximated by some isotopy of space (which corresponds to the impossibility of self-intersection of a paper sheet during the folding process).

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