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# The Difficulties of Folding a Map A rectangular map is a rectangular piece of paper folded along $n-1$ horizontal and $m-1$ vertical, evenly-spaced lines forming an $n \times m$ matrix of new identical rectangles.

Creating a Map of Arbitrary Dimensions

An interesting question of Map-Folding is how to divide the length or width of a piece of paper into $n$ equal parts without a ruler. For $n = 2^x$ where $x$ is a non-negative integer, this is trivial.  Because dividing a piece of paper in half is easy, so is dividing those halves into halves and so on.  If one can divide a piece of paper into three parts then $n = 2^x3^y$ where $y$ is a non-negative integer also are trivial.  Clearly, by focusing on obtaining methods to achieve $n = p$ where $p$ is prime all values can be generated.

The Fujimoto Approximation Method can be used to construct $p$ equal sections. For example, when dividing a piece of paper into 5 equal parts, estimate $\frac{1}{5}$ and make a preliminary crease. Then fold the remaining part of the paper in half and each of those segments in half again. Lastly fold the original end back in on itself.

With each additional fold the error decreases. Initially the value of the first fold is $\frac{1}{5} \pm \epsilon$. The value of the second fold can described by $\frac{1}{5} \pm \frac{\epsilon}{2}$. The next two folds have values $\frac{4}{5} \pm \frac{\epsilon}{4}$ and $\frac{2}{5} \pm \frac{\epsilon}{8}$ respectively. The fi nal corrected mark has value $\frac{1}{5} \pm \frac{\epsilon}{16}$. By iterating this process, the partitions will become
more accurate.

Folding Maps and Stamps

Stanislaw M. Ulam proposed an unsolved problem in combinatorics. He asked the
following question: Given a rectangular map and by folding only along the creases, how
many unique ways are there to refold the map completely?

For simplicity, we can number the rectangles of the matrix on the front and back and
then record the sequences (reading the sequence from the top of the stack to the bottom).  We can rephrase the question as: How many permutations of $n$ are possible?

The simplest case is analyzing a $1 \times n$ strip of $n$ squares; this is also known as the Stamp-Folding problem. For $n = 2$ and $n = 3$ all $n!$ permutations are possible, but for $n >3$ this is not the case. For example when $n = 4$ only 16 of the 4! possible permutations are possible. Note that in this case the sequences 1, 2, 3, 4 and 4, 3, 2, 1 are both counted as unique folds.

Many recreational paper-folding games rely on the fact that $2 \times 4, 3 \times 3,$ and $3 \times 4$ maps can be difficult to manipulate.

Cutting and Folding Maps

When cutting along creases is allowed, some interesting constructions can be achieved.

In 1939 Arthur Stone, a graduate student at Princeton, invented Flexagons, objects made out of paper which can be manipulated to bring up more (previously hidden) faces. His first model was a six sided figure he called a Hexaflexagon. He showed his friends at Princeton and a “Flexagation Committee” was quickly established.

Tetraflexagons are figures with four sides when properly assembled. One Tetraflexagon can be formed by altering a $3 \times 4$ map. References

1. T. Hull, “Dividing a Length into Equal Nths: Fujimoto Approximation”, Project
Origami, A.K.Peters, Ltd. (2006), pp. 15-26.

2. M. Gardner, “Paper-Folding”, The Colossal Book of Mathematics, W.W.Norton &
Company (2001), pp. 423-436.

3. M. Gardner, “Tetraflexagons”, Mathematical Puzzles and Diversions, Simon and Schuster(1961), pp. 24-31.