# Bedsheet Problem

(Difference between revisions)
 Revision as of 13:32, 1 June 2011 (edit)← Previous diff Current revision (14:58, 24 June 2011) (edit) (undo) (41 intermediate revisions not shown.) Line 2: Line 2: |ImageName=Bedsheet Problem |ImageName=Bedsheet Problem |Image=12fold.jpg |Image=12fold.jpg - |ImageIntro=Take a piece of paper. Now try to fold it in half more than 7 times. Is it possible? This leads into the problem: what is the ultimate number of folds a flat piece of material can achieve? + |ImageIntro=Take a piece of paper. Now try to fold it in half more than 7 times. Is it possible? What is the ultimate number of folds a flat piece of material can achieve? This image shows Britney Gallivan’s success at folding a sheet 12 times. - :This question raises an issue. Theoretically if we are given an infinitely long sheet of paper, then there should be no absolute folding limit. Yet, when the experiment is conducted the sheet of paper does not behave accordingly. There is a limited amount of fold! Britney Gallivan in December 2001 created a mathematical representation of the bedsheet problem in which she describes the interaction between the thickness, length, and number of limited folds. Gallivan's mathematical model describes the reality of the physical system. Her derivation gives the loss function for folding a piece of paper in half. + |ImageDescElem=The bedsheet problem is an urban legend that states the following: any piece of paper (no matter the dimensions) cannot fold more than 7 times . All who claimed the myth was valid could only cite empirical evidence. They could not explain or prove it mathematically. The puzzle was both mysterious and inexplicable. - |ImageDescElem=The bedsheet problem is an urban legend as well as the following: any piece of paper (no matter the dimensions) cannot fold more than 7 times . All who claimed the myth was valid could only cite empirical evidence. They could not explain or prove it mathematically. The puzzle was both mysterious and inexplicable. Gallivan has discovered the wonder that limits the folding. This image shows Gallivan’s success at folding a sheet 12 times. + - This well-known problem was not solved before for many reason. First, many believed the puzzle was true because no one understood or proved mathematically the geometric limitations. Most people folded a sheet of paper and no matter what material was used people usually got around 5, 6, and 7 folds. Because 7 was the highest number it became the folding limit. Also, there was little creativity in disproving the myth. No one realized that the thickness, width, flexibility of the material, strength of the person and crease lines all play a role in how many folds are achievable. Nevertheless, a sheet of paper was folded 12 times. + Theoretically if we are given an infinitely long sheet of paper, then there would be no absolute folding limit. However, this result does not seem true when the experiment is conducted. In December 2001, Britney Gallivan created a mathematical representation of the bedsheet problem in which she described the interaction between the thickness, length, and number of folds that were possible. Gallivan's mathematical model describes the reality of the physical system. Her derivation gives the loss function for folding a piece of paper in half. - ====Britney Gallivan==== + [[Image: Britney11.jpg|right|250px]] -
Britney Gallivan, who was at the time a junior in high school, solved this well-known problem. She was asked by her teacher to fold a sheet of paper 12 times and as an incentive she would get extra credit. She failed multiples times. Later she succeeded after using a thin gold sheet and proved the assumption wrong. Gallivan was able to achieve 12 folds by folding a roll of thin toilet paper that stretched over three-fourths of a mile. It took seven hours in a shopping mall with her parents, but Gallivan was able to bust a myth as well as derive a formula relating the width, thickness of a paper and the number of folds achievable. The urban legend of 7 folds was disproved in 2001.
+ - [[Image: Britney11.jpg|left|250px]] + ===Debunking the Myth=== + Britney Gallivan, who was at the time a junior in high school, solved this well-known problem. She was asked by her teacher to fold a sheet of paper 12 times and as an incentive she would get extra credit. She failed multiples times. Later she succeeded after using a thin gold sheet and proved the assumption wrong. Gallivan was able to achieve 12 folds by folding a roll of thin toilet paper that stretched over three-fourths of a mile. It took seven hours in a shopping mall with her parents, but Gallivan was able to bust a myth as well as derive a formula relating the width, thickness of a paper and the number of folds achievable. The urban legend of 7 folds was disproved in 2001. Gallivan wrote a 40 page pamphlet on her discovery in which she explained the mathematics, the story and other information about her project. - ====Explanation==== + ===What's Happening=== - The length of a fold adds up to the length of the flat top and bottom and the circumference of the semi-circle once the ends of the paper have a noticeable curved. This is shown in image 2. We will call this the nth fold. The next fold which is called the n+1th fold can be achieved if and only if its length is greater than or equal to the circumference of the previous fold's curved section. When the n+1th fold is performed the flat top and bottom is halved and the thickness of the previous fold is included in this new length. At this point in the thickness of the n+1th fold is double the thickness of the nth fold. + When we fold a piece of paper what exactly is happening? Say we start with a piece of paper where the length is greater than the width. Fold the paper in half so the opposite sides of the width meet. With one fold, the sheet has a thickness of two paper, for two folds it has a thickness of 4 sheets of paper and so on. See the image on the left. In fact, the thickness of a sheet of paper doubles after each fold. - ====King's Problem==== + [[Image:3folds.jpg|thumb|This image describes the thickness of the sheet of paper increasing exponentially|left|300px]] - Overall the paper folding connects to exponential growth. This is the essence of exponential growth: very small amounts rapidly become astronomically large through simple doubling. There is a math folktale about a clever merchant who asked the King to pay him with grains of wheat on a chessboard. The merchant asked the King to place one grain on the first square, two on the second, four on the third, eight on the fourth, sixteen on the fifth, and so on until the 64 boxes were filled. The king was too proud to admit that he could not calculate the sum of the grains. He foolishly granted the wish, not knowing that it would wipe out his stock and he would be in debt. If it isn't clear yet imagine the 20th box where the number 2 is multiplied by itself 20 times. This means that just on the 20th box the merchant would have more than a million grains. This is an example of exponential growth. + - [[Image: Chessboard.jpg| left| 250px]] + But there's more. With each fold the sheet fails to remain flat and long. The paper begins to crumple and it does not fold smoothly. With each fold the additional layers make it hard to lay one end of the sheet of paper over the opposite side. The paper begins to curve and puff at the ends. After a certain number of folds (this varies based on the thickness and width of the paper) we begin to notice something interesting happening around the edges. - The bed sheet problem though becomes a little bit more complex. The thickness does not simply double after each fold. When a sheet of paper is folded, one end of the paper is placed onto the opposite end. The paper begins to crumple and it does not fold smoothly. With each fold the additional layers make it hard to lay one end over the opposite side. The paper begins to curve and puff at the ends. After a certain number of folds (this varies based on the thickness and width of the paper) the paper will reach a limit and any additional folds will no longer keep the paper long and flat but rather it will be a semi-circle. In addition to the thickness doubling, the sheet of paper begins to curve. The circumference of the semicircle at the edges are the reason why the bedsheet problem does not imitate the King’s Problem. + The middle of the previous fold becomes a round edge. The radius of the edge is one half the new thickness. We begin to notice this round edge once the thickness of the paper is thick compared to the length of the paper. - ====Different Methods of Folding==== + The radius section is important because it begins to take up a greater percentage of the paper's volume. The folding limit is reached when there is not enough volume of the remaining paper to fill the radius section. At this point an additional fold will cause the entire sheet of paper to transform into a semi-circle. + [[Image: radiusfoldedsection.jpg|right|150px]] + + + ===Folding Directions=== + There are many ways we can fold a sheet of paper. However, we will only focus on two: linear folding and alternative folding. + + Linear folding requires that the length of the paper is much greater than the width. Take the ends of one side of width and place the vertices on top of the opposite ends with carefulness and accuracy. Continue to fold in this manner until it cannot be done anymore. This way of folding is shown in the image on the bottom right. [[Image: singlefold.jpg|thumb|Single linear Folding|right|200px]]. + + Using this method, the thickness of the sheet of paper limits the amount of times the paper can be folded. After each fold the thickness doubles. For example after the third fold the thickness is the same as eight sheets. This brings up the issue with folding a sheet of paper in half: it ends up being very thick really fast. The thickness will double each time while the length is halved. Gallivan completed the 12th fold using the linear folding. + + The second method to fold is the alternative direction. Do one fold following the linear folding method. Make the vertices of the width's side meet. Then rotate the paper and take the vertices of the now length sides, which is half the original length, and place it onto of the opposite vertices. These vertices should still be on the same side. The folding repeats until it reaches it limit. The image on the bottom left [[Image: altfold.jpg|thumb|Alternative Folding|left|250px]] depicts the alternative folding method. + + The alternative method's has either the width and length halved every fold. Thus meaning the width by itself is halved every two folds. The thickness doubles every fold. Folding a sheet in alternative direction requires that the sheet is wide. Folding in alternative direction has one advantage: the sheet will not unravel a previous fold. + + Gallivan created her derivation based on these two methods of folding. + + ====King's Problem==== + [[Image: Chessboard.jpg|right|220px]] + + In the larger perspective, the paper folding experiment connects to exponential growth. This is the essence of exponential growth: very small amounts rapidly become astronomically large through simple doubling. + + There is a math related fairytale about a clever merchant who asked the King to pay him with grains of wheat on a chessboard. The merchant asked the King to place one grain on the first square, two on the second, four on the third, eight on the fourth, sixteen on the fifth, and so on until the 64 boxes were filled. The king was too proud to admit that he could not calculate the sum of the grains. He foolishly granted the wish, not knowing that it would wipe out his stock and he would be in debt. Imagine the 20th box where the number 2 is multiplied by itself 20 times. This means that just on the 20th box the merchant would have more than a million grains. This is an example of exponential growth. + + The bed sheet problem though becomes a little bit more complex. The thickness does not simply double after each fold. When a sheet of paper is folded, we saw that we get a radius section. The key word is radius. It is clear that there a large semicircle begins to form after a large number of folds. The circumference of the semicircle at the edges are the reason why the bedsheet problem does not imitate the King’s Problem. + |ImageDesc=====Derivations==== |ImageDesc=====Derivations==== - If you keep folding a sheet of paper the thickness doubles after each fold. With one fold, you have a thickness of two paper, for two folds it has a thickness of 4 sheets of paper and so on. The layers increase by $2^N$ where N is the number of folds. This is somewhat similar to the chessboard example. + [[Image:foldedsection.jpg|left|250px]] + + The thickness of a sheet of paper doubles after each fold. With one fold, the sheet has a thickness of two sheets of paper, for two folds it has a thickness of 4 sheets of paper and so on. The layers increase by $2^N$ where N is the number of folds. With each fold the sheet fails to remain flat and long. It begins to curve at the edge. The radius of the entire radius section is half the overall thickness. + + The volume and thickness in the curved section grows exponentially. Again, the folding limit is reached when there is not enough volume or length of the remaining paper to fill the radius section. To be specific we consider the folded section to be the region that has two times the number of folds of the straight layers. The curved ends are not counted as part of the folded section. - However, the sheet does not remain flat and long. It begins to curve at the edge. Each time a fold takes place the edges become round and the radius of the curved section. The radius of each layer is a half of the thickness. [[Image: radiusfoldedsection.jpg|left|150px]] + The length of a fold adds up to the length of the flat top and bottom and the circumference of the semi-circle once the ends of the paper have a noticeable curve. This is shown in [[Image:.jpg|.jpg]]. We can perform another fold if and only if the next fold's length is greater than or equal to the circumference of the previous fold's radius section. When the next fold is performed the flat top and bottom of the previous fold is halved and the thickness of the previous fold is doubled. Using these relationships Gallivan was able to construct an equation where L, is the minimum possible length of the material, t is the thickness of the material, and n is the possible number of folds. - With each layer the radius is different. The curved section of the folded sheet is noticeable when the thickness of the paper is equal or greater than the width. With an increase in folds, the radius section begins to take up a greater percentage of the paper's volume. The volume and thickness in curved section grows exponentially. The volume squares with each fold while the thickness doubles. The folding limit is reached when there is not enough volume or length of the remaining paper to fill the curved section. Any folds pass the absolute limit the entire sheet will become a semi-circle. To be specific we consider the folded section to be the region that has two times the number of folds of the straight layers. The curved ends are not counted as part of the folded section. [[Image:foldedsection.jpg|left|250px]] + The information below reveals the detailed description and formula of the folding limit equation derived based on the two folding methods mentioned. =====Single Direction===== =====Single Direction===== - {{HideShowThis|ShowMessage=Click to reveal Single Direction|HideMessage=Click to conceal Single Direction|HiddenText=The length of paper required to fold a sheet of paper N times in a single direction, meaning the folding occurs in only one direction, includes how much paper has been lost after a certain number of folds. For a single fold there is a minimum length of the thickness times /pi, the length of the semicircle. + {{HideShowThis|ShowMessage=Click to reveal Single Direction|HideMessage=Click to conceal Single Direction|HiddenText=Adding more here later on (Going to explain in word the formation of the summations below) - The following definition are used in the equation. + The following definition are used in the derivation for the single fold equation. *L=length of material needed *L=length of material needed Line 44: Line 71: *i=dummy variable *i=dummy variable - L=$\pi t * \sum_{i=1}^n {\sum_{k=1}^{2^{i-1}} 2^{i-1}-(k-1)}$ + L=$\pi t * \sum_{i=1}^n {\sum_{k=1}^{2^{i-1}} \lbrack 2^{i-1}-(k-1) \rbrack}$ - The first summation is for the actual folds. The second is for the amount of sheet loss in a particular fold. The expression $(k-1)$ the $2^{n-1}$ is the radius of the outer folded layer. + The outer summation represents the actual folds. The inner summation represents the amount of sheet loss in a particular fold. The expression $(k-1)$ the $2^{n-1}$ is the radius of the outer folded layer. - The following are the steps to simplify the equation. The ${\pi}*t$ term is excluded in the following steps, but the final answer will have this term. + The following steps show how to expand the summation such that we get a algebraic equation. The ${\pi}*t$ term is excluded in the following steps, but the final answer will have this term. - :$\sum_{k=1}^n k-1$=$\sum_ {k=0}^{n-1} k$=$\cfrac{n-1}{n}\cfrac{2}\$. + We rewrite the summation in a simpler form and then we distribute it along the right side of the expression. - In our case we get + :$\sum_{k=1}^n k-1$=$\sum_ {k=0}^{n-1} k$=${1/2} \cfrac{n-1}{n}$. + + We get :$\frac{(2^{i-1}-1)(2^{i-1})}{2}$ :$\frac{(2^{i-1}-1)(2^{i-1})}{2}$ - Simplify the expression + Relying on algebra we simplify the expression :$-2^{2i-3}+2^{i-2}$ :$-2^{2i-3}+2^{i-2}$ Line 63: Line 92: :$\sum_{i=1}^n \frac{4^i}{8}+\sum_{i=1}^n \frac{2^i}{4}$ :$\sum_{i=1}^n \frac{4^i}{8}+\sum_{i=1}^n \frac{2^i}{4}$ + + Looking at the general case, we use series to find an expression for the two terms we solved above. :$\mathrm{S}$=$\sum_{i=1}^n b^i$ :$\mathrm{S}$=$\sum_{i=1}^n b^i$ Line 82: Line 113: We simplify and get We simplify and get - :$\frac{4^{n+1}-4}{24}$ = $\frac{(4^{n}-1)}{6}$ = $frac{2^{2n}-1}{6}$ + :$\frac{4^{n+1}-4}{24}$ = $\frac{(4^{n}-1)}{6}$ = $\frac{2^{2n-1}}{6}$ :$\sum_{i=1}^n \frac{2^i}{4}$ = $\frac{2^{n+1}-2}{1*4}$ :$\sum_{i=1}^n \frac{2^i}{4}$ = $\frac{2^{n+1}-2}{1*4}$ - - We simplify and get :$\frac{2^{n+1}-2}{4}$ = $\frac{2^{n}-1}{2}$ = $\frac{3*2^n-3}{6}$ :$\frac{2^{n+1}-2}{4}$ = $\frac{2^{n}-1}{2}$ = $\frac{3*2^n-3}{6}$ Line 104: Line 133: =====Alternative Direction===== =====Alternative Direction===== - {{HideShowThis|ShowMessage=Click to reveal Alternative Direction|HideMessage=Click to conceal Alternative Direction|HiddenText=Alternate direction folding uses both flat directions of the paper, so the width decreases after every fold. The alternative direction derivation is complicated because there are two folding limits. For the last fold to be achieved the previous fold's length has the be $\pi$ times the thickness. This relationship applies for both methods of folding. The sides of the fold are $\pi t 2^{(n-1)}$ and the thickness at this stage is $t 2^{(n-1)}$. The total area of all sheet is the area of a single sheet times the number of sheets. + {{HideShowThis|ShowMessage=Click to reveal Alternative Direction|HideMessage=Click to conceal Alternative Direction|HiddenText=The alternative direction derivation is complicated because there are two folding limits one for the width and one for the length. For the last fold to be achieved the previous fold's length again has the be $\pi$ times the thickness. The sides of the fold are $\pi t 2^{(n-1)}$ and the thickness at the soon to be last stage is $t 2^{(n-1)}$. The total area of all sheet is the area of a single sheet times the number of sheets. - Area=$\pi^2 t^2 2^{3(n-1)}$. + $\mathrm{A} \mathrm{r} \mathrm{e} \mathrm{a}$ = ${\pi t 2^{(n-1)}}^2$ + + We then multiply the area by the number of sheets. The number of sheets is $2^{n-1}$. + + The final area is $\pi^2 t^2 2^{3n-3}$ Taking the square root of the area will give the limiting width of the original sheet. Taking the square root of the area will give the limiting width of the original sheet. Line 119: Line 152: [[Image:nthfoldaxis.jpg|200px|right]] [[Image:nthfoldaxis.jpg|200px|right]] + {{EquationRef2|Eq. 2}}W=$\pi t* 2^{3(n-1)/2}$ - {{EquationRef2|Eq. 2}}W=$\pi t* 2^{3(n-1)/2}$}} - + The equation though isn't completely accurate because it does not include the materials lost in the radii of previous folds. For odd number of folds there is lost if the paper is folded in the odd fold direction. The odd fold do not contribute to the loss in the even fold direction. For large number of folds there is a large gap in accuracy. In practice a sheet of material could not fold that many times or could be that tight. Equation 2 accurately demonstrates the limiting width in reality though.}} - The equation though isn't completely accurate because it does not include the materials lost in the radii of previous folds. For odd number of folds there is lost if the paper is folded in the odd fold direction. The old fold do not contribute to the loss in the even fold direction. For large number of folds there is a possible limit that is 60% less than the actual limit. In practice a sheet of material could not fold that tight so equation 2 accurately demonstrates the limiting width. + ====Limitations==== ====Limitations==== Line 145: Line 177: ====Related Problem==== ====Related Problem==== - Rather than how many folds are achievable, a similar problem to the bed sheet problem focuses on the thickness of the sheet of paper. If you were to take a large sheet of paper with thickness 1/400 inch and fold it in half 50 times, how tall would it be (if it was theoretically possible to fold a sheet of paper 50 times)? Most people in their minds would imagine it to be as thick as a phone book. Some might even being daring and say a few feet. After 3 folds the sheet of paper would be as thick as your fingernail. And, if you continue to fold until the paper is at 50 folds then it will be as thick as 40 million miles. The paper would be able to reach the sun and return back to earth. This sounds ridiculous, but it isn't. This is an example in mathematics called geometric progression. Each number in the sequence is multiplied by a fixed number to get the next term. + Rather than how many folds are achievable, a similar problem to the bed sheet problem focuses on the thickness of the sheet of paper. If you were to take a large sheet of paper with thickness 1/400 inch and fold it in half 50 times, how tall would it be (if it was theoretically possible to fold a sheet of paper 50 times)? Most people in their minds would imagine it to be as thick as a phone book. Some might even being daring and say a few feet. After 3 folds the sheet of paper would be as thick as your fingernail. And, if you continue to fold until the paper is at 50 folds then it will be as thick as 40 million miles. The paper would be able to reach the sun and return back to earth. This sounds ridiculous, but it isn't. + [[Image:folds2sun.jpg|thumb|This image shows that on the 42nd fold the stack of paper will reach the moon. At such a rate the thickness of the stack of paper can become very large.|right|300px]] + |SiteName=Pomona Historical Society |SiteName=Pomona Historical Society |SiteURL=http://pomonahistorical.org/12times.htm |SiteURL=http://pomonahistorical.org/12times.htm |Field=Algebra |Field=Algebra - |WhyInteresting=This problem is interesting because Gallivan showed that any person can do the impossible. She solved a problem that many mathematicians attempted but also failed to solve. It is interesting that a high school student with such fervor demonstrated a person should not accept anything to be true without evidence. + |WhyInteresting=This problem is interesting because Gallivan showed that any person can do the impossible. She solved a problem that many mathematicians attempted but also failed to solve. It is interesting that a high school student with such fervor demonstrated that mathematics is a puzzle for everyone for all kinds of ages. Gallivan showed that a person should not accept anything to be true without evidence. Gallivan has inspired others to break the record for most folds. Late April 2011 some students from Massachusetts claimed the world record in achieving 13 folds. They used the thinnest type of toilet paper that stretched over 2 miles. To hear more on their story click [http://www.necn.com/04/07/11/Students-claim-world-record-folding-toil/landing_scitech.html?blockID=499544&feedID=4213 here]. Gallivan has inspired others to break the record for most folds. Late April 2011 some students from Massachusetts claimed the world record in achieving 13 folds. They used the thinnest type of toilet paper that stretched over 2 miles. To hear more on their story click [http://www.necn.com/04/07/11/Students-claim-world-record-folding-toil/landing_scitech.html?blockID=499544&feedID=4213 here].

## Current revision

Bedsheet Problem
Take a piece of paper. Now try to fold it in half more than 7 times. Is it possible? What is the ultimate number of folds a flat piece of material can achieve? This image shows Britney Gallivan’s success at folding a sheet 12 times.

# Basic Description

The bedsheet problem is an urban legend that states the following: any piece of paper (no matter the dimensions) cannot fold more than 7 times . All who claimed the myth was valid could only cite empirical evidence. They could not explain or prove it mathematically. The puzzle was both mysterious and inexplicable.

Theoretically if we are given an infinitely long sheet of paper, then there would be no absolute folding limit. However, this result does not seem true when the experiment is conducted. In December 2001, Britney Gallivan created a mathematical representation of the bedsheet problem in which she described the interaction between the thickness, length, and number of folds that were possible. Gallivan's mathematical model describes the reality of the physical system. Her derivation gives the loss function for folding a piece of paper in half.

### Debunking the Myth

Britney Gallivan, who was at the time a junior in high school, solved this well-known problem. She was asked by her teacher to fold a sheet of paper 12 times and as an incentive she would get extra credit. She failed multiples times. Later she succeeded after using a thin gold sheet and proved the assumption wrong. Gallivan was able to achieve 12 folds by folding a roll of thin toilet paper that stretched over three-fourths of a mile. It took seven hours in a shopping mall with her parents, but Gallivan was able to bust a myth as well as derive a formula relating the width, thickness of a paper and the number of folds achievable. The urban legend of 7 folds was disproved in 2001. Gallivan wrote a 40 page pamphlet on her discovery in which she explained the mathematics, the story and other information about her project.

### What's Happening

When we fold a piece of paper what exactly is happening? Say we start with a piece of paper where the length is greater than the width. Fold the paper in half so the opposite sides of the width meet. With one fold, the sheet has a thickness of two paper, for two folds it has a thickness of 4 sheets of paper and so on. See the image on the left. In fact, the thickness of a sheet of paper doubles after each fold.

This image describes the thickness of the sheet of paper increasing exponentially

But there's more. With each fold the sheet fails to remain flat and long. The paper begins to crumple and it does not fold smoothly. With each fold the additional layers make it hard to lay one end of the sheet of paper over the opposite side. The paper begins to curve and puff at the ends. After a certain number of folds (this varies based on the thickness and width of the paper) we begin to notice something interesting happening around the edges.

The middle of the previous fold becomes a round edge. The radius of the edge is one half the new thickness. We begin to notice this round edge once the thickness of the paper is thick compared to the length of the paper.

The radius section is important because it begins to take up a greater percentage of the paper's volume. The folding limit is reached when there is not enough volume of the remaining paper to fill the radius section. At this point an additional fold will cause the entire sheet of paper to transform into a semi-circle.

### Folding Directions

There are many ways we can fold a sheet of paper. However, we will only focus on two: linear folding and alternative folding.

Linear folding requires that the length of the paper is much greater than the width. Take the ends of one side of width and place the vertices on top of the opposite ends with carefulness and accuracy. Continue to fold in this manner until it cannot be done anymore. This way of folding is shown in the image on the bottom right.
Single linear Folding
.

Using this method, the thickness of the sheet of paper limits the amount of times the paper can be folded. After each fold the thickness doubles. For example after the third fold the thickness is the same as eight sheets. This brings up the issue with folding a sheet of paper in half: it ends up being very thick really fast. The thickness will double each time while the length is halved. Gallivan completed the 12th fold using the linear folding.

The second method to fold is the alternative direction. Do one fold following the linear folding method. Make the vertices of the width's side meet. Then rotate the paper and take the vertices of the now length sides, which is half the original length, and place it onto of the opposite vertices. These vertices should still be on the same side. The folding repeats until it reaches it limit. The image on the bottom left
Alternative Folding
depicts the alternative folding method.

The alternative method's has either the width and length halved every fold. Thus meaning the width by itself is halved every two folds. The thickness doubles every fold. Folding a sheet in alternative direction requires that the sheet is wide. Folding in alternative direction has one advantage: the sheet will not unravel a previous fold.

Gallivan created her derivation based on these two methods of folding.

#### King's Problem

In the larger perspective, the paper folding experiment connects to exponential growth. This is the essence of exponential growth: very small amounts rapidly become astronomically large through simple doubling.

There is a math related fairytale about a clever merchant who asked the King to pay him with grains of wheat on a chessboard. The merchant asked the King to place one grain on the first square, two on the second, four on the third, eight on the fourth, sixteen on the fifth, and so on until the 64 boxes were filled. The king was too proud to admit that he could not calculate the sum of the grains. He foolishly granted the wish, not knowing that it would wipe out his stock and he would be in debt. Imagine the 20th box where the number 2 is multiplied by itself 20 times. This means that just on the 20th box the merchant would have more than a million grains. This is an example of exponential growth.

The bed sheet problem though becomes a little bit more complex. The thickness does not simply double after each fold. When a sheet of paper is folded, we saw that we get a radius section. The key word is radius. It is clear that there a large semicircle begins to form after a large number of folds. The circumference of the semicircle at the edges are the reason why the bedsheet problem does not imitate the King’s Problem.

# A More Mathematical Explanation

#### Derivations

The thickness of a sheet of paper doubles [...]

#### Derivations

The thickness of a sheet of paper doubles after each fold. With one fold, the sheet has a thickness of two sheets of paper, for two folds it has a thickness of 4 sheets of paper and so on. The layers increase by $2^N$ where N is the number of folds. With each fold the sheet fails to remain flat and long. It begins to curve at the edge. The radius of the entire radius section is half the overall thickness.

The volume and thickness in the curved section grows exponentially. Again, the folding limit is reached when there is not enough volume or length of the remaining paper to fill the radius section. To be specific we consider the folded section to be the region that has two times the number of folds of the straight layers. The curved ends are not counted as part of the folded section.

The length of a fold adds up to the length of the flat top and bottom and the circumference of the semi-circle once the ends of the paper have a noticeable curve. This is shown in Image:.jpg. We can perform another fold if and only if the next fold's length is greater than or equal to the circumference of the previous fold's radius section. When the next fold is performed the flat top and bottom of the previous fold is halved and the thickness of the previous fold is doubled. Using these relationships Gallivan was able to construct an equation where L, is the minimum possible length of the material, t is the thickness of the material, and n is the possible number of folds.

The information below reveals the detailed description and formula of the folding limit equation derived based on the two folding methods mentioned.

##### Single Direction

Adding more here later on (Going to explain in word the formation of the summations below)

The following definition are used in the derivation for the single fold equation.

• L=length of material needed
• t=thickness of one sheet of material
• n=number of folds
• k=dummy variable
• i=dummy variable

L=$\pi t * \sum_{i=1}^n {\sum_{k=1}^{2^{i-1}} \lbrack 2^{i-1}-(k-1) \rbrack}$

The outer summation represents the actual folds. The inner summation represents the amount of sheet loss in a particular fold. The expression $(k-1)$ the $2^{n-1}$ is the radius of the outer folded layer.

The following steps show how to expand the summation such that we get a algebraic equation. The ${\pi}*t$ term is excluded in the following steps, but the final answer will have this term.

We rewrite the summation in a simpler form and then we distribute it along the right side of the expression.

$\sum_{k=1}^n k-1$=$\sum_ {k=0}^{n-1} k$=${1/2} \cfrac{n-1}{n}$.

We get

$\frac{(2^{i-1}-1)(2^{i-1})}{2}$

Relying on algebra we simplify the expression

$-2^{2i-3}+2^{i-2}$
$\sum_{i=1}^n \frac{2^{2i}}{8}+\sum_{i=1}^n \frac{2^i}{4}$
$\sum_{i=1}^n \frac{4^i}{8}+\sum_{i=1}^n \frac{2^i}{4}$

Looking at the general case, we use series to find an expression for the two terms we solved above.

$\mathrm{S}$=$\sum_{i=1}^n b^i$
$\mathrm{S}$=$b+b^2+\cdots+b^n$
$\mathrm{S} \mathrm{b}$=$b^2+\cdots+b^{n+1}$
$\mathrm{S}*\mathrm{b}-\mathrm{S}$=$b^{n+1}-b$
$\mathrm{S}(\mathrm{b}-1)$=$b^{n+1}-b$
$\mathrm{S}$=$\frac{b^{n+1}-b}{b-1}$

We then apply this general relationship to the two terms.

$\sum_{i=1}^n \frac{4^i}{8}$=$\frac{4^{n+1}-4}{3*8}$

We simplify and get

$\frac{4^{n+1}-4}{24}$ = $\frac{(4^{n}-1)}{6}$ = $\frac{2^{2n-1}}{6}$
$\sum_{i=1}^n \frac{2^i}{4}$ = $\frac{2^{n+1}-2}{1*4}$
$\frac{2^{n+1}-2}{4}$ = $\frac{2^{n}-1}{2}$ = $\frac{3*2^n-3}{6}$

Then we add the two terms.

$\frac{(2^{2n}-1)+(3*2^n-3)}{6}$
$\frac{(2^{2n})+(3*2^n-4)}{6}$
$\frac{(2^n+4)(2^n-1)}{6}$

The following equation gives the loss function for folding a paper in half in one direction.

Eq. 1        $\frac{\pi t * (2^n+4)(2^n-1)}{6}$

##### Alternative Direction

The alternative direction derivation is complicated because there are two folding limits one for the width and one for the length. For the last fold to be achieved the previous fold's length again has the be $\pi$ times the thickness. The sides of the fold are $\pi t 2^{(n-1)}$ and the thickness at the soon to be last stage is $t 2^{(n-1)}$. The total area of all sheet is the area of a single sheet times the number of sheets.

$\mathrm{A} \mathrm{r} \mathrm{e} \mathrm{a}$ = ${\pi t 2^{(n-1)}}^2$

We then multiply the area by the number of sheets. The number of sheets is $2^{n-1}$.

The final area is $\pi^2 t^2 2^{3n-3}$

Taking the square root of the area will give the limiting width of the original sheet.

• W=Limiting width
• t=Thickness of material
• n=Number of folds

Equation 2 refers to the limiting paper width based on the last fold.

Eq. 2        W=$\pi t* 2^{3(n-1)/2}$

The equation though isn't completely accurate because it does not include the materials lost in the radii of previous folds. For odd number of folds there is lost if the paper is folded in the odd fold direction. The odd fold do not contribute to the loss in the even fold direction. For large number of folds there is a large gap in accuracy. In practice a sheet of material could not fold that many times or could be that tight. Equation 2 accurately demonstrates the limiting width in reality though.

#### Limitations

In order to compute the length of the paper required and the thickness of each sheet it is important to understand what exactly limits the number of folds.

• Smoothness

In particular, the smoothness of the paper determines if the paper can slide around the curved sections. As the thickness increases so does the stiffness and the resistance to folding. At this point no human or machine effort can produce another fold. The stiffness issue occurs when the folded section is length is less than π times the thickness. In order to do another successful fold the length to thickness ratio has to be greater than π.

Another important factor is the difference in length of layers. At the round ends each layer has a different radius and circumference. This can happen when paper is pushed out the folded section and is not included in the curved section. The radius section then takes the majority of the paper’s volume until it reaches a point from making the folded materials into a semi-circle.

• Folding Technique

After each fold, it becomes more difficult to set each layer flatly on top of each other. This happens when layers are not wrinkle free and lumps begin to form in the inner portions. From this, paper extends beyond the folded section.

• Miscellaneous
1. The strength of a person can affect how many fold are achievable. The stronger the person the flatter the edges.
2. The paper is not uniform.
3. There's a difference in the limited number of folds based on whether the paper is being folded in the single direction or the alternative direction.

#### Related Problem

Rather than how many folds are achievable, a similar problem to the bed sheet problem focuses on the thickness of the sheet of paper. If you were to take a large sheet of paper with thickness 1/400 inch and fold it in half 50 times, how tall would it be (if it was theoretically possible to fold a sheet of paper 50 times)? Most people in their minds would imagine it to be as thick as a phone book. Some might even being daring and say a few feet. After 3 folds the sheet of paper would be as thick as your fingernail. And, if you continue to fold until the paper is at 50 folds then it will be as thick as 40 million miles. The paper would be able to reach the sun and return back to earth. This sounds ridiculous, but it isn't.

This image shows that on the 42nd fold the stack of paper will reach the moon. At such a rate the thickness of the stack of paper can become very large.

# Why It's Interesting

This problem is interesting because Gallivan showed that any person can do the impossible. She solved a problem that many mathematicians attempted but also failed to solve. It is interesting that a high school student with such fervor demonstrated that mathematics is a puzzle for everyone for all kinds of ages. Gallivan showed that a person should not accept anything to be true without evidence.

Gallivan has inspired others to break the record for most folds. Late April 2011 some students from Massachusetts claimed the world record in achieving 13 folds. They used the thinnest type of toilet paper that stretched over 2 miles. To hear more on their story click here.

# References

Folding Paper in Half Twelve Times. Retrieved May 17 2011, from http://pomonahistorical.org/12times.htm
Kruszenlnicki, Karl. Folding Paper Science Bits Retrieved May 23, 2011, from http://sciencebits.wordpress.com/2008/09/06/folding-paper/
Gallivan, B. C. 2002. "How to Fold Paper in Half Twelve Times: An 'Impossible Challenge' Solved and Explained." Pomona, CA: Historical Society of Pomona Valley.
Folding--from wolfram MathWorld. Wolfram Mathworld: The Web's Most Extensive Mathematic Resource. Retrieved May 17 2011, from http://mathworld.wolfram.com/Folding.html

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