Britney Gallivan, who was at the time a junior in high school, solv...
In the larger perspective, the paper folding experiment connects to...
To understand better how the length is lost as we fold the paper le...
$$
L=\frac{\pi t}{8}(\sum_{i=1}^{n}2^{2i}+2\sum_{i=1}^{n}2^{i})
...
Another way to see this is:
$$
L=\sum_{i=1}^{n}\sum_{j=1}^{2^{i...
As we can see below the function $\frac{1}{6}(2^n+4)(2^n-1)$ grows ...
In 2012, students at St. Mark's School in Southborough, Massachuset...
If we don't consider the length lost in radii of earlier folds it m...
Alternate folding is special since folding in one direction doesn't...
At Right Angles | Vol. 4, No. 3, Nov 2015
9
Gaurish Korpal
Folding
Paper in Half
feature
Vinay Nair
Miles Please
When I was a small child I read the following claim about
paper folding: “It is impossible to fold any piece of paper in half
more than eight times no matter how big, small, thin or thick
the paper is” ([1], pp. 32-43). At that time I accepted the fact
after some experimentation. But last month, to my amazement,
I discovered that a grade 11 student in California, Britney C.
Gallivan, had mathematically disproved the above statement
in 2002 [1]. This article discusses the proof given by Gallivan
regarding “folding paper in half, n times”.
Say Crease!
Vol. 4, No. 3, November 2015 | At Right Angles 20
20 At Right Angles | Vol. 4, No. 3, November 2015
Gaurish Korpal
Folding
Paper in Half
When I was a small child I read the following claim about
paper folding: “It is impossible to fold any piece of paper in half
more than eight times no matter how big, small, thin or thick
the paper is” ([1], pp. 32-43). At that time I accepted the fact
after some experimentation. But last month, to my amazement,
I discovered that a grade 11student in California, Britney C.
Gallivan, had mathematically disproved the above statement
in 2002 [1]. This article discusses the proof given by Gallivan
regarding “folding paper in half, times”.
Introducing the Problem: Gallivan’s Rules
We �irst state the rules to be followed while folding a sheet of
paper in half, as enunciated by Britney Gallivan:
1. A single rectangular sheet of paper of any size and uniform
thickness can be used.
2. The fold has to be in the same direction each time. (Hence the
fold lines are all parallel to each other.)
3. The folding process must not tear the paper. (That is, it must
not introduce any discontinuities.)
4. When folded in half, the portions of the inner layers which
face one another must almost touch one another.
5. The average thickness or structure of material of paper must
remain unaffected by the folding process.
6. A fold is considered complete if portions of all layers lie in one
straight line (called folded section).
Keywords: Gallivan, paper folding, half fold, geometric progression, creep
section, radius section, series summation, quadratic equation
feature
Vol. 4, No. 3, November 2015 | At Right Angles 21
21 At Right Angles | Vol. 4, No. 3, November 2015
Following the rules, we claim that: The length of
the given sheet of paper decides the number of times
we can fold it in half. Thus if we have a sheet of
paper of given length, we can calculate the number
of times we can fold it theoretically (allowing a
reasonable amount of manpower and time).
HOW TO REACH THE SUN …ON A PIECE OF PAPER
A poem by Wes Magee
Take a sheet of paper and fold it,
and fold it again,
and again, and again.
By the 6th fold it will be 1-centimeter thick.
By the 11th fold it will be
32-centimeter thick,
and by the 15th fold - 5-meters.
At the 20th fold it measures 160-meters.
At the 24th fold - 2.5-kilometers,
and by fold 30 it is 160-kilometers high.
At the 35th fold it is 5000-kilometers.
At the 43rd fold it will reach the moon.
And by the fold 52
will stretch from here
to the sun!
Take a piece of paper.
Go on.
TRY IT!
Absolute folding limit
Geometric series. There is a short poem by Wes
Magee titled “How to reach the sun …on a piece of
paper” ([2], page 19), which illustrates the
geometric series involved in folding paper in half.
After folds (if possible), the width of the folded
paper will be approximately equal to the distance
between the sun and the earth!
Every time we fold the paper in half, we double
the number of layers involved. We have to fold
sheets of paper for the
th
fold. Thus for each
successive fold we need more and more energy.
Initially this was thought to be the reason for our
inability to fold a piece of paper more than times.
But, as stated earlier, the strength of the arm is
not the limiting factor for the number of folds.
Understanding folds. After each fold, some part
of the middle section of the previous layer
becomes a rounded edge. The radius of the
rounded portion is one half of the total thickness
of folded paper; see Figure 1.
Initially the radius is small as compared with the
length of the remaining part of sheet. As the folds
Figure 1. Paper folded in half 12 times illustrating the
decrease in folded section and increase in radius
section caused due to continued folding which leads to
fold losses [©Britney C. Gallivan]; source: [1], [4]
begin nearing their �inal thickness, the curved
portion becomes more prominent and begins
taking up a greater percentage of the volume of
the paper. The radius section is the part of the
paper ‘wasted’ in connecting the layers.
The section that projects past the folded section
on the side opposite the radius section is called
the creep (Figure 1). It is caused by the difference
in lengths of layers due to the rounded section of
the fold layers having different radii and
circumferences.
The limit to the number of folds is reached when a
fold has been completed but there is not enough
volume or length in the folded section of the
paper to �ill the entire volume needed for the
radius section of the next fold. Thus while making
folds there is loss of paper in the form of radius
section and creep section.
Limit formula. Since the radius and creep
sections are semicircular, the length to height
ratio of the paper being folded has to be greater
than to allow one more successful fold to occur.
If a folded section’s length is less than times the
height, the next fold cannot be completed.
Let be the thickness of a sheet of paper. On the
�irst fold, we lose a semicircle of radius , so the
length lost is (‘lost’ in the sense of ‘not
contributing to the length’). On the second fold,
we lose a semicircle of radius and another
semicircle of radius , so the length lost is
2
At Right Angles Vol. 4, No. 3, November 2015