V ol. 4, No. 3, November 2015 | At Right Angles 23

23 At Right Angles | V ol. 4, No. 3, November 2015

Figure 3. Britney Gallivan holding the first sheet of

paper ever to be folded twelve times [ © Britney C.

Gallivan]; source: [1], [4]

thickness  inch) would hav e been enough

to accomplish the task of folding paper in half

twel ve times. For a nice writeup on this episode,

please see [5]. See also [6].

Ex ercise

After the above analy sis, the following question

may arise: “How many times can we fold a sheet

of paper in half, by folding in alternate directions,

keeping all other rules the same?”

As expected, the answer is: The length and width

of the given sheet decide the number of times we

can fold the paper in half .

A good idea to proceed will be to start with a

square sheet of paper and calculate its limiting

width. Without considering the effects of

material lost in radii of earlier folds, we get a

crude bound for the width  of a square sheet of

paper of thickness  required for folding in half 

times as:

   





I invite readers to deriv e the above formula.

Surel y , the above formula does not giv e any

minimum limit . Using analy sis seen in one

direction folding , we can deriv e a Limit Formula

for alternate folding . But in alternate folding , we

get separate equations for odd and even folds.

Note that odd folds accumulate losses in an odd

fold direction, and even folds accumulate losses in

an even fold direction. Also, each fold in an odd

direction doubles the amount of paper for the

next even direction, and vice versa.

Refer ences

1. Britney C. Galliv an, How to Fold Paper in Half Twelv e Times , Historical Society of Pomona V alley Inc. (2002); see

http://pomonahistorical.org/12times.htm

2. Arvind Gupta, Hands on Ideas and Activities , Vigyan Prasar (An autonomous organization under the Department of

Science and T echnology , Gov ernment of India), ISBN : 87-7480-118-9 (2005)

3. http://mathforum.org/libr ary/drmath/view/60675.html

4. http://mathforum.org/mathimages/inde x.php/Bedsheet_Problem

5. http://www .abc.net .au/science/articles/2005/12/21/1523497.htm

6. W eisstein, Eric W . ”F olding. ” From MathW orld–A W olfram W eb Resource. http://mathworld.w olfram.com/F olding.html

4 At Right Angles  V ol. 4, No. 3, November 2015

GAURISH K ORP AL is an undergraduate student at National Institute of Science Education and Research (NISER),

Jatni; he is currently in his second year . He loves mathematics and is a regular blogger at gaurish4math.