Folding Paper in Half 12 Times Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen.
Through the Looking Glass, L. Carroll
Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using paper of any size or shape.
In April of 2005 Britney's success was mentioned on the prime time CBS television show Numb3rs.
The task was commonalty known to be impossible. Over the years the problem has been discussed by many people, including mathematicians and has been demonstrated to be impossible on TV.
For extra credit in a math class Britney was given the challenge to fold anything in half 12 times. After extensive experimentation, she folded a sheet of gold foil 12 times, breaking the record. This was using alternate directions of folding. But, the challenge was then redefined to fold a piece of paper. She studied the problem and was the first person to realize the basic cause for the limits. She then derived the folding limit equation for any given dimension. Limiting equations were derived for the case of folding in alternate directions and for the case of folding in a single direction using a long strip of paper. The merits of both folding approaches are discussed, but for high numbers of folds, single direction folding requires less paper.
The exact limit for single direction folding case was derived. It is based on the accumulative limiting effects induced by each and every fold in the folding process. Considering the intricacy of the problem the equation has a relatively simple form.
For the single direction folding case the exact limiting equation is:
where L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction.
L and t need to be expressed using the same units.
Stringent rules and definitions were defined by Britney for the folding process. One rule is: For a sheet to be considered folded n times it must be convincingly documented and independently verified that (2n ) unique layers lie in at least one straight line. Sections that do not meet this criteria are not counted as a part of the folded section.
Diagram showing part of a rotational sliding folding sequence
In some web pages the limits found by Britney are described as being due to thickness to width ratios of the final folds or attributed to the folder not being strong enough to fold any more times. Both explanations for the mathematical limits are incorrect and misses the actual detailed reason for the physical mathematical limit.
In one day Britney was the first person to set the world record for folding paper in half 9, 10, 11 or 12 times.
Some incorrectly talk about wetting paper to get more folds, squeezing the sheet of paper like a telephone book to half its thickness, or stretching the paper when wet or dry. Wetting the paper only allows it to tear easier. Tearing and cutting are not folding.
The Historical Society of Pomona Valley is now selling Britney's booklet. It contains over 40 pages of her account of solving the problem with interesting stories and has comments from others who have also tried to solve the problem. It shows the development of the limiting equations. The booklet gives a detailed explanation of what physically causes the limit and tabulates the number of fold limits for different size sheets.
The booklet tackles the problem both in detail and breath. It narrates interesting aspects and consequences of the problem such as how it relates to practical fractals..
and she is HOT!
