Everyone has been dealing with paper for many years, and I don't know how much paper has been folded, and even all kinds of patterns can be folded. But have you tried another origami challenge?
Take a piece of ordinary A4 paper, fold it in half and then fold it in half, how many times do you think you can fold it in half at most?
Blind guess: It shouldn't be a problem to fold in half a dozen times, right?
However, when you try it out, you'll find:
After the A4 paper is folded in half 7 times|Courtesy of the author
It is difficult to fold an ordinary A4 paper in half more than 7 times!
7 times, is it really the upper limit of the origami challenge? Is it possible to break through this limitation? Actually, it's a simple math problem
01 Mathematical description of origami
Assuming that the original length, width, and thickness of the paper are A, B, and T respectively, the changes in the dimensions of the three after folding n times are shown in the table below.
The relationship between the number of folds of A4 paper and the size of the paper|Courtesy of the author
As you can see in the table, the length or width is halved for each half fold, while the thickness is doubled. There is a power of 2^n relationship between them, which means that the size relationship changes very quickly. Many stories involve power-of-2^n relationships. For example, the salary problem: you will be given 1 point on the first day, and it will be doubled every day from the second day, and the total salary will exceed 10 million yuan after a month. Another example is the king's checkerboard rice grain problem: the first grid is 1 grain of rice, and each subsequent square is doubled, and when the 64 squares are finished, the entire kingdom and even the whole world will be hollowed out.
Chessboard rice grain multiplication relationship|Courtesy of the author
Returning to the origami problem, when folded in half for the nth time, the thickness of the paper becomes 2^n times the original. A sheet of A4 paper is about 0.1 mm thick, and if folded in half 23 times, it will reach a thickness of about 839 meters, and the tallest building on earth, the Burj Khalifa, is only 828 meters high.
The tallest building in the world|Source: Photo.com
When folded in half 42 times, the thickness of the paper will reach 439805 km, which is already more than the average distance between the Earth and the Moon (384,403.9 km). If it continues to fold in half to 58 times, the thickness will exceed the diameter of the solar system (about 30 billion kilometers); Fold it in half 83 times and you can rush out of the Milky Way (about 160,000 light-years in diameter).
Maybe one day we can reach the sun with origami, land on Mars, and the little A4 paper will be able to look at the universe mathematically. If it is folded in half, there will be no need for Chang'e-6 at all. However, such an obvious big bug will naturally be limited by the "rules of the universe", and this rule is mechanics and materials.
02 The Universal Rules of Origami - Mechanics and Materials
In reality, regular A4 paper can usually only be folded in half 7 times. After 7 folds in half, the theoretical size of the paper becomes:
Length: 18.6 mm
Width: 26.3 mm
Thickness: 12.8 mm
At this time, the thickness size is the smallest, and it does not seem difficult to fold it in half once. However, after folding it in half once, the thickness increases to 25.6 mm, which is much larger than the length and width of the size, and it is easy to understand that it is not easy to bend at this time. In reality, however, the "rules of the universe" intervened much earlier. After folding it in half 7 times, the appearance of the whole origami looks very "twisted", there are many warps in many places, and the gaps between the layers of the paper have become larger. At this point, it becomes very difficult to fold it in half for the eighth time. How exactly does this explain?
Reason 1: The rule of force
As the number of origami increases, the size of the paper becomes smaller and the thickness increases. From a mechanical point of view, the smaller the size and the thicker the object, the greater the force required to bend.
On the one hand, the reduction in size reduces the bending moment generated by the force, and more force is required to bend the paper. On the other hand, the increase in thickness makes the laminated paper more rigid and less prone to deformation.
Comparison of bending forces|Courtesy of the author
I is the moment of inertia, b is the width of the section, h is the height of the section Reason 2: the rule of size
Clever students may have noticed some clues: if the A4 paper becomes thinner, the folded paper will also be thinner. Then according to the rules of force, the thinner laminated paper is less stiff and easier to bend. Indeed, if a thinner napkin is used, it is relatively easy to fold it in half 8 times. However, the thickness of the paper cannot be thinned indefinitely, so choosing a larger size paper can also solve this problem. Before you know it, origami becomes a challenge.
The first origami record was made in 2002. At that time, a high school junior named Britney Gallivan in California, USA, successfully folded a piece of paper in half 12 times in a row, setting a Guinness World Record.
图源：Science Engineering Technology
In 2012, students at St. Mark's High School in Texas, USA, folded a piece of paper more than 4 kilometers in half 13 times. Due to the use of many ancillary facilities in the last fold, this record was not recognized by Guinness.
13 records that have not been recognized|Source: Baijiahao
The famous "Mythbusters" team spliced the paper into a large paper of 55 x 75 meters, and with the help of a road roller, finally folded it in half 11 times.
The Mythbusters team completed 11 folds in half|Source: YouTube
Reason 3: The rules of the structure
The paper is folded in half and stacked into a single piece, forming a structure similar to laminate. This interlayer-free structure bends so that the inner layer remains almost unchanged, while the outer layer is elongated. The thicker the paper, the more elongated the outer layer is, and therefore the greater the external force required. If the layers are completely independent, it will not be enough to resist the bending force by friction alone, and it is easy to produce interlayer slippage after bending. In the case of intact paper, the creases are connected to each other, and this connection force, combined with the friction between the layers, makes it difficult for the layers to slide and requires greater force to overcome the bending force.
Even, when the paper is thick enough, the pulling force of the outer layer of paper generated by bending will exceed its tensile limit, causing the paper to crack and the challenge fails. In addition, the single layer of ordinary A4 paper also has a certain structural strength, which is why softer napkins can be folded in half many times. If it is the same soft thin silk, it can also be easily folded in half more than 7 times.
The thin and soft silk is folded in half 8 times|Courtesy of the author
The multiplication relationship of the number of folds may break through your imagination, and no more than 100 folds can easily break through the galaxy and look at the universe. However, the rules of the universe limit the folding limit of A4 paper. With the increase of the number of folds, the required external force increases sharply due to the decrease of the force arm and the increase of the crease connection force. To fold in half more times, a thinner, softer, larger flat material is required.
Hurry up and fold the paper in half to try, work hard, maybe the new Guinness World Origami record holder is you!
Author: Wang Yongjian, associate professor of Nanjing Agricultural University, deputy secretary-general of the Science Popularization Committee of Jiangsu Mechanics Society, and chief science and technology communication expert of Jiangsu Province
Curated & Edited: Little Dandelion
Cover image source: TED-Ed