From Earth to the moon with a piece of paper

The power of exponential growth

Grab a piece of paper and fold it from the center, do the same thing again, and again. How many times do you think you can fold a paper continuously?

Using mathematics, we can prove that an A4 paper can’t be folded more than 7 times. But what is the problem?

When we fold a piece of paper from the center, we halve its length and double its thickness. And that’s why after 7 times it is impossible to fold the paper.

In 2002, 16-year old Britney Gallivan discovered a formula that can be used to determine the maximum number of times a piece of paper can be folded continuously.

The formula is L = Πt∕6 (2^(n )+ 4) (2^(n) -1) in which is the L length of the paper, t is the thickness of it and n is the number of times it can be folded continuously. The reason we have pie in this formula is that each time we fold a paper a half-circle shape part of the paper is being used in the corners and it doesn’t give us a perfect square (as shown below)

She used this formula and folded a 1200 m long paper 12 times and breaks the world's record.

But what is the point of that, why folding a piece of paper is even important? Let assume that there isn’t any limit to the number of times we can fold a piece of paper and we can that as many times as we want.

Normal papers have a thickness of 10^(-2) cm, but let take a thinner one, like a newspaper with a thickness of 10^(-3) cm, let's start folding it. As mentioned, each time we fold a paper its thickness doubles, so after 10 times we will get a thickness of 10^(-3) * 2^(10) = 1.024 cm. After 17 times it will be 131 cm thick, and after 25 folds it will reach a thickness of 335m which is equivalent to the height of skyscrapers. After 30 folds it will reach a height in which airplanes fly or 11 km, after 40 continuous folds its thickness becomes 11000 km which is the distance of satellites from the earth. DO you think how many folds are required to reach the moon, the answer is quite shocking, with only 45 folds we can reach a thickness of 384000km which is the distance of the moon from Earth, and after 53 folds we reach the sun. To that end, can you imagine how thick the paper is going to be after 100 times? Probably not, because it is going to be longer than the whole discovered universe.

This type of growth is called exponential growth. In exponential growth and is in the form of where is a constant number, at first, the slope (dy∕dx) is quite small but when the value of x increases the growth in value of y becomes massive!!

The paper example is one of the slowest growing exponential terms since it grows with a factor of 2 each time and 2^x after x times.

Exponential growth has many applications in biology, physics, economics, finance, and computer science.

Link to computer science:

In computational complexity theory, an algorithm with an exponential order of complexity requires a huge amount of resources (time and memory) for little increases in the problems’ size. The reason for that is clear since we know how exponential growth work (the amount of resources required grows exponentially). To illustrate, imagine an algorithm with a complexity of 3^x, if this algorithm needs 5 seconds to a complete a problem with a size of 10, it needs 15 seconds for a problem with a size of 11, and so on (for each increase in the problem size, the resources required triples). To that end, algorithms with exponential order of complexity are not efficient, and an algorithm like the previous example will not be able to complete tasks with a size of more than 40, but most of the computers need to solve much larger problems.

On the other hand, algorithms with a complexity order of log(x), are considered to be quite efficient, and the reason is that the logarithm function is the inverse of the exponential function (a function g(x) is called to be inverse of f(x), if and only if g(f(x)) = x)

The graph of log(x) (base 2) (red)and 2^x (blue)are compared below:

Moore’s law:

Another interesting thing about exponential growth in computer science is Moore’s law, which states that the number of transistors in a dense integrated circuit double about every two years. Consequently, the number of transistors follow an exponential growth with a factor of two.

With the examples provided in this article, hopefully, we get a better understanding of exponential growth and what does it mean to grow exponentially.

Key takeaways:

· Definition of exponential growth

· How exponential growth works

· Exponential order of complexity (Computer science)

· Logarithmic order of complexity (Computer science)

· Moore’s law