How many times can you fold a standard paper?
You can fold a standard A4-sized piece of paper in half 7 times. An A4 piece of paper measures 8.3 x 11.7 inches (210 x 297 mm). You can fold it in half 7 times, but the 7th fold will be tough (you may even need a friend to help you bend and crease the paper stack). However, the total number of times you can fold any piece of paper in half depends on its size. Larger and/or thinner sheets of paper can be folded in half more than 7 times! A4-sized paper is the standard document size in most countries outside of North America. If you’re in the US, you can also only fold a standard 8.5 x 11 inch (216 x 279 mm) piece of paper up to 7 times.
What limits the number of folds you can make?
The thickness and length of the paper Basically, the longer and thinner the paper, the more times you can fold it in half. Each time you fold the paper, the thickness of the paper doubles while the length gets cut in half. Once the paper stack is thicker than the remaining length of paper, you can’t fold it anymore (there’s just not enough material). Example: Imagine a jumbo-sized roll of toilet paper rolled out into one very long “sheet” of paper about 170,000 inches (2.68 miles or 4.32 kilometers). Calculations show you could fold this sheet of paper 13 times! (More on how to calculate the maximum number of folds you can get below.)
Exponential growth Each time you fold the paper in half, its thickness doubles (for reference, a page of printer paper is usually 0.004 inches (0.1 mm) thick). When you start with such a slim sheet, doubling its thickness doesn’t seem so consequential at first. However, after just 10 folds, that same ultra-thin paper will be stacked as thick as your hand is wide. The crazy exponential growth of your paper stack just keeps going from there: After 20 folds, your paper is 10 kilometers (6.2 miles) thick (that’s taller than Mt. Everest). After 42 folds, your paper stack could reach the moon (that’s 238,900 miles, or 384,400 kilometers). After 51 folds, you’ve made it to the sun (91.97 million miles or 148.01 million kilometers). After 103 folds, you’ve created a stack of paper as thick as the entire observable universe (and that’s a staggering 93 billion light-years in diameter!).
The distortion of the paper Of course, if you try to fold paper continuously in half for real, the actual paper will fight you a bit. After a few folds, the creases aren’t so crisp and flat; they start to get thicker and more curved, making it harder to make another fold. The paper also becomes more rigid the thicker it gets, meaning it takes more strength or energy to fold it each time.
Equations to Find the Maximum Number of Folds
If you’re alternating the direction you’re folding… Use n = 0.96 l n ( w / t ) {\displaystyle n=0.96ln(w/t)} {\displaystyle n=0.96ln(w/t)} to solve for the maximum number of folds ( n {\displaystyle n} n) you can get based on the paper’s width ( w {\displaystyle w} w) and thickness ( t {\displaystyle t} t). This equation only works if you’re folding top-to-bottom, then side-to-side, then top-to-bottom, etc. (like you would for a regular sheet of printer paper). It also only works if w {\displaystyle w} w is the smaller of the paper’s 2 measurements and you start with a top-to-bottom fold. Mathematically, this means that you’re cutting the width in half every 2 folds. If we solve this equation for the dimensions of an 8.5 x 11 inch paper with a width of 0.004 inches: n = 0.96 l n ( 8.5 / 0.004 ) {\displaystyle n=0.96ln(8.5/0.004)} {\displaystyle n=0.96ln(8.5/0.004)} n = 0.96 l n ( 2125 ) {\displaystyle n=0.96ln(2125)} {\displaystyle n=0.96ln(2125)} n = 0.96 ( 7.66 ) {\displaystyle n=0.96(7.66)} {\displaystyle n=0.96(7.66)} n = 7.36 {\displaystyle n=7.36} {\displaystyle n=7.36}, or 7 possible folds (since you can’t make 0.36 of a fold, you round down to the nearest whole number). Math tip: l n {\displaystyle ln} {\displaystyle ln} represents the natural logarithm, which tells you how many times you need to multiply a value by itself to get a certain number.
If you’re folding in the same direction… Use n = 0.72 l n ( w / t ) {\displaystyle n=0.72ln(w/t)} {\displaystyle n=0.72ln(w/t)} instead. Imagine you’re folding up that very long and narrow roll of toilet paper from a previous example. It makes more sense to keep folding it in the same direction, rather than alternating like you would with a piece of paper where the dimensions are much closer together. This separate equation accounts for how the width is halved on every fold, rather than every 2 folds. If we solve this equation for the dimensions of a 170,000 inch roll of toilet paper with a width of 0.004 inches, we can verify that you can only fold that paper up to 13 times. Both of these equations were formulated by Britney Gallivan, who set the world record for the most number of folds in 2002.
The Paper Folding World Record
Britney Gallivan folded a paper 12 times in 2002, securing the world record. In order to achieve this, the then-teenaged Gallivan had to use a sheet of very thin tissue paper that was 4,000 feet (1,219 meters) long, or about ¾ of a mile! The idea came from an extra credit assignment where students were challenged to fold anything in half 12 times. After successfully completing the assignment with a piece of ultra-thin gold foil, Gallivan’s teacher changed the challenge to include folding a thicker piece of paper. The assignment is what pushed Gallivan to come up with the equations that identify how many times you can fold a piece of paper based on its width and thickness. The endeavor also made Gallivan the first person to ever fold a sheet of paper 9, 10, 11, and 12 times. If you’re curious about Gallivan and her experience mathematizing and folding paper, you can read about it in her booklet published by the Historical Society of Pomona Valley, How to Fold Paper Twelve Times.
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