How many times can a sheet of paper be folded?

How many times can an A4 sheet be folded?

Six, by Gallivan's formula, if every fold goes the same way. An ordinary A4 sheet weighs 80 g/m² and is about 0.1 mm thick. After six folds the packet has 64 layers, is 6.4 mm thick and at most 4.6 mm long. A seventh fold would need a strip at least 87.8 cm long, and the sheet is 29.7 cm.

If you alternate directions, once along the length and once across the width, Gallivan's formula for a square sheet asks for sides of at least 16.1 cm for seven folds, and A4 is 21 cm on its short side. The formula gives a minimum, so in practice a sheet may give up earlier.

The recognised record is 12 folds. It took a roll of paper 1,219 m long.

Who folded a sheet of paper 12 times?

Britney Gallivan, a high-school student from Pomona, California. In a maths class in her junior year, the third of four years of American high school, a challenge went out to the class: fold anything in half 12 times and earn extra credit. People said it could not be done more than seven or eight times.

She began with a square of gold foil, 10 cm across and 0.28 micrometres thick, and managed to fold it 12 times. She was then asked for paper. In December 2001 she worked out the formula that says how long a strip must be for a given number of folds.

On 27 January 2002 she unrolled a 4,000-foot (1,219 m) roll of toilet paper in a shopping mall in Pomona, with her parents, and folded it 12 times. She has said it took eight hours of crawling along the floor. Guinness World Records recognises her as the first person to fold a single sheet 9, 10, 11 and 12 times.

For 12 folds, 1,219 m of paper is enough only if the paper is at most 0.139 mm thick. Gallivan searched for a special toilet paper that would suit; its thickness is not published.

Others have tried for more. In December 2011, pupils at St. Mark's School in Massachusetts announced 13 folds, at MIT, with rolls of toilet paper taped end to end. NPR called the record unofficial, and Gallivan objected that taped paper is not a single sheet. The Guinness World Records page still shows her 12 folds as the record.

Why does the paper needed grow almost fourfold with every fold?

At each fold the paper bends around the stack. Every layer at the edge wraps around all the layers inside it, like a loop, and the thicker the stack, the longer the loop. The paper in the loop no longer lies along the stack.

Gallivan added up loop after loop. For n folds in the same direction, the strip must be at least L = (π · t / 6) · (2ⁿ + 4) · (2ⁿ − 1), where t is the thickness of the paper. The stack has 2ⁿ layers, but 2ⁿ appears twice in the formula, so the paper needed grows like 4ⁿ: at each step the thickness doubles and the strip grows almost fourfold. Gallivan puts it the same way: the next fold needs four times the length (Guinness World Records, 2025).

The last folds use almost all the paper: the final fold alone takes about three quarters of the strip.

What if you fold in the other direction too?

You can, and it gets more folds out of the same paper. Alternating directions, once along the length and once across the width, Gallivan derived for a square sheet a minimum side of W = π · t · 2^3(n−1)/2, with the exponent 3(n−1)/2. The side grows by almost three times per fold, not four as the strip does, and the stack still doubles.

For A4 the formula gives seven alternating folds: the seventh needs a side of 16.1 cm and the short side is 21 cm; the eighth needs 45.5 cm. In one direction it is six. Press “Alternating” in the sheet above to make them.

For 12 folds, a square sheet 29.1 m across would be enough by the formula, against a strip 879 m long. A square as wide as the Earth's circumference (40,075 km) lasts for 25 folds, and one as wide as the way to the Moon (384,400 km) for 27 folds, making a stack 13.4 km tall.

Alternating does not save paper: the sheet comes out shorter and wider. For 12 folds the square 29.1 m across has an area of 848 m²; a strip 879 m long and 10 cm wide has 88 m². The length needed falls and the area grows.

Even so the stack does not reach the Moon with paper we can make: for 42 folds the square sheet would be 0.11 light-years across, about 6,800 times the distance from the Earth to the Sun. The sheet above starts with one direction because Gallivan's 2002 record was in one direction, and the 107,052-light-year strip is the one-direction figure.

Does the stack reach the Moon? And the paper?

The stack does. At 42 folds there are 2⁴² layers of 0.1 mm, which is 439,805 km, more than the 384,400 km between the Earth and the Moon (the mean distance, according to NASA).

The paper does not. In one direction, the formula asks for a strip 107,052 light-years long for 42 folds; alternating directions, a square sheet 0.11 light-years across. The disc of the Milky Way is about 100,000 light-years wide, according to ESA and NASA; a 2018 study puts it at about 200,000, but the strip stays in the same order of magnitude.

Turned round: a strip as long as the distance to the Moon (384,400 km) lasts for 21 folds and gives a stack 210 m tall, lower than the Eiffel Tower. One as long as the Earth's circumference (40,075 km) lasts for 19 folds and gives 52.4 m.

The result depends directly on thickness. With paper 0.05 mm thick, the strip for 42 folds would be 53,500 light-years long and the stack only 220,000 km, too little for the Moon. At 0.12 mm the strip would be 128,000 light-years.

Trying it on a real sheet

1. Take a sheet of A4 printer paper.
2. Fold it in half along its length, then again and again in the same direction, pressing each crease hard with a fingernail.
3. At the end, count how many folds you managed.

The formula assumes ideal paper, so a real sheet may give up before the limit.

Sources and limits

The figures on this page are calculated with Gallivan's formula, for paper 0.1 mm thick, with perfectly tight layers. They are for one direction, except in the section on other directions and in Alternating mode, which use her formula for a square sheet. Two manufacturers' data sheets give 0.105 and 0.110 mm for 80 g/m² paper; I use 0.1 mm.

• The formula: Wolfram MathWorld, Folding and OEIS A076024.
• The loops behind the formula: Korpal, G., “Folding paper in half”, At Right Angles 4(3), November 2015, in the copy on Fermat's Library.
• The record: Guinness World Records, Most times to fold a piece of paper (also the formula for alternating directions) and Guinness News, 22 October 2025.
• The gold foil, the roll and the mall: ABC Science, Karl Kruszelnicki, 21 December 2005.
• The belief in a limit of seven or eight folds: Ivars Peterson, Science News.
• The 13 folds of December 2011 and Gallivan's objection: MySouthborough, 12 December 2011 and NPR (WWNO), 5 December 2011.
• The class, the mall and the hours: Damn Interesting, 2006.
• The Moon: NASA, Moon Fact Sheet. The Milky Way: ESA, Gaia and NASA, Imagine the Universe; the 200,000 estimate: IAC.
• Paper thickness: data sheets from Navigator (110 µm) and HP Office (105 µm); the thickness of Gallivan's paper is not published.
• Landmarks: the A4 sheet, ISO 216; the bus, Solaris Urbino 12; the tennis court, ITF; the football pitch, UEFA; the Eiffel Tower (330 m), Radio World; the Burj Khalifa, CTBUH; the Golden Gate Bridge, goldengate.org; the marathon, World Athletics; Moldoveanu Peak in Romania's Făgăraș Mountains, Radio Romania; Everest, Al Jazeera; the edge of space, FAI; the space station, NASA; geostationary orbit, NASA; the Earth's circumference, Caltech IPAC; the astronomical unit and the light-year, IAU.