How many times must you fold a paper to reach the Moon? | by Ethan Siegel | Starts With A Bang! | Jan, 2024

If you fold a piece of paper in half enough times, you could eventually cover enough distance that you could reach the Moon. How many “folds” of this piece of paper would it take? (Credit: Adrian Paenza/TED/YouTube)

Each time you fold a piece of paper, you double the paper’s thickness. It doesn’t take all that long to even reach the Moon.

The Moon is the closest natural object to Earth.

Japan’s Kaguya probe went to and orbited the Moon, which enabled magnificent views of the Earth seen over the lunar surface. Here, the Moon is photographed along its day/night boundary, the terminator, while Earth appears in a half-full phase. From the near side of the Moon, the Earth is always visible; both are the result of the aftermath of an early, giant impact between a Mars-sized protoplanet and a proto-Earth. (Credit: JAXA/NHK)

Its orbital distance ranges from 356,000 to 407,000 km.

A perigee full Moon compared with an apogee full Moon, where the former is 14% larger and the latter is 12% smaller than the other. The longest lunar eclipses possible correspond to the smallest apogee full Moons of all. At apogee, the Moon is not only farther and appears smaller, but also moves at its slowest in its orbit around Earth, and takes the longest amount of time for a round-trip signal to traverse that distance. (Credit: Tomruen/Wikimedia Commons)

Simply folding a paper in half enough times would eventually reach the Moon.

Whenever you take a sheet of paper and fold it in half, you double its thickness and double the number of sheets in the stack. This happens not only for the first fold, but for each subsequent fold, compounded atop all prior ones. (Credit: Sxeptomaniac/Wikimedia Commons)

But how many? The answer lies in the mathematics of exponential growth.

After one “doubling time” has passed, the initial population of any exponentially growing collection has increased by a factor of 2. After another doubling time, there’s another doubling, for a total factor of 4. After 16 doublings, the initial population would have increased by a factor of 2 to the 16 power, or 65,536. Exponential growth, whenever it occurs and regardless of whether it occurs in time, space, or by any other metric, is as catastrophic as it is relentless. (Credit: Paul Hewitt, NSTA)

To start, you first need to know how thick a single sheet of paper is.

A ream of paper is normally packaged in a stack of 500 sheets. Each such ream is normally about 2 inches, or 5 centimeters, thick. (Credit: Sage Ross (WMF)/Wikimedia Commons)

Paper is sold in reams of 500 pages, typically ~5 cm (2 inches) thick.

A single sheet of paper is quite thin: typically around just 0.1 millimeters (or 0.004 inches) thick. This is comparable to the thickness of a single human hair. (Credit: Tero Vesalainen/FreeRangeStock)

That implies a single sheet is ~0.1 mm (0.004 inches) thick.

Each time you fold a sheet of paper, you increase the number of sheets in the stack by a factor of two, while simultaneously increasing the thickness of the stack by two as well. (Credit: Fred the Oyster/Wikimedia Commons)

Each time you fold a piece of paper, you double its thickness.

A sheet of paper folded anywhere from one through six times, with the relative smaller area and increased thickness corresponding to the number of folds inherent to the paper. (Credit: Echo Romeo/Physics Buzz, ret. 2022)

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