How many times must you fold a paper to 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.

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.

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.

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.

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.

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.

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.

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.

Fold it once, twice, and then three times, and it becomes 0.2, 0.4, and then 0.8 mm thick.

This animation features satellite images of the far side of the Moon, illuminated by the Sun, as it crosses between the DSCOVR spacecraft’s Earth Polychromatic Imaging Camera (EPIC) and telescope, and the Earth — one million miles (1.6 million km) away. The far side of the Moon is vastly different from the near side. The Moon itself is located an average of 384,000 km away from Earth, but due to its elliptical orbit, can get more than 20,000 km closer or farther than that figure.

To reach the Moon, however, it must become at least 356,000 km thick.

This unfamiliar view shows the size of the Earth and Moon, plus the distance from the Earth to the Moon, to actual scale. The Earth is about 12,700 km in diameter with the Moon being a little over a quarter of the Earth’s size, but the present Earth-Moon distance averages out to an enormous 384,000 km: just over 30 times the Earth’s diameter.

That’s a factor of 3.56 trillion thicker than a single sheet of paper.

To fold a paper as many times as possible, the best strategy is to take the largest single sheet of paper you can access and fold it upon itself as many times as you can.

Each successive fold cumulatively doubles its previous thickness.

The greater the number of times you fold a piece of paper onto itself, the greater the thickness and number of sheets will be, with each fold further doubling the thickness and number of sheets in the stack over the prior fold.

After 10 folds, its thickness increases by a factor of 1024 (just over 1000).

Each successive folding of a paper (or set of papers) will double the number of sheets in the stack, and will double the thickness of the stack from the prior fold. Once 10 folds have accumulated, the number of sheets will have risen from 1 to 1024: since 2^10 = 1024.

After 20, 30, and 40 folds, the paper becomes over a million, a billion, and then a trillion times thicker than the original.

The exponential function, e^x, where e is the transcendental number that is the base of natural logarithms, is the only function whose slope at every point along the curve, as shown here, is equal to the value of the function itself. Numbers other than e can be exponentiated as well, and although “exponential growth” is a property common to all exponentiated numbers greater than 1, but the slope of such a curve will not be exactly equal to the value of the function.

Only 42 folds equates to a thickness of 439,804.6511104 km: enough to reach the Moon.

Although the Earth might be large and massive compared to the Moon, both bodies are very small compared to the distance between them. It takes about 1.25 seconds for light to travel one-way from the Earth to the Moon, and the Earth-Moon separation is about 30 times the Earth’s diameter. A paper folded 42 times would be thicker than the distance that separates the Earth from the Moon, even at its farthest.

Mostly Mute Monday tells an astronomical story in images, visuals, and no more than 200 words. Thanks to Lewis & Clark College’s Prof. Michael Broide for teaching the author this lesson in 2009.