Fold a 0.1 mm sheet 42 times and the stack passes the Moon. Drag the thickness across its whole range and that number barely moves. So which slider is doing the work?
In January 2002 Britney Gallivan folded a sheet of paper in half twelve times. She was a high-school student in Pomona, California. The standing claim was that no sheet folds more than seven or eight times. She worked out her own equations for the limit first, then went and did it.
A second story about paper runs the other way. Fold a sheet forty-two times, people say, and the stack reaches the Moon. Both stories are about one sheet. Both are arithmetic anyone can check.
They are about two different quantities. One is thickness and the other is length. The chart and the rail below follow the thickness; the length shows up once, as a single readout.
There are two sliders. Starting thickness sets how thick the flat sheet is. Folds sets how many times it is folded in half. Three buttons above them jump to preset settings, and one more control, Vertical axis, draws the same numbers two ways.
At 0.10 mm and 42 folds the model puts the stack at 4.398 × 10⁸ m. That is about 439,805 km, which the readout rounds to 439,800 km. The Moon is at 384,400 km, so the stack is past it. Fold 41 falls short, at 2.199 × 10⁸ m.
Now leave Folds alone and drag Starting thickness from end to end. The whole range is a factor of ten. The crossing moves from fold 43 to fold 40, and no further. For the Moon, ten times the paper buys three folds.
So the model does not make this a fact about paper being thin. In this model the answer is moved mostly by the number of folds; across its whole range, the sheet you start with moves it by three.
Nothing speeds up along the way, either. The doubling rate at fold 3 and at fold 43 is the same. What changes is the axis under the curve, and here is how to read it.
Almost everything on screen is one number — the height of the stack — drawn three ways. The Length needed card is the one figure from a different formula, explained below. The large number beside the badge gives the height. A badge names the last thing the stack has grown past, or the first one it has yet to pass. Drag Folds and the number doubles at every step. The badge moves far more slowly. It changes only when the stack crosses a marked height, up or down.
Under the readout, the rail puts that same height on a fixed ruler. The ruler is marked in powers of ten and never rescales. A position on it means the same thing at every setting. Each power of ten on it is ten times the one before; on a narrow screen only some of them carry a label. So every fold slides the marker the same distance. The ticks above the rail are the flat sheet and each fold made so far. The dots are the marked heights, filled in once the stack has passed them.
The chart plots that same height against fold count. The Vertical axis toggle is the point of the page. On Linear the axis has to reach the largest value on screen. Once the fold count is high, the early folds are pinned flat against zero. Many of those values are not small: at the default setting, fold 30 is already 107 km. They are small next to the last one. On Logarithmic the same numbers become a straight line with no bend anywhere.
The table at the end lists the first fold that passes each height. One line of arithmetic produces all of it.
A fold doubles the number of layers. That is the whole model.
The layer count is what doubles. Thickness follows, because the layers sit on one another. Nothing else enters: not the width of the sheet, not the material, not the direction of the folds.
Multiplying t0 by ten is worth log2(10) ≈ 3.32 folds. The whole thickness slider is a factor of ten from end to end. So it moves most answers by three folds. Sometimes four, where that leftover 0.32 crosses one more doubling: a standing adult, a 10-storey building and Mercury's orbit shift by four. The grain of sand shifts by one, because the thickest sheet is already past it.
Each fold adds the same amount to log2(t). The exponent grows in a straight line. The thickness does not. A logarithmic axis is spaced by the exponent, so the model draws a ramp with no bend. A linear axis is spaced by the thickness, so the same numbers look like nothing and then a wall. The exponent is easier to trust once one sheet is counted by hand.
Take the default sheet, 0.10 mm, and follow one number.
Seven folds gives 2⁷ = 128 layers. That is 128 × 0.1 mm = 12.8 mm, about a paperback. Fifteen folds is 32,768 layers, or 3.28 m, the first fold past a standing adult. Twenty-seven folds reaches 13.4 km and clears Mount Everest at 8.85 km. Thirty-two folds reaches 429.5 km and passes the model's 400 km mark for the space station.
Each fold is the one before it doubled. Fold 15 and fold 27 differ by a factor of four thousand. On a linear chart that must also fit fold 42, the two sit in the same row of pixels.
Ten more doublings from fold 32 gives fold 42: 429.5 km × 2¹⁰ ≈ 439,805 km.
The exponent gets there in one step and lands on the same number. The readout shows 4.398 × 10⁸ m, and the Layers card shows 4.4 trillion. That is 2⁴² sheets standing on each other. All of it rests on four rules.
Every thickness on this page follows these four. Drop any of the first three and the thicknesses worked out above stop meaning what they say.
No simulation. Every thickness is that one line of arithmetic worked out directly. Each multiplication is by a power of two. Only the exponent of the stored number changes, and no rounding error builds up.
The eighteen heights. The marks run from a grain of sand to the Sun. There is one for every power of ten in between. Several are set by a standard, and the model uses the standard's figure as written, among them:
Where no body has fixed it, the mark says so on screen. A grain of rice is a round 6 mm and varies by variety. The space station is entered at about 400 km. At the default sheet, a mark entered a little differently can move its fold count by one. A grain of rice longer than 6.4 mm takes a 7th fold, and a station above 429.5 km takes a 33rd. Length is a different matter.
Whether the folding is possible. This is the largest gap on the page. It is not a correction but a different model. The formula Gallivan derived for folding in one direction is
L is the length of material needed and t its thickness, in the same units. At 0.10 mm that puts twelve folds at 879 m of paper. Forty-two folds need 1.013 × 10²¹ m. That is around a hundred thousand light years.
The two formulas take the same sheet and the same fold count. Neither one's answer feeds the other. The thickness model has no length term at all. It will report fold 42 for a sheet the size of a postage stamp. Nothing here argues that a sheet can be folded any particular number of times.
Folds that alternate. Folding along alternate sides, rather than always along the same side, is a separate case. Later work has derived lost-length formulas for it, on square and rectangular paper. Only the single-direction case appears here; that extension is not in this model.
The folklore, and the slider. No primary source was found for the seven-fold claim. This page does not attribute it to anyone. The Folds slider stops at 50. Far past that, the stack outgrows what a stored number can hold.
Every computed figure here comes from the two formulas above, worked out directly rather than simulated. How I check things.
Sources.
Scale
Path loss by band and wall material. With the same power and antenna gain the model puts 5 GHz below 2.4 GHz with no walls at all, and the wall decides which way the gap moves from there.
7.07 dB the model’s 5 GHz shortfall with no walls, at equal power and antenna gain
4.82× the model’s swing in one-way delay to Mars, closest against farthest
Chance
Move the door count and the number of doors the host opens. The model puts switching ahead of staying by the doors you did not pick, divided by the doors you could still swap to — two to one at three doors. Opening more doors raises that edge; adding doors without opening more lowers it.
0.667 the model's switch win rate at three doors