Understanding the Science of Folding
Folding a piece of paper seems simple, but it is one of the most powerful demonstrations of exponential growth. For decades, it was a commonly held myth that a piece of paper could not be folded more than seven times. However, in 2002, high school student Britney Gallivan proved that with enough material and a specific folding strategy, it is possible to fold paper 12 times.
The Math of Exponential Growth
When you fold a sheet of paper once, you have 2 layers. Fold it again, and you have 4. By the third fold, you have 8. The formula for the number of layers is 2n, where n is the number of folds. This leads to a rapid increase in thickness. If you could fold a standard 0.1mm piece of paper 42 times, the stack would be thick enough to reach the Moon.
Britney Gallivan's Equation
Folding isn't just about thickness; it's about the physical length of the material required to complete the turn. For folding in a single direction, Gallivan derived a formula to determine the minimum length (L) of material required based on the thickness (t) and the number of folds (n):
L = (πt / 6) * (2n + 4) * (2n – 1)
This calculator utilizes this physics formula to estimate how many meters (or miles) of paper you would need to achieve a high number of folds.
Folding Milestones
- 10 Folds: The stack is roughly the width of a human hand.
- 17 Folds: The stack is taller than a 2-story house.
- 23 Folds: The stack is 1 kilometer thick.
- 51 Folds: The stack would reach the Sun.
function calculateFoldingMath() {
var t = parseFloat(document.getElementById('initialThickness').value);
var n = parseInt(document.getElementById('foldCount').value);
if (isNaN(t) || isNaN(n) || t <= 0 || n = 1000000) {
thicknessDisplay = (totalThicknessMm / 1000000).toFixed(4) + " km";
} else if (totalThicknessMm >= 1000) {
thicknessDisplay = (totalThicknessMm / 1000).toFixed(4) + " meters";
} else {
thicknessDisplay = totalThicknessMm.toFixed(2) + " mm";
}
// Gallivan Length Formula (in mm)
// L = (pi * t / 6) * (2^n + 4) * (2^n – 1)
var pi = Math.PI;
var part1 = (pi * t) / 6;
var part2 = Math.pow(2, n) + 4;
var part3 = Math.pow(2, n) – 1;
var lengthMm = part1 * part2 * part3;
var lengthDisplay = "";
if (lengthMm >= 1000000) {
lengthDisplay = (lengthMm / 1000000).toFixed(4) + " km";
} else if (lengthMm >= 1000) {
lengthDisplay = (lengthMm / 1000).toFixed(4) + " meters";
} else {
lengthDisplay = lengthMm.toFixed(2) + " mm";
}
// Display Results
document.getElementById('resLayers').innerText = layers.toLocaleString();
document.getElementById('resThickness').innerText = thicknessDisplay;
document.getElementById('resLength').innerText = lengthDisplay;
document.getElementById('foldResults').style.display = "block";
// Fun comparisons
var compText = "";
if (n >= 42) {
compText = "Comparison: This stack would reach the Moon!";
} else if (n >= 26) {
compText = "Comparison: This stack is taller than Mount Everest!";
} else if (n >= 14) {
compText = "Comparison: This stack is taller than an average human.";
} else if (n >= 7) {
compText = "Comparison: You have surpassed the traditional paper folding limit.";
} else {
compText = "Keep folding to see the power of exponential growth!";
}
document.getElementById('comparisons').innerText = compText;
}