The Thickness of Folded Notebook Paper Explained

Have you ever wondered how thick a single piece of notebook paper would become if it were folded in half 13 times? This article delves into the mathematical calculations and provides a detailed explanation of the phenomena involved.

Understanding the Basics

A standard piece of notebook paper is approximately 0.1 millimeters (mm) thick, which is equivalent to 0.01 centimeters (cm) or 0.00394 inches. This initial thickness is crucial when determining the thickness after multiple folds.

Folding Process and Thickness Calculation

Each time a piece of paper is folded in half, its thickness doubles. This doubling effect can be calculated using the formula for exponential growth: Thickness after n folds Initial thickness × 2n.

Let's calculate the thickness after 13 folds:

Initial thickness: 0.01 cm Folds: 13 Calculation: 0.01 cm × 213 0.01 cm × 8192 81.92 cm

Converting the thickness in centimeters to inches using the conversion factor 1 cm ≈ 0.393701 inches:

81.92 cm × 0.393701 ≈ 32.25 inches

Alternative Fold Calculation

Interestingly, if you were to consider the scenario where a piece of paper is 0.004 inches thick, folding it 13 times would result in a thickness of approximately 32.8 inches or 83.23 cm. This is a theoretical scenario because it's highly unlikely to achieve such an initial thickness with regular notebook paper.

Physical Constraints and Limitations

No matter how many times you fold a piece of paper, the result cannot exceed the maximum dimensions of the original sheet due to the limitations of paper properties. The formula for the minimum initial length L required to make n folds of thickness t is given by:

L (frac{2^{2n 1} pi t}{6})

Given this formula, if you aim to make 13 folds, the necessary initial thickness for an A4 sheet (21 x 29.7 cm or 8.3 x 11.7 inches) would be:

t (frac{6L}{2^{2n 1} pi})

For an A4 sheet, the initial thickness would be approximately 0.000000844720283 cm or 0.000000332567040 inches, which is 8.447 nanometers. This thickness is incredibly thin and far below the practical limits of paper.

For comparison, 10 nanometers is the thickness of the cell wall of gram-negative bacteria, highlighting just how thin this value is.

Conclusion

The thickness of a single piece of notebook paper after 13 folds is approximately 81.92 cm or 32.25 inches. This is a fascinating demonstration of exponential growth and a great example of the limitations imposed by physical properties.