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A4 Paper: The 2\sqrt{2} Ratio Secret Guide

MathematicsBeginnerReading time: 3 min

Overview

The dimensions of A4 paper (297mm×210mm297mm \times 210mm) are not arbitrary numbers; they are a perfect combination of mathematical beauty and industrial standards. The core secret behind it lies in the 2 \sqrt{2} ratio (Silver Ratio). This unique ratio ensures that the paper maintains its aspect ratio when cut in half, forming a perfect self-similarity.

Background

If the paper ratio is not 2\sqrt{2}, such as 1:11:1 (square) or 3:23:2, the shape will change drastically after folding. Only the 2\sqrt{2} ratio ensures "similarity after folding", which allows content of the same layout to be scaled freely without distortion.

Background

  • 1786: German scientist Georg Christoph Lichtenberg first proposed the advantages of the 2\sqrt{2} ratio in paper folding in a letter to a friend.
  • 1922: Walter Porstmann formulated this concept into the German DIN 476 standard, establishing the A-series, B-series, and C-series paper sizes.
  • 1975: This standard was officially adopted as the international standard ISO 216 and is currently used by most countries in the world (except North America).

Key Concepts

Silver Ratio

LengthWidth=2\frac{\text{Length}}{\text{Width}} = \sqrt{2}

The ratio of the length to the width of a rectangle is 2≈1.414\sqrt{2} \approx 1.414. This is the only rectangular ratio that maintains its original aspect ratio after being cut in half.

Self-Similarity

An∼An+1A_n \sim A_{n+1}

The local shape of an object is similar to the whole. Whether you look at the largest A0 or the smallest A8, their "shape" (aspect ratio) is exactly the same.

ISO 216 Standard

Area(A0)=1m2Area(A0) = 1 m^2

International paper size standard based on German industry standard DIN 476. It specifies that the area of A0 is 1m21m^2 and the aspect ratio is 2\sqrt{2}.

Formulas & Derivation

Constant Aspect Ratio Derivation

LW=WL/2  ⟹  L2=2W2  ⟹  LW=2\frac{L}{W} = \frac{W}{L/2} \implies L^2 = 2W^2 \implies \frac{L}{W} = \sqrt{2}
Assuming the original aspect ratio is equal to the aspect ratio after folding (original width becomes new length, half of original length becomes new width), solving the equation yields 2\sqrt{2}.

Area Recurrence Formula

Area(An)=12n×1m2Area(A_n) = \frac{1}{2^n} \times 1 m^2
The area of A0 is 1, and for each increase in number, the area is halved.

Experiment Steps

  1. 1

    Observe the Whole (A0)

    At the beginning of the experiment, a complete A0 paper is displayed. Please note that its area is a standardized 1m21m^2. Can you observe roughly what the ratio of its long side to its short side is?
  2. 2

    First Split (Observe Shape Change)

    Click the "Split" button to cut A0 in half into two A1 sheets. Please observe carefully: Does the relative relationship between the long and short sides of the newly generated A1 paper look very similar to the original A0 paper?
  3. 3

    Recursive Division (Finding Patterns)

    Continue clicking "Split", from A1 to A2, then to A3, A4. As the paper gets smaller, please pay attention to the "Aspect Ratio" value in the right control panel. Has this value ever changed significantly?
  4. 4

    Ratio Verification

    Continue splitting the paper and observe the aspect ratio value in the control panel. No matter what level you split to (up to A6), what pattern did you find? Think about it, what condition must a ratio satisfy to achieve this "unchanged shape after folding" effect?

Learning Outcomes

  • Deeply understand the core role of the 2\sqrt{2} ratio in paper size standardization.
  • Intuitively feel the self-similarity and recursive division process of geometric figures.
  • Understand why the image is neither stretched nor left with white space when a copier reduces A3 to A4.

Real-world Applications

  • Copier Scaling: When reducing two sheets of A4 paper side by side onto one sheet of A4 paper, or reducing A3 to A4, the scaling ratio is exactly 71%(1/2)71\% (1/\sqrt{2}), and the content is perfectly filled without distortion.
  • Paper Weight Calculation: Since the area of A0 is 1m21m^2, if the paper density is 80g/m280g/m^2, then one A0 sheet weighs 80g80g. One A4 is 1/161/16 of A0, so the weight can be calculated by simple division (5g5g), which is very convenient for postage calculation.
  • Engineering Drawing and Microphotography: Standardized scaling and archiving of technical drawings rely on this invariance of aspect ratio.

Common Misconceptions

Misconception
The size of A4 paper is an integer (e.g., 30cm x 20cm).
Correct
No. The size of A4 297mm×210mm297mm \times 210mm is to get the closest integer millimeter value to the 2\sqrt{2} ratio, which is an approximation of an irrational number ratio.
Misconception
US Letter paper is also 2\sqrt{2} ratio.
Correct
No. The ratio of Letter paper (8.5×118.5 \times 11 inches) is about 1.291.29. After folding, the shape becomes wider and cannot be perfectly reduced like A4.

Further Reading

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