Paper sizes, geometry and irrationality

Triggered by a comment on the dancing professor’s blog about weird British paper sizes, I had to look up the details of the A system (ISO 216). And found I was among the confused people who thought it was based on the Golden ratio. So as an aide memoire to myself:

The A series paper sizes are all in the ratio 1:√2, which has the property that when you cut a piece of A-sized paper in half the two halves are of the same aspect ratio as the original piece of paper. And this ratio is related to the Silver ratio, not the Golden Ratio. The actual sizes used start from A0, the largest, which has an area of 1 square meter (thank you internet – I didn’t know that bit before!), so that’s why the absolute dimensions may seem irrational or arbitrary. (They are irrational of course, as √2 is irrational.) The arbitrariness comes from the meter being defined originally in terms of the circumference of the Earth (putting the ‘geo’ into geometry).

To construct a rectangle of this ratio, construct a square and take an arc from the diagonal to determine the long side of the rectangle (good old Pythagoras: the square on the hypotenuse and all that). (See diagram A – a throwback to secondary school geometry; I was pleased to discover I had not forgotten how to construct right angles, bisect a line etc. using a straight edge and pair of compasses.)

Diagram A
Diagram A

The Golden Ratio (another irrational number – related to the Fibonacci numbers), on the other hand, is approximately 1:1.618 (or phi), and it has the property that when you cut the largest possible square off the rectangle you are left with another rectangle of the same aspect ratio as the one you started with. To construct a Golden rectangle:

Construct a square. Draw a line from the midpoint of one side of the square to either of the opposite corners. Use that line as the radius to draw an arc that defines the long side of the rectangle. See diagram B.

Diagram B
Diagram B

This is the ratio that crops up all over the place in natural forms like shells and pine cones, and also has been used as a compositional device by artists and architects for centuries.

(This little excursion into geometry, as mentioned, arose from a passing comment on differences in paper sizes. It might seem that the mathematics behind the A series means it will take over the world (it is an ISO standard after all). But, while standardisation may yield benefits, like knowing you can get a ready-made frame to fit your drawing, or being able to ship goods around the world in ISO sized containers, variety is the spice of life they say – and perverse artists will always be cutting their paper (or canvas) to fit their subject, or even sticking extra bits on when they run out of room, thereby keeping the custom framers in business.)

Copyright ©2015 F. Watts

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4 thoughts on “Paper sizes, geometry and irrationality

    1. One of my pleasures when teaching intro to philosophy a very long time ago was going through the proof in the Meno about the square on the diagonal – it made me feel like I understood partly why Pythagoras theorem is true 🙂

  1. This is very satisfying—in that it’s one of the many, many, (many), subjects in the fog of the very back of my mind—And I love to hear an artist, who has both the instinct in seeing and the need to find an unambiguous explanation, spell it out in numbers, so to speak (to apply three forms of communication in this wobbly metaphor of my wobblier understanding).
    I keep finding you wonderfuller and wonderfuller.

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