Question

What does cutting squares from an A4 (`297"mm"xx210"mm"`) sheet of paper tell you about `sqrt(2)`?

Answer 1

It illustrates the continued fraction for `sqrt(2)`

`sqrt(2) = 1+1/(2+1/(2+1/(2+...)))`

Explanation:

If you start with an accurate sheet of A4 (`297"mm" xx 210"mm"`) then in theory you can cut it into `11` squares:

• One `210"mm"xx210"mm"`
• Two `87"mm"xx87"mm"`
• Two `36"mm"xx36"mm"`
• Two `15"mm"xx15"mm"`
• Two `6"mm"xx6"mm"`
• Two `3"mm"xx3"mm"`

In practice, it only takes a small error (say `0.2"mm"`) to mess up this dissection, but in theory we end up with a visual demonstration that:

`297/210 = 1+1/(2+1/(2+1/(2+1/(2+1/2))))`

The dimensions of a sheet of A4 are designed to be in a `sqrt(2):1` ratio, to the nearest millimetre. The advantage of such a ratio is that if you cut a sheet of A4 in half, then the resulting two sheets are very similar to the original. The resulting size is A5 to the nearest millimetre.

In fact A0 has area very close to `1"m"^2` and sides in ratio as close as possible to `sqrt(2)` rounded to the nearest millimetre. To achieve that, it has dimensions:

`1189"mm" xx 841"mm" ~~ (1000*root(4)(2))"mm" xx (1000/root(4)(2))"mm"`

Then each smaller size is half the area of the previous size (rounded down to the nearest millimetre):

• A0 `841"mm" xx 1189"mm"`
• A1 `594"mm" xx 841"mm"`
• A2 `420"mm" xx 594"mm"`
• A3 `297"mm" xx 420"mm"`
• A4 `210"mm" xx 297"mm"`
• A5 `148"mm" xx 210"mm"`
• A6 `105"mm" xx 148"mm"`

etc.

So A4 has area very close to `1/16"m"^2`

The terminating continued fraction for `297/210` points to the non-terminating continued fraction for `sqrt(2)`...

`sqrt(2) = 1+1/(2+1/(2+1/(2+1/(2+1/(2+...))))) = [1;bar(2)]`