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

##### 1 Answer

It illustrates the continued fraction for

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

#### Explanation:

If you start with an accurate sheet of A4 (

- 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

#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

In fact A0 has area very close to

#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

The terminating continued fraction for

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