Math Is Fun Forum

hi PHRU,

Welcome to the forum.

This is how I've introduced this to classes in the past.

Start with a rectangle, say 4 x 1.  Chop off a square ( 1x1 ) and you're left with a differently proportioned rectangle, 3 x 1

Start with a different rectangle, let's say 2 x 1.  Chop off the 1 x 1 square and you're left with a rectangle 1 x 1.  Again the proportion of length to width has changed ( from 2:1 into 1:1)

So, can you find a rectangle in a certain proportion, say x:1, so that when you cut off a square, what's left is still in the same ratio?

See diagram below.

The start rectangle is x by 1.  After a square is removed, the rectangle that is left now has measurements 1 by (x-1).  If these must be in the same ratio then

If you use the quadratic formula to solve this the larger root is

which is the golden ratio.  The other root is the reciprocal of the golden ratio.

And if you start with that rectangle and chop off a square you get another rectangle in the ratio; and if you start with that rectangle ......... for ever. 

That's why you get a neat spiral if you join up corresponding points in all the recatngles.

But how do snails know this?

Bob