The Golden Rectangle
Mathematics

Tutor: None Selected  Time limit: 1 Day 
Take the golden rectangle and attach a square to the longer side so that you create a new larger rectangle. Is this new rectangle a golden rectangle?
First of all, the definition of a golden rectangle is one with the side ratios of 1 to about 1.618 (the golden ratio).
Part 1:
Yes, it will create a new golden rectangle. You are dividing each side by 2:
1/2:1.618/2
0.5:0.809
This new rectangle (when you multiply by 2), has a ratio of 1:1.618, so it is golden.
Part 2:
If you have a rectangle that is 1 by 1.618, you can add a 1.618 by 1.618 square to it, creating a new rectangle that is:
1.618 by 2.618
Let's see what the ratio is here:
1.618/1.618 to 2.618/1.618
1 to 1.618
It is another golden ratio. This process can be repeated indefinitely with the same results.
Part 3:
b = 1.618, h = 1
Remove a 1 by 1 square:
b' = 0.618, h' = 1
Area ratio:
bh/b'h'
= 1*1.618/0.618*1
= 2.618 = 1 + golden ratio
Let's use variables to see that this is always true:
b' = b1, h' = h
area ratio = bh/(b1)h
= b/(b1)
Assuming that the base is *any* multiple of the golden ratio:
= 1.618c/(0.618c)
= 2.618 = 1 + golden ratio
The answer does not depend on the size of g.
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