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?

May 10th, 2015

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' = b-1, h' = h

area ratio = bh/(b-1)h

= b/(b-1)

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.




May 10th, 2015

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