Frames:

The golden rectangle

We now look at a two-dimensional geometric interpretation of the golden ratio condition c = a + b and as shown in the sketch alongside.
This condition implies that the two rectangles in the sketch are similar. We want to construct such a rectangle, prove that or explain why the construction works, and find the dimensions of such a rectangle.

Click on the button below to see how Euclid (300 BCE) constructed a golden rectangle. Now prove that the constructed rectangle is golden, i.e. that the ratio of the sides of this rectangle is in the golden ratio. Click on the button below for the solution:

The Golden Spiral

From a mathematical point of view, the interest in the golden rectangle is not so much because its dimensions are pleasing to the eye, but in the fact that a golden rectangle can be partitioned into a square and a new rectangle, which is again a golden rectangle, similar to the original. And this process can be continued to infinity! Formulated differently: the golden rectangle has the property that when a square is removed from it, a smaller similar rectangle remains. Thus a smaller square can be removed from it, and this can be continued …

Open the Ad Infinitum applet below. In the applet, click on Rectangle 1, etc, to see the sequence of successive golden rectangles.

If the dimensions of Rectangle 1 is by 2, prove that the 8 rectangles shown in the applet are all golden rectangles, by finding the dimensions of each rectangle and showing that the ratio of the longer side to the shorter side is f.

Write down the lengths of the longer sides of the 8 rectangles as a sequence. What kind of sequence is it? Prove it! Write down the general term Tn for the sequence, and check your formula by using it to calculate the first four terms of the sequence.

Write down the lengths of the shorter sides of the 8 rectangles as a sequence. What kind of sequence is it? Prove it!

 


The Golden Spiral is based on the pattern of squares that can be constructed within the golden rectangle as shown above. Click on the button below to see the golden spiral.

These spirals occur frequently in nature, for example in a Nautilus shell or the centre of a sunflower: