The golden ratio

Investigation of the golden ratio and the human body which can be done individually or in the classroom.

Age

11 to 14

Challenge level

Project

The Golden Ratio

and the human body

This exercise is divided into 3 parts:

A. The golden ratio

Measure the following:

Distance from the ground to your belly button
Distance from your belly button to the top of your head
Distance from the ground to your knees
Distances A, B and C
Length of your hand
Distance from your wrist to your elbow

Now calculate the following ratios:

Distance from the ground to your belly button / Distance from your belly button to the top of your head
Distance from the ground to your belly button / Distance from the ground to your knees
Distance C / Distance B
Distance B / Distance A
Distance from your wrist to your elbow / Length of your hand

Write all your results on the following table:

Student name	Ratio 1	Ratio 2	Ratio 3	Ratio 4	Ratio 5

...
Average

Can you see anything special about these ratios?

B. The fibonacci sequence

Now look at the following sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The following number is the sum of the previous two. This is Fibonacci's sequence.

Now do the following ratios on a calculator and give answers in non-fraction numbers:

1/2 =

3/2 =

5/3 =

8/5 =

13/8 =

21/13 =

34/21 =

55/34 =

89/55 =

As you go on and on dividing a number in the sequence by the previous number you get closer and closer to the number you discovered in the first part of the exercise, phi = $\phi$ = 1.6180339887498948482.

C. The golden rectangle

We can also draw a rectangle with the fibonacci number's ratio. From this rectangle we can then derive interesting shapes.

First colour in two 1x1 squares on a piece of squared paper:

Then draw a 2x2 square on top of this one:

Then draw a 3x3 square to the right of these:

Then draw a 5x5 square under these:

Then draw a 8x8 square to the left of these:

Then draw a 13x13 square on top of these:

We could go on like this forever, making bigger and bigger rectangles in which the ratio of length/ width gets closer and closer to the Fibonacci number.

Let's try making a more interesting shape, going back to our first 1x1 squares and using a compass, place the compass tip on the top right hand corner of the right hand square and draw a semi circle like this: