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:
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:
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:
Then place the compass tip on the bottom left corner of the 2x2 square and draw an arc like this:
Then place the compass tip on the left hand, top corner of the 3x3 square and do the same:
Do the same for the other three squares to obtain:
This shape is widely found in nature, can you find any other examples?