The golden section in the natural sciences
The golden section has not lost its power to attract either artists or scientists today. It is interesting to see that experts can even be divided about whether the golden section is a unique number or not.
According to the American mathematician, George Markowsky, it is not self-evident that 'the golden proportion' is especially harmonious, and he refuses to accord the number any special status.
On the other hand, the German physicist, Peter Richter, well-known for his book The Beauty of Fractals, harbours quite a different respect for the golden section. He proves that the number (1.618034) is "the most noble of noble numbers" and "the most irrational of all irrational numbers" (this is specialist terminology, which I make no pretence to understand). His arguments are also based on his research into non-linear relationships in nature, in which he finds the golden section.
A further argument for the fact that the golden section 'must' be found in nature is provided by the interesting and informative website of the University of Surrey.
The Fibonacci sequence
Fibonacci, or Leonard of Pisa, lived from 1170 to 1250. He was born in Pisa, but grew up in North Africa. He played a central role in the development of mathematics, not least due to his rediscovery of the mathematics of antiquity. Hans Liber abaci introduced the Arabic numerals (instead of Roman numerals) and the decimal system to Europe .
The Fibonacci sequence is a row of numbers closely related to the golden section. The series is:
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 |
89 |
144 |
233 |
377 |
610 |
987 |
1597 |
... |
One begins by taking the two numbers 1 and 1 - and then adding them to make the next number, 2. This is added to 1 to make 3. Thus, each number in the series is the sum of the two preceding numbers.
The mathematical relationship between any two adjacent quickly becomes a number close to the golden section.
| 3:2 | is thus | 1.5 |
| 5:3 | is | 1.6666 |
| 8:5 | is | 1.6 |
| 13:8 | is | 1.625 |
| 21:13 | is | 1.6153 |
| ... | ||
| 1597:987 | is | 1.610344 ... |
The product is alternately too high or too low, but the further one moves up along the series the more precisely one approaches the golden section.
Moreover, one can also start with two quite arbitrary numbers, such as 3 and 11 - and carry out the same procedure. In this case, too, the relationship between two adjacent numbers in the series will approach the golden section. After a time we reach the two numbers 1851 and 1144, which divided by each other make 1.618007. However, the familiar Fibonacci sequence most quickly approaches the golden section, and therefore occupies a special position.
A more detailed argument for the connection between the golden section and the Fibonacci sequence may be found in the website of the University of Surrey.
