the golden number or ratio
Created | Updated Jan 28, 2002
There is a special constant considered by the Greeks, founders of geometry that when a rectangle with sides in the ratio 1 to (1+sqrt(5))/2, is aesthetically pleasing to the eye. This ratio of approximately 1:1.618 is called the golden ratio and the number 1.618... or the formula given (1+sqrt(5))/2 is called the golden number, or phi and is represented by the Greek letter phi.
It is a solution to the quadratic equation
x^2-x-1=0, and the number can also be shown by the infinite continued fraction:
1/(1+(1/(1+(1/(1+ ...
Other mathematical properties of phi is that
phi^2 = phi + 1
1/phi = phi - 1
This number has many other interesting properties, not only in pretty rectangles, but crops up in rabbits, flowers, shells, art, even the human body.
A mathematician by the name of Fibonacci discovered a certain series of numbers while studying the populations of rabbits. This series is derived by taking 0, 1, then each following number is the sum of the previous two. So we get 0, 1, 1, 2, 3, 5, 8, 13, 21,...
Where does phi enter into this series? Divide each sucessive number by the previous and the results converge to phi:
1/1=1, 2/1=2, 3/2=1.5, 8/5=1.6, 13/8=1.625, 21/13=1.6153... and so on.
The Fibonacci numbers are present in flowers as well. The number of petals is frequently a Fibonacci number. The number of seeds in one of the spirals on a sunflower is a Fibonacci number.
If you have a golden rectangle (a rectangle which sides are in the golden ratio) and you draw a line so you have a square on one side of the line and a rectangle on the other, the rectangle will also be a golden rectangle. This process can be continued infinitely. If you draw a curve from the far right corner of the square to the diagonally opposite corner, and continue this curve into the next square and so on you draw a spiral. This spiral is special as the nautilus shell is curved in this exact same way.
As before, the Greeks thought that this ratio was pretty special and beautiful, and they incorporated this into their art and buildings.
The Parthenon's front face is a rectangle in approximately the golden ratio. In many paintings certain features are in the golden ratio. Even in music, Bach, Mozart, and other composers had sections in length that related to the golden ratio.
Finally, if you take one finger and measure from your knuckle to the second joint, and if you divide this length by the length from the first to the second joint, this value is approximately phi.
Like other numbers such as pi, e, sqrt(2), this number is irrational (it cannot be expressed as a finite fraction), and the decimal value does not terminate.
The number phi is approximately =1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891....
Enter the golden ratio into a search engine for other information on this remarkable constant.
It is a solution to the quadratic equation
x^2-x-1=0, and the number can also be shown by the infinite continued fraction:
1/(1+(1/(1+(1/(1+ ...
Other mathematical properties of phi is that
phi^2 = phi + 1
1/phi = phi - 1
This number has many other interesting properties, not only in pretty rectangles, but crops up in rabbits, flowers, shells, art, even the human body.
A mathematician by the name of Fibonacci discovered a certain series of numbers while studying the populations of rabbits. This series is derived by taking 0, 1, then each following number is the sum of the previous two. So we get 0, 1, 1, 2, 3, 5, 8, 13, 21,...
Where does phi enter into this series? Divide each sucessive number by the previous and the results converge to phi:
1/1=1, 2/1=2, 3/2=1.5, 8/5=1.6, 13/8=1.625, 21/13=1.6153... and so on.
The Fibonacci numbers are present in flowers as well. The number of petals is frequently a Fibonacci number. The number of seeds in one of the spirals on a sunflower is a Fibonacci number.
If you have a golden rectangle (a rectangle which sides are in the golden ratio) and you draw a line so you have a square on one side of the line and a rectangle on the other, the rectangle will also be a golden rectangle. This process can be continued infinitely. If you draw a curve from the far right corner of the square to the diagonally opposite corner, and continue this curve into the next square and so on you draw a spiral. This spiral is special as the nautilus shell is curved in this exact same way.
As before, the Greeks thought that this ratio was pretty special and beautiful, and they incorporated this into their art and buildings.
The Parthenon's front face is a rectangle in approximately the golden ratio. In many paintings certain features are in the golden ratio. Even in music, Bach, Mozart, and other composers had sections in length that related to the golden ratio.
Finally, if you take one finger and measure from your knuckle to the second joint, and if you divide this length by the length from the first to the second joint, this value is approximately phi.
Like other numbers such as pi, e, sqrt(2), this number is irrational (it cannot be expressed as a finite fraction), and the decimal value does not terminate.
The number phi is approximately =1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891....
Enter the golden ratio into a search engine for other information on this remarkable constant.
