Golden ratio facts for kids

With one number a and another smaller number b, the ratio of the two numbers is found by dividing them. Their ratio is a/b. Another ratio is found by adding the two numbers together and dividing this by the larger number a. The new ratio is (a+b)/a. If these two ratios are equal to the same number, then that number is called the golden ratio. The Greek letter (phi) is usually used as the name for the golden ratio.

For example, if b = 1 and a/b = , then a = . The second ratio (a+b)/a is then . Because these two ratios are equal, this is true:

One way to write this number is

is like any number which, when multiplied by itself, makes 5 (or which number is multiplied): .

The golden ratio is an irrational number. If a person tries to write it, it will never stop and never make a pattern, but it will start this way: 1.6180339887... An important thing about this number is that a person can subtract 1 from it or divide 1 by it. Either way, the number will still keep going and never stop.

Golden rectangle

The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. For any such rectangle, and only for rectangles of that specific proportion, if we remove square B, what is left, A, is another golden rectangle; that is, with the same proportions as the original rectangle.

If the length of a rectangle divided by its width is equal to the golden ratio, then the rectangle is a "golden rectangle". If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because . The blue part (B) is a square, and the pink part by itself (A) is another golden rectangle because . The big rectangle and the pink rectangle have the same form, but the pink rectangle is smaller and is turned.

Fibonacci numbers

The Fibonacci numbers are a list of numbers. A person can find the next number in the list by adding the last two numbers together. If a person divides a number in the list by the number that came before it, this ratio comes closer and closer to the golden ratio.

Fibonacci number	divided by the one before	ratio
1
1	1/1	= 1.0000
2	2/1	= 2.0000
3	3/2	= 1.5000
5	5/3	= 1.6667
8	8/5	= 1.6000
13	13/8	= 1.6250
21	21/13	= 1.6154...
34	34/21	= 1.6190...
55	55/34	= 1.6177...
89	89/55	= 1.6182...
...	...	...
		= 1.6180...

Golden ratio in nature

Using the golden angle will optimally use the light of the sun. This is a view from the top.

A leaf of common ivy, showing the golden ratio

In nature, the golden ratio is often used for the arrangement of leaves or flowers. These use the golden angle of approximately 137.5 degrees. Leaves or flowers arranged in that angle best use sunlight.

Related pages

• Fibonacci numbers
• Phi

Images for kids

• 

  Michael Maestlin, the first to write a decimal approximation of the ratio

• 

  Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.

• 

  The golden ratio \varphi and its negative reciprocal -\varphi^{-1} are the two roots of the quadratic polynomial x^2 - x - 1. The golden ratio's negative -\varphi and reciprocal \varphi^{-1} are the two roots of the quadratic polynomial x^2 + x - 1.

• 

  Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

• 

  The 600-cell, a regular 4-polytope with H4 symmetry whose vertices form the roots of the E8 group, has an edge-to-circumradius ratio of 1:\varphi, like the icosidodecahedron and decagon.

• 

  The 421 polytope Coxeter plane projection of roots of E8, containing two copies of the vertices of a 600-cell with H4 symmetry. They are scaled in 1:\varphi golden ratio, depicted here with orange and green hulls. It's Coxeter-Dynkin diagram is \text{ } .

• 

  Da Vinci's illustration of a dodecahedron from Pacioli's Divina proportione (1509)

• 

  Detail of the saucer plant, Aeonium tabuliforme, showing the multiple spiral arrangement (parastichy)

• 

  Nautilus shells are often erroneously claimed to be golden-proportioned.

• 

  The Great Pyramid of Giza

• 

  Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.

• 

  Albert Gleizes, Les Baigneuses (1912)

See also

In Spanish: Número áureo para niños