An Elementary
Look At The Golden Ratio
The golden ratio (symbolized by
the Greek letter “phi”) is a special number approximately equal to 1.618. It
appears many times in geometry, art, architecture and other areas.
The Math Behind It
If you divide a line into two
parts so that:
the longer part (a)
divided by the smaller part (b)
is also equal to
the whole length (a+b)
divided by the longer part (a)
then you will have the golden
ratio (1.618…)
Beauty
This rectangle below has been
made using the Golden Ratio. It looks like a typical frame for a painting, does
it not?
Some believe the golden ratio
makes the most pleasing and beautiful shape; in human faces, as well.
Many buildings and artworks
have the golden ratio incorporated within them. The Parthenon in Greece, the
Taj Mahal in India and Bottocelli’s
“Birth of Venus” are some examples.
The Actual Value
The Golden Ratio is equal to:
1.61803398874989484820… (etc.)
The digits just keep on going,
with no pattern. In fact the Golden Ratio is known to be an Irrational
Number…more about this in a bit.
Calculating It
You can calculate it yourself
by starting with any number and following these steps:
• Divide 1 by your number (=1/number)
• Add 1
• That is your new number
• Start again at step A
With a calculator, just keep
pressing “1/x”, “+”, “1”, “=”, around and around. If you start with 2 you will
get this:
# 1/# Add 12 1/2=0.5 0.5+1=1.5
1.5 1/1.5=0.666...
0.666...+1=1.666...1.666...1/1.666...=0.60.6+1=1.61.61/1.6=0.6250.625+1=1.6251.6251/1.625=0.6154...0.6154...+1=1.6154...1.6154…
But it takes a long time to get
even close; however with a computer it can be calculated to thousands of
decimal places quite quickly; with the same result.
Drawing It
Here is one way to draw a
rectangle with the Golden Ratio:
• Draw a square (of size “1”).
• Place a dot half way along
one side.
• Draw a line from that point
to an opposite corner (it will be √>5/2 in length).
• Turn that line so that it
runs along the square’s side.
• Extend it to create a
rectangle based on the Golden Ratio.
The Formula
Looking at the rectangle we
just drew, you can see that there is a simple formula for it. If one side is 1,
the other side will be:
The square root of 5 is
approximately 2.236068, so The Golden Ratio is approximately (1+2.236068)/2 =
3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.
Fibonacci Sequence
There is a special relationship
between the Golden Ratio and the Fibonacci Sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21,
34, …
(The next number is found by
adding up the two numbers before it.)
And here is a surprise: if you
take any two successive (one after the other) Fibonacci Numbers, their ratio is
very close to the Golden Ratio.
In fact, the bigger the pair of
Fibonacci Numbers, the closer the approximation. For example:
ABB/A231.5351.666666666...581.6813.625.........1442331.6118055556...2333771.618025751............
You don’t even have to start
with 2 and 3; as example start: with 192 and 16 (and get the sequence 192, 16,
208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, …):
ABB/A192160.08333333...16208132082241.07692308...2244321.92857143............7408119841.61771058...11984193921.61815754............
The Most Irrational
The Golden Ratio may be the
most irrational number. Here is why …
One of the special properties
of the Golden Ratio is that it can be defined in terms of itself, like this:
(In numbers: 1.61803… = 1 +
1/1.61803…)
That can be expanded into this
fraction that goes on forever (called a “continued fraction”):
So, it neatly slips in between
simple fractions.
Whereas many other irrational
numbers are reasonably close to rational numbers (for example Pi = 3.141592654
is pretty close to 22/7 = 3.1428571…)
Other Names
The Golden Ratio is also
sometimes called the golden section, golden mean, golden number, phi ratio,
divine proportion, divine section and golden proportion.