The document discusses the golden ratio (approximately 1.618), which appears frequently in geometry and art. Ancient Greek mathematicians studied it due to its appearance in pentagrams and pentagons. While its exact value is irrational, dividing a line into "extreme and mean ratio" yields the golden ratio, where the whole line relates to the larger segment as the larger relates to the smaller. Some artists believe the golden ratio produces the most aesthetically pleasing shapes, though its intentional use in architecture is debated. A formula demonstrates that the golden ratio φ satisfies φ = 1 + 1/φ.

1. And it’s relation with Fibonacci Numbers

2. What is golden ratio?
 The golden ratio (symbol is the Greek
letter phi “φ”) is a special number
approximately equal to 1.618
 It appears many times in geometry, art,
architecture and other areas.

3. History
 Ancient Greek mathematicians first studied the golden
ratio because of its frequent appearance in geometry; the
division of a line into "extreme and mean ratio" (the
golden section) is important in the geometry of
regular pentagrams and pentagons. According to one
story, 5th-century BC mathematician Hippasus discovered
that the golden ratio was neither a whole number nor a
fraction (an irrational number), surprising Pythagoreans.
Eucid’s Elements (300 BC) provides
several propositions and their proofs employing the
golden ratio, and contains its first known definition which
proceeds as follows:
A straight line is said to have been cut in extreme and mean
ratio when, as the whole line is to the greater segment, so
is the greater to the lesser.

4. The Idea Behind It
 We find the golden ratio when we
divide a line into two parts so that:
the whole length divided by the long part
is also equal to
the long part divided by the short part

5.  This rectangle has been made using the
Golden Ratio.
 Some artists and architects believe the
Golden Ratio makes the most pleasing and
beautiful shape.

6.  Many buildings and artworks have the
Golden Ratio in them, such as the
Parthenon in Greece, but it is not really
known if it was designed that way.

7. The Actual value
 The Golden Ratio is equal to:
1.61803398874989484820...
 The digits just keep on going because
the Golden Ratio is an Irrational
Number.

8. Formula
We saw above that the Golden Ratio has this
property:
We can split the right-hand fraction like this:

9. a/b is the Golden Ratio “φ”, a/a=1 and
b/a =1/φ, which gets us:
φ = 1 + 1/φ