The document discusses the golden ratio φ (approximately 1.618). It has been called the golden mean, golden section, divine proportion, and extreme and mean ratio. The ratio is exhibited in geometry, such as the construction of the golden rectangle. It appears throughout nature, art, architecture like the Parthenon, and even the human body. The ratio can be derived mathematically through equations involving quantities in the ratio. While some see it as divinely inspired, others view it as an interesting irrational number with no special significance.

1. The Golden Number
“All life is biology. All biology is physiology.
All physiology is chemistry.
“ All chemistry is physics. All physics is math.”
Dr. Stephen Marquardt

2. ϕ
His multiple names ...
• Golden Section
• Golden mean
• Extreme and mean ratio
• Medial section
• Divine Proportion
• Divine Section
• Golden Proportion
• Mean of Phidias... 1 =Φ
φ

3. Euclid
• "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 less."
Elements
(4th. Century B.C.)

4. a+b = a = ϕ
Mathematically:
a b

5. Aesthetically pleasing
The Golden rectangle

6. The Golden rectangle
A golden rectangle with longer side a and shorter side b,
when placed adjacent to a square with sides of length a, will
produce a similar golden rectangle with longer side a + b
and shorter side a. This illustrates the relationship .

7. How to draw it
We must proceed:
a Drawing a square
Draw the middle point of one of its
sides.
a
=ϕ With centre in this point and ratio to
b b
the opposite vertex, trace an arch
cutting the extension of the side.

8. The Golden rectangle
Mathematically: a+b = a = ϕ
a b

9. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b

10. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b
One method for finding the value of φ is to start with the
left fraction.

11. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b
One method for finding the value of φ is to start with the
left fraction.
a +b = a
a a b

12. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b
One method for finding the value of φ is to start with the
left fraction.
a +b = a
a a b
And we know a =ϕ
b

13. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b
One method for finding the value of φ is to start with the
left fraction.
a +b = a
a a b
And we know a =ϕ So, substituting,
b

14. Two quantities a and b are said to
be in the golden ratio φ if:
a+b = a = ϕ
a b
One method for finding the value of φ is to start with the
left fraction.
a +b = a
a a b
And we know a =ϕ So, substituting,
b
1
1+ φ =φ

15. From this equation
1
1+ φ =φ

16. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain

17. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2

18. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2
Ordering,

19. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2
Ordering,
φ2 − φ − 1= 0

20. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2
Ordering,
φ2 − φ − 1= 0
And solving:

21. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2
Ordering,
φ2 − φ − 1= 0
And solving:
φ =1+ 5
2

22. From this equation
1
1+ φ =φ
Multiplying both sides by phi, we obtain
φ +1= φ2
Ordering,
φ2 − φ − 1= 0
And solving:
φ =1+ 5 =1.6180339887...
2

23. So, this is the particular value of the
number Phi
ϕ= 2
1+ 5 =1.6180339887...

24. ϕ= 1+ 5
2
This is called the “algebraic form”

25. ϕ= 1+ 5
2
This is called the “algebraic form”
• There are others:

26. ϕ= 1+ 5
2
This is called the “algebraic form”
• There are anothers:
This is called the “continued fraction”

27. ϕ= 1+ 5
2
This is called the “algebraic form”
• There are anothers:
This is called the “continued fraction”
And this is the “infinite series”

29. Appearences in geometry

30. Appearences in geometry
BC =Φ = 1
BD φ
Relation of side and diagonal

31. Appearences in geometry
bigger part =φ
smaller part
Relation between the two sections of a diagonal

33. Pyramids

34. Parthenon

35. Fibonacci sequence

36. Logarithmic Spiral
It is bulit over the golden rectagles, dividing them successsively.

37. Leonardo Da Vinci

39. Paint

40. Paint

41. Detractors
• “In the Elements (308 BC) the Greek
mathematician merely regarded that
number as an interesting irrational
number, in connection with the middle and
extreme ratios. It is indeed exemplary that
the great Euclid, contrary to generations of
mystics who followed, would soberly treat
that number for what it is, without
attaching to it other than its factual
properties”
Midhat J. Gazalé
(XX Century)

42. Detractors
• "Certainly, the oft repeated assertion that
the Parthenon in Athens is based on the
golden ratio is not supported by actual
measurements. In fact, the entire story
about the Greeks and golden ratio seems
to be without foundation. The one thing we
know for sure is that Euclid, in his famous
textbook Elements, written around 300
BC, showed how to calculate its value."
Keith Devlin
(XIX Century)

43. In our Solar System

44. In our Universe

45. Physical phenomenon
In the atmosphere, low pressure areas spin counter-clockwise in the
north hemisphere and its form is nearly a logarithmic spiral.

46. Biology
• Phyllotaxis
• Seeds of plants
• Different fruits

47. Biology
• Animals
• Human body

48. Geometry has two great treasures: one is the theorem
of Pythagoras, the other the division of a line into mean
and extreme ratio. The first we may compare to a
mass of gold, the second we may call a precious jewel.
— Johannes Kepler