The Golden Number        “All life is biology. All biology is physiology.                 All physiology is chemistry.    ...
ϕ         His multiple names ...•   Golden Section•   Golden mean•   Extreme and mean ratio•   Medial section•   Divine Pr...
Euclid• "A straight line is said to have been cut  in extreme and mean ratio when,  as the whole line is to the greater  s...
a+b = a = ϕMathematically:                   a    b
Aesthetically pleasing  The Golden rectangle
The Golden rectangleA golden rectangle with longer side a and shorter side b,when placed adjacent to a square with sides o...
How to draw itWe must proceed:            a          Drawing a square                       Draw the middle point of one o...
The Golden rectangle Mathematically:   a+b = a = ϕ                    a    b
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    b
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    bOne method for fi...
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    bOne method for fi...
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    bOne method for fi...
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    bOne method for fi...
Two quantities a and b are said to   be in the golden ratio φ if:           a+b = a = ϕ            a    bOne method for fi...
From this equation       1    1+ φ =φ
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2          Or...
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2          Or...
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2          Or...
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2          Or...
From this equation              1           1+ φ =φMultiplying both sides by phi, we obtain           φ +1= φ2          Or...
So, this is the particular value of thenumber Phiϕ= 2  1+ 5 =1.6180339887...
ϕ=   1+ 5      2            This is called the “algebraic form”
ϕ=   1+ 5         2                  This is called the “algebraic form”• There are others:
ϕ=   1+ 5         2                  This is called the “algebraic form”• There are anothers:                  This is cal...
ϕ=   1+ 5         2                  This is called the “algebraic form”• There are anothers:                  This is cal...
Appearences in geometry
Appearences in geometry                                BC =Φ = 1                                BD      φRelation of side ...
Appearences in geometry                             bigger part =φ                             smaller partRelation betwee...
Pyramids
Parthenon
Fibonacci sequence
Logarithmic SpiralIt is bulit over the golden rectagles, dividing them successsively.
Leonardo Da Vinci
Paint
Paint
Detractors• “In the Elements (308 BC) the Greek  mathematician merely regarded that  number as an interesting irrational  ...
Detractors• "Certainly, the oft repeated assertion that  the Parthenon in Athens is based on the  golden ratio is not supp...
In our Solar System
In our Universe
Physical phenomenonIn the atmosphere, low pressure areas spin counter-clockwise in thenorth hemisphere and its form is nea...
Biology• Phyllotaxis• Seeds of plants• Different fruits
Biology• Animals• Human body
Geometry has two great treasures: one is the theoremof Pythagoras, the other the division of a line into meanand extreme r...
Golden number
Golden number
Golden number
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Golden number

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Presentation at class 2012, November 28th.

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Golden number

  1. 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. 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. 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. 4. a+b = a = ϕMathematically: a b
  5. 5. Aesthetically pleasing The Golden rectangle
  6. 6. The Golden rectangleA golden rectangle with longer side a and shorter side b,when placed adjacent to a square with sides of length a, willproduce a similar golden rectangle with longer side a + band shorter side a. This illustrates the relationship .
  7. 7. How to draw itWe 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. 8. The Golden rectangle Mathematically: a+b = a = ϕ a b
  9. 9. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a b
  10. 10. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a bOne method for finding the value of φ is to start with theleft fraction.
  11. 11. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a bOne method for finding the value of φ is to start with theleft fraction. a +b = a a a b
  12. 12. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a bOne method for finding the value of φ is to start with theleft fraction. a +b = a a a b And we know a =ϕ b
  13. 13. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a bOne method for finding the value of φ is to start with theleft fraction. a +b = a a a b And we know a =ϕ So, substituting, b
  14. 14. Two quantities a and b are said to be in the golden ratio φ if: a+b = a = ϕ a bOne method for finding the value of φ is to start with theleft fraction. a +b = a a a b And we know a =ϕ So, substituting, b 1 1+ φ =φ
  15. 15. From this equation 1 1+ φ =φ
  16. 16. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain
  17. 17. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2
  18. 18. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering,
  19. 19. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0
  20. 20. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0 And solving:
  21. 21. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0 And solving: φ =1+ 5 2
  22. 22. From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0 And solving: φ =1+ 5 =1.6180339887... 2
  23. 23. So, this is the particular value of thenumber Phiϕ= 2 1+ 5 =1.6180339887...
  24. 24. ϕ= 1+ 5 2 This is called the “algebraic form”
  25. 25. ϕ= 1+ 5 2 This is called the “algebraic form”• There are others:
  26. 26. ϕ= 1+ 5 2 This is called the “algebraic form”• There are anothers: This is called the “continued fraction”
  27. 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”
  28. 28. Appearences in geometry
  29. 29. Appearences in geometry BC =Φ = 1 BD φRelation of side and diagonal
  30. 30. Appearences in geometry bigger part =φ smaller partRelation between the two sections of a diagonal
  31. 31. Pyramids
  32. 32. Parthenon
  33. 33. Fibonacci sequence
  34. 34. Logarithmic SpiralIt is bulit over the golden rectagles, dividing them successsively.
  35. 35. Leonardo Da Vinci
  36. 36. Paint
  37. 37. Paint
  38. 38. 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)
  39. 39. 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)
  40. 40. In our Solar System
  41. 41. In our Universe
  42. 42. Physical phenomenonIn the atmosphere, low pressure areas spin counter-clockwise in thenorth hemisphere and its form is nearly a logarithmic spiral.
  43. 43. Biology• Phyllotaxis• Seeds of plants• Different fruits
  44. 44. Biology• Animals• Human body
  45. 45. Geometry has two great treasures: one is the theoremof Pythagoras, the other the division of a line into meanand extreme ratio. The first we may compare to amass of gold, the second we may call a precious jewel. — Johannes Kepler

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