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# Golden number

## by laruamc on Dec 01, 2012

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Presentation at class 2012, November 28th.

Presentation at class 2012, November 28th.

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## Golden numberPresentation Transcript

• 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
• 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
• 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.)
• 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 of length a, willproduce a similar golden rectangle with longer side a + band shorter side a. This illustrates the relationship .
• 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.
• 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 finding the value of φ is to start with theleft fraction.
• 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
• 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
• 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
• 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
• 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 Ordering,
• From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0
• From this equation 1 1+ φ =φMultiplying both sides by phi, we obtain φ +1= φ2 Ordering, φ2 − φ − 1= 0 And solving:
• 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
• 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
• 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 called the “continued fraction”
• 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”
• Appearences in geometry
• Appearences in geometry BC =Φ = 1 BD φRelation of side and diagonal
• Appearences in geometry bigger part =φ smaller partRelation between the two sections of a diagonal
• 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 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)
• 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)
• In our Solar System
• In our Universe
• Physical phenomenonIn the atmosphere, low pressure areas spin counter-clockwise in thenorth hemisphere and its form is nearly a logarithmic spiral.
• 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 ratio. The first we may compare to amass of gold, the second we may call a precious jewel. — Johannes Kepler