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					       The Golden Ratio
Aim to answer the following:
•What is the Golden Ratio?
•How is it related to Fibonacci?
•When does it appear in geometry?
•How has it been used by mankind?
•How does it occur naturally in life?
                 Facts about Phi
•The golden ratio, Ф, is an irrational number and
therefore cannot be expressed fully in decimal form.

•Ф=1.618033988749895… or         1+√5

•Just as pi (Π) is the ratio of the
circumference of a circle to its
diameter, Ф is simply the ratio of the
line segments that result when a line is
divided in one very special and unique
•The total length a+b is to the longer segment a
as a is to the shorter segment b.

•Phi is the solution to a quadratic equation.
a+b = a = Ф
 a    b


  bФ+b = bФ
   bФ     b

   Ф+1 = Ф
    Ф           Ф = 1 + √5
   Ф²-Ф-1 = 0
                       History of Phi
• In the 12th century, Leonardo Fibonacci
discovered a simple numerical series that is the
foundation for an incredible mathematical
relationship behind phi.

•Starting with 0 and 1, each new number in
the series is simply the sum of the two before
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
•The ratio of each successive pair of numbers in the series
approximates phi (1.618. . .). For instance 5:3 is 1.666..., 8:5 is 1.60,
13:8 is 1.625, 21:13 is 1.615...
The Golden Rectangle and Spiral
                  Phi in Geometry
•Insert an equilateral triangle
inside a circle. Find the midpoints
of two sides at A and B, then
connect with a line and extend the
line to the circle.
The ratio of AB to BG is Phi.
      •Insert a square inside a
      The ratio of AB to BG is Phi.

•Insert a pentagon inside a circle.
Connect three of the five points
to cut one line into three sections.
The ratio of AB to BG is Phi.
             Phi in Architecture
•Its use started as early as with the Egyptians in
the design of the pyramids:
•It was used in the design    •The CN Tower in
of the Notre Dame in Paris.   Toronto, the tallest
                              tower and
                              freestanding structure
                              in the world, contains
                              the golden ratio in its
                              The ratio of
                              observation deck at
                              342 meters to the total
                              height of 553.33 is
                              0.618, the reciprocal
                              of Phi!
                        Phi in Art
 Leonardo da Vinci

Georges-Pierre Seurat
                Phi in Humans

•Each section of your index finger,
from the tip to the base of the wrist,
                                         •The ear reflects
is larger than the preceding one by
                                         the shape of a
about the Fibonacci ratio of 1.618,
                                         Fibonacci spiral.
also fitting the Fibonacci numbers
2, 3, 5 and 8.
•The ratio of your forearm
to hand is Phi.
                    Phi in Animals
•The eyes, fins and tail all fall
at golden sections of the length
of a dolphin's body.

•The eye-like markings of this moth
fall at golden sections of the lines
that mark its width and length.

•The spiral growth of sea
shells provide a simple,
but beautiful, example . . .
      Summary of how the
      Golden Ratio could be
         used in schools
• Sequences       0, 1, 1, 2, 3, 5, ...

• Geometry

• Limits

• Cockcroft Report stated:‘there is a need for teachers to devote more
time to the use of mathematics in applications taken from real life’
The End

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