6.1 Golden Section

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Further Applications (1)
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Contents
6.1 Golden Section
6.2 More about Exponential and Logarithmic
Functions
6.3 Nine-Point Circle
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6.1 Golden Section
A. The Golden Ratio
A golden section is a certain length that is
divided in such a way that the ratio of the
longer part to the whole is the same as the ratio
of the shorter part to the longer part.
Fig. 6.2
This specific ratio is called the golden ratio.
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6.1 Golden Section
Definition 6.1
Examples
(a) The largest pyramid in the world, Horizon
of Khufu (?????), is a right pyramid with height
146 m and a square base of side 230 m. The ratio
of its height to the side of its base is 146
230 ? 1 1.58.

1. Another famous pyramid, Horizon of Menkaure
   (?????) , is also a right pyramid with height 67
   m and a square base of side 108 m. The ratio of
   its height to the side of its base is 67 108 ?
   1 1.61.

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6.1 Golden Section
Consider a line segment PQ with length (1 x)cm.
Fig. 6.5(a)
Divide the line segment into two parts such that
PR 1 cm and RQ x cm.
Fig. 6.5(b)
According to the definition of the golden
section, we have
Therefore,
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6.1 Golden Section
B. Applications of the Golden Ratio
(i) The Parthenon The Parthenon (?????), which is
situated in Athens (??), Greece, is one of the
most famous ancient Greek temples.
Fig. 6.8
L1 W1 is close to the golden ratio.
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6.1 Golden Section
(ii) The Eiffel Tower The tower is 320 m high.
The ratio of the portion below and above the
second floor (l1 l2 as shown in Fig. 6.9) is
equal to the golden ratio.
Fig. 6.9
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6.1 Golden Section
C. Fibonacci Sequence
The Fibonacci sequence is a special sequence that
was discovered by a great Italian mathematician,
Leonardo Fibonacci (????). This sequence was
first derived from the trend of rabbits growth.
Suppose a newborn pair or rabbits A1 (male) and
A2 (female) are put in the wild.
1st month A1 and A2 are growing.
2nd month A1 and A2 are mating at the age of
one month. Another pair of rabbits B1 (male)
and B2 (female) are born at the end of this month.
3rd month A1 and A2 are mating, another pair
of rabbits C1 (male) and C2
v
(female) are born at the end of this
month. B1 and B2 are growing.
If the rabbits never die, and each female rabbits
born a new pair of rabbits every month when she
is two months old or elder, what happens later?
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6.1 Golden Section
Fig. 6.12
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6.1 Golden Section
Definition 6.2
The Fibonacci sequence is a sequence that
satisfies the recurrence formula
According to the definition of the Fibonacci
sequence, the first ten terms of the sequence are
1, 1, 2, 3, 5, 8, 13, 21, 34,
55.
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6.1 Golden Section
Consider that seven squares with sides 1 cm, 1
cm, 2 cm, 3 cm, 5 cm, 8 cm, 13 cm respectively.
Arrange the squares as in the following diagram
Fig. 6.13
If we measure the dimensions of the rectangles,
each successive rectangle has width and length
that are consecutive terms in the Fibonacci
sequence
Then the ratio of the length to the width of the
rectangle will tend to the golden ratio.
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6.1 Golden Section
D. Applications of the Fibonacci Sequence
(a) In Music
The piano keyboard of a scale of 13 keys as shown
in Fig. 6.14, 8 of them are white in colour,
while the other 5 of them are black in colour.
The 5 black keys are further split into groups of
3 and 2.
Note that the numbers 1,2,3,5,8,13 are
consecutive terms of the Fibonacci sequence.
Fig. 6.14
In musical compositions, the climax of songs is
often found at roughly the phi point (61.8) of
the song, as opposed to the middle or end of the
song.
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6.1 Golden Section
(b) In Nature
Number of petals in a flower is often one of the
Fibonacci numbers such as 1, 3, 5, 8, 13 and 21.
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6.2 More about Exponential and Logarithmic
Functions
Applications
(a) In Economics
Suppose we deposited P in a savings account and
the interest is paid k times a year with annual
interest rate r, then the total amount A in the
account at the end of t years can be calculated
by the following formula
In this case, the earned interest is deposited
back in the account and also earns interests in
the coming year, so we say that the account is
earning compound interest.
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6.2 More about Exponential and Logarithmic
Functions
(b) In Chemistry
The concentration of the hydrogen ions is
indirectly indicated by the pH scale, or hydrogen
ion index.
pH Value of a solution
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6.2 More about Exponential and Logarithmic
Functions
(c) In Social Sciences
Some social scientists claimed that human
population grows exponentially.
Suppose the population P of a city after n years
obeys the exponential function
where 20 000 is the present population of the
city.
From the equation, the population of the city
after five years will be approximately 29 000.
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6.2 More about Exponential and Logarithmic
Functions
(d) In Archaeology
Scientists have determined the time taken for
half of a given radioactive material to
decompose. Such time is called the half-life of
the material.
We can estimate the age of an ancient object by
measuring the amount of carbon-14 present in the
object.
Radioactive Decay Formula
The amount A of radioactive material present in
an object at a time t after it dies follows the
formula
Where A0 is the original amount of the
radioactive material and h is its half-life.
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6.3 Nine-point Circle
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Theorem 6.1
In a triangle, the feet of the three altitudes,
the mid-points of the three sides and the
mid-points of the segments from the three
vertices to the orthocentre, all lie on the same
circle.
Fig. 6.17