6.1 Golden Section

6.2 More about Exponential and Logarithmic 　 　　　 Functions

6.3 Nine-Point Circle

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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.

A. The Golden Ratio

6.1 Golden Section

. have wethen section, golden the in

rod the divides point thesay weif 6.2, Fig. in shown as rod the Consider

AC

CB

AB

AC

CAB

This specific ratio is called the golden ratio.

Fig. 6.2

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Definition 6.1:

6.1 Golden Section

Examples:

(a) The largest pyramid in the world, Horizon of Khufu (柯孚王之墓 ), 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.

(b) Another famous pyramid, Horizon of Menkaure (高卡王之墓 ) , 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.

.2

15:11:

2

156180:11:6181

or is ratio golden the of

value exact The . or to equalsely approximat ratio Thisrectangle. appealingvisually a of sides the of raio a is ratio golden The

..

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Consider a line segment PQ with length (1 + x)cm.

Divide the line segment into two parts such that PR = 1 cm and RQ = x cm.

6.1 Golden Section

According to the definition of the golden section, we have

.PR

RQ

PQ

PR

Therefore,

formulaquadratic By

10

)1(111

1

2 xx

xx

x

x

(rejected)or2

51

2

51 x

Fig. 6.5(a)

Fig. 6.5(b)

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6.1 Golden Section

B. Applications of the Golden Ratio

L1 : W1 is close to the golden ratio.

(i) The Parthenon

The Parthenon (巴特農神殿 ), which is situated in Athens (雅典 ), Greece, is one of the most famous ancient Greek temples.

Fig. 6.8

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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 (斐波那契 ). 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.

,321 nTTT nnn for

sequence. Fibonacci the of termth the for stands and where nTTT n1,1 21

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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:

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.

Fig. 6.13

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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.

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.

Note that the numbers 1,2,3,5,8,13 are consecutive terms of the Fibonacci sequence.

Fig. 6.14

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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.

kt

k

rPA

1001

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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.

is value pH its then ,mol/dm

is solution a of ions hydrogen of ionconcentrat the if example, For3810

8

10log)8(

10logpH

logpH8

H

pH Value of a solution

H

H

logpH

3 then ,mol/dm in ionconcentrat ion hydrogen the denotes If

abab loglog

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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.

,)08.1(00020 nP

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

Where A0 is the original amount of the radioactive material and h is its half-life.

,20h

t

AA

The amount A of radioactive material present in an object at a time t after it dies follows the formula:

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6.3 Nine-point Circle

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.

6

Fig. 6.17