Nature`s number Lesson 1

Nature’s number
A.
1.
Lesson 1
Retangles
Draw a retangle on a blank sheet of paper.
Measure its length and its width, and determine the ratio of the length to the
width, rounded o to two deimal plaes.
2.
Make the same alulation using a postard, a book, a TV sreen or any other
retangle you an nd.
3.
How ould you desribe a perfet retangle ?
B.
The Golden Ratio
Today's programme by Simon Singh is about a number that an be found in paintings and arhiteture, in the struture of DNA and of snail shells, in pineapples and
in sunowers, in the rhythm of heart beats and in loads of mathematial problems ;
however, very few people know about it nowadays.
So Simon Singh starts by going bak to the origins of a forgotten number.
I. The origins
1
But rst, where does this number ome from ? Well, imagine I've got a
2
line and I divide it into two parts, suh that the ratio of the short bit to
3
4
entire length.
That only works with a ratio of 1.618 to 1. That's the Golden Ratio :
the long bit is idential to the ratio of the long bit to the
5
1.618.
6
And, as Ian Stewart, author of Nature's Numbers, points out, the
7
loved it.
8
They saw it I think as one of the fundamentals of geometry.
9
The Greeks were very hot on this Platonist onept of the
Greeks
ideal world,
10
with the perfet irle, the perfet line, and the Golden Number as a sort
11
of perfet ratio. They thought of it as two
12
the other.
13
So just as in the same way as
14
and a diameter. . .
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lengths,
one about 1.6 times
is the ratio between
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a irumferene
Tuesday, September 15
th,
2009
1.
2.
Can you translate the last sentene into a formula ?
Let
a and b be the lengths of two line segments that are in the Golden Ratio.
Use the denition of the Golden Ratio to nd an equation depending only on
r=
3.
b
.
a
Is 1.618 an exat solution to this equation ?
II. An approximation
The mathematiian Ron Knott gives us a way of approximating the Golden Ratio
using a sequene :
reiproal. So
1
Usually on a alulator there is a 1/x button, or take the
2
if you tap in any number, then add 1 to it, and then hit the 1/x button,
3
and then add 1 to that, and hit the 1/x button, then add 1 to that, hit
4
the 1/x button, keep doing that, you'll nd very soon it'll
5
to the Golden setion number 0.618 or 1.618.
1.
settle down
Fill in the following table, starting with any number you like (round o to the
thousandth).
a
2.
b=a+1
=
Use your alulator to nd out an approximation of the Golden Ratio to ve
deimal plaes.
III. An exat value
The mathematiian Robin Wilson then explains how we an nd out an exat value
for the Golden Ratio.
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Tuesday, September 15
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2009
1
It's a number whih is about 1.618. But if you happen to square it, then
2
what you get is 2.618, and if you happen to take its
3
the number, you get 0.618. So it's a very strange number : if you square
4
it it's the same as adding 1, and if you take 1 over it it's the same as
reiproal, 1
over
5
subtrating 1.
6
The reason it has these properties is that it is atually a number whih
satises a ertain quadrati equation : x2 = x + 1.
7
1.
a) Do we already know of an equation whih has the Golden Ratio as a solution ?
b) Is this equation a quadrati equation, as Robin Wilson seems to say ?
p
p
5 and 1 + 5 are two solutions to this equa2
2
) Can you transform our equation into a quadrati equation ?
1
2. a) Chek that the numbers
tion.
b) Whih one of these numbers is the Golden Ratio ?
) What kind of number is the Golden Ratio ?
3.
Can you use the equations we found to explain why if you square it it's the same
as adding 1, and if you take 1 over it it's the same as subtrating 1 ?
C.
Fibonai's sequene
Amongst other things, the Golden Ratio is very interesting beause you an stumble
upon it in quite unexpeted plaes.
th entury
Italian known as Fibonai, who unwittingly diso-
1
Meet a 12
2
vered a hitherto hidden aspet of the Golden Ratio.
3
Start with a list ontaining just 1 and 1, and add them. So now our list
4
is 1, 1, 2.
5
Add the last two numbers 1+2=3 and keep doing this, adding to the list :
6
2+3=5, 3+5=8, and you an keep doing this. Add the last two numbers
7
on the list to get the next one. Any number on this list is a Fibonai
8
number. 1, 1, 2, 3, 5, 8, 13,
9
What's fantasti is that the numbers from this sequene turn up in
10
21, 34, 55 and so on.
the most unlikely plaes.
I. In a ar park
Simon Singh then interviews the mathematiian Ron Knott.
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Tuesday, September 15
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2009
1
Supposing you're in a ar park and the mahine takes ¿1 and ¿2 oins
2
and you want a tiket say for 3 pounds. How many
3
an you put in to make your 3 pounds ? And you've only got ones and
4
two pounds
5
put in a ¿2 oin and then a 1, or you ould put the ¿1 oin in rst and
6
then the 2. So there are three ways to make a 3 pound tiket.
7
Now supposing I wanted to stay a bit longer, and it was ¿4. How many
8
ways ould you do it ?
1.
sequenes
of oins
available. Well you ould put in three ¿1 oins, or you ould
Can you nd out how many ways there are to pay for a ¿4 tiket with ¿1 a ¿2
oins ?
2.
And what about a ¿5 tiket ?
II. Simon Singh's onlusion : Fibonai and the Golden Ratio
1
So far I haven't said expliitly how Fibonai numbers are linked to the
2
Golden Ratio whih is where we started. Well, remember, the Golden Ra-
3
tio is roughly 1.618. Now, take any two neighbouring Fibonai numbers,
4
divide
the larger by the smaller, and you get roughly 1.618.
Can you explain the link between Fibonai numbers and the Golden Ratio ?
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Tuesday, September 15
th,
2009