Nature’s number A.
1.
Lesson 1
Re tangles
Draw a re tangle 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 de imal pla es.
2.
Make the same al ulation using a post ard, a book, a TV s reen or any other re tangle you an nd.
3.
How ould you des ribe a perfe t re tangle ?
B.
The Golden Ratio
Today's programme by Simon Singh is about a number that an be found in paintings and ar hite ture, in the stru ture of DNA and of snail shells, in pineapples and in sun owers, in the rhythm of heart beats and in loads of mathemati al problems ; however, very few people know about it nowadays. So Simon Singh starts by going ba k 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, su h 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 identi al 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 on ept of the
Greeks
ideal world,
10
with the perfe t ir le, the perfe t line, and the Golden Number as a sort
11
of perfe t ratio. They thought of it as two
12
the other.
13
So just as in the same way as
14
and a diameter. . . Première Euro
lengths,
one about 1.6 times
is the ratio between
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Tuesday, September 15
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2009
1. 2.
Can you translate the last senten e into a formula ? Let
a and b be the lengths of two line segments that are in the Golden Ratio.
Use the de nition of the Golden Ratio to nd an equation depending only on
r=
3.
b . a
Is 1.618 an exa t solution to this equation ?
II. An approximation The mathemati ian Ron Knott gives us a way of approximating the Golden Ratio using a sequen e :
re ipro al. So
1
Usually on a al ulator 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 se tion 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 al ulator to nd out an approximation of the Golden Ratio to ve de imal pla es.
III. An exa t value The mathemati ian Robin Wilson then explains how we an nd out an exa t value for the Golden Ratio.
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1
It's a number whi h 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
re ipro al, 1
over
5
subtra ting 1.
6
The reason it has these properties is that it is a tually a number whi h
satis es a ertain quadrati equation : x2 = x + 1.
7
1.
a) Do we already know of an equation whi h 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) Che k that the numbers tion.
b) Whi h 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 subtra ting 1 ?
C.
Fibona
i's sequen e
Amongst other things, the Golden Ratio is very interesting be ause you an stumble upon it in quite unexpe ted pla es.
th entury
Italian known as Fibona
i, who unwittingly dis o-
1
Meet a 12
2
vered a hitherto hidden aspe t 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 Fibona
i
8
number. 1, 1, 2, 3, 5, 8, 13,
9
What's fantasti is that the numbers from this sequen e turn up in
10
21, 34, 55 and so on.
the most unlikely pla es.
I. In a ar park Simon Singh then interviews the mathemati ian Ron Knott. Première Euro
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Tuesday, September 15
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1
Supposing you're in a ar park and the ma hine takes ¿1 and ¿2 oins
2
and you want a ti ket 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 ti ket.
7
Now supposing I wanted to stay a bit longer, and it was ¿4. How many
8
ways ould you do it ?
1.
sequen es
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 ti ket with ¿1 a ¿2
oins ?
2.
And what about a ¿5 ti ket ?
II. Simon Singh's on lusion : Fibona
i and the Golden Ratio 1
So far I haven't said expli itly how Fibona
i numbers are linked to the
2
Golden Ratio whi h is where we started. Well, remember, the Golden Ra-
3
tio is roughly 1.618. Now, take any two neighbouring Fibona
i numbers,
4
divide
the larger by the smaller, and you get roughly 1.618.
Can you explain the link between Fibona
i numbers and the Golden Ratio ?
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