Download Year 7 The Golden Number

1
Leonardo of Pisa
(1170 – 1250 AD)
was an Italian
mathematician. He
is sometimes called
Fibonacci.
Fibonacci is famous
for helping to spread
the use of Hindu
Arabic numbers in
Europe. These
numbers replaced
the Roman number
system.
2
Italy
3
There is a special sequence of numbers
called the Fibonacci sequence because
Fibonacci wrote about this sequence.
0,1,1, 2 , 3,
5 , 8 , 13 , 21 …
4
Number
Fibonacci Number
Calculation
1
0
2
1
3
1
1  1
1
4
2
2  1
2
5
3
3  2
1.5
6
5
5  3
1.666666666667
7
8
8  5
1.6
8
13
13  8
1.625
9
21
21  13
1.615384615385
10
34
34  21
1.619047619048
11
55
55  34
1.617647058824
12
89
89  55
1.618181818182
13
144
144  89
1.617977528090
14
233
233  144
1.618055555556
15
377
377  233
1.618025751073
16
610
610  377
1.618037135279
17
987
987  610
1.618032786885
18
1597
1597  2987
1.618034447822
19
2584
2584  1597
1.618033813400
20
4181
4181  2584
1.618034055728
Golden Number ()
5
The Fibonacci numbers can be used to
calculate the value of a special number
called the golden number.
The golden number cannot be written down
exactly because it is a decimal number that
goes on for ever without any recurring
pattern.
Because the golden number cannot be
written down exactly we use the 21st letter of
the Greek alphabet,  or phi to represent the
golden number.
6
The Greek
Alphabet
7
This is  to 20 decimal places
1.618 033 988 749 894 848 20
8
This is  to 1000 decimal places
1.618033988749894848204586834365638117720309179805762862
135448622705260462818902449707207204189391137484754088075
386891752126633862223536931793180060766726354433389086595
939582905638322661319928290267880675208766892501711696207
032221043216269548626296313614438149758701220340805887954
454749246185695364864449241044320771344947049565846788509
874339442212544877066478091588460749988712400765217057517
978834166256249407589069704000281210427621771117778053153
171410117046665991466979873176135600670874807101317952368
942752194843530567830022878569978297783478458782289110976
250030269615617002504643382437764861028383126833037242926
7526311653392473167111211588186385133162038400522216579128
667529465490681131715993432359734949850904094762132229810
172610705961164562990981629055520852479035240602017279974
717534277759277862561943208275051312181562855122248093947
123414517022373580577278616008688382952304592647878017889
921990270776903895321968198615143780314997411069260886742
96226757560523172777520353613936
9
Finding the Golden Number
2.2
2.0
1.8
Golden Number
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
5
10
Number
15
20
Egypt
It is thought that the Egyptians may have
known about  over 2500 years
11
Some people think that the Egyptians used
 in the design of the Great pyramid of
Giza which was completed about 2500
years ago.
12
13
Any rectangle that has a length
that is  times the width is called a
golden rectangle
l
w
This is a golden rectangle because
l=w
14
Width Length
Rectangle
(cm)
(cm)
  Width
(cm)
Golden
rectangle?
A
3.0
4.9
1.618  3.0 = 4.854
Yes
B
3.5
6.7
1.618  3.5 = 5.663
No
C
2.0
3.2
1.618  2.0 = 3.236
Yes
D
3.5
5.7
1.618  3.5 = 5.663
Yes
E
2.2
7.0
1.618  2.2 = 3.5596
No
F
5.0
8.1
1.618  5.0 = 8.09
Yes
15
Greece
People in Greece knew about golden
rectangles about 2500 years ago.
16
The Parthenon is a temple that was built
about 2500 years ago on the Acropolis
in Athens, Greece.
17
Many golden rectangles can be found in
the design of the Parthenon.
18
19
France
The west face of the cathedral of Notre Dam
in Paris which was completed in the 13th
century contains many golden rectangles.
20
21
22
India
The Taj Mahal In Agra in India was complet
in around 1648 AD. Many golden rectangle
can be found in the Taj Mahal.
23
24
25
26
Draw a square of any size.
27
Mark the midpoint of one side of the square.
Extend the same side of the square.
28
Put the point of the
compasses on the
midpoint
Put the pencil
on this corner
Draw an arc from the corner of the square
to the extend line.
29
Draw the golden rectangle
30
Mexico
Guatemala
Honduras
The Mayan people have lived in the countries
that are now Mexico, Guatemala and
Honduras for thousands of years.
31
It is thought that the Mayan people used golden
rectangles in their architecture. This is part of
32
the Maya ruins of Copan in Honduras.
33
Angles
Right angle Acute angle
Less than 90º
90º
Obtuse angle
Between 90º
and 180º
34
Triangles
Equilateral Triangle
Right Angled Triangle
Isosceles Triangle
Scalene Triangle35
There are two special isosceles triangles that
are called golden triangles. In both the length
of the longer side is  times the length of the
shorter side.
Golden triangle with
the two equal sides
shorter than the
third side.
Golden triangle with
the two equal sides
longer than the third
side.
36
This golden
triangle has one
obtuse angle and
two acute angles.
This golden triangle
has three acute
angles
37
38
Constructing a golden triangle with
three acute angles
39
Draw a line 5 cm
long
Multiply the
length of the line
by 1.618 ()
Your answer
should be 8.09
which is 8.1 to
1 decimal place
Write down your
answer correct to
1 decimal place.
40
Set the compasses
so that the point
and the pencil are
8.1cm apart.
Put the point of the
compasses on the
right hand side of
the line and draw
an arc above the
middle of the line.
41
Keep the
compasses at the
same setting
Put the point on
the left hand side
of the line and
draw an arc to
cross the first arc.
42
Carefully join the ends
of the line to the point
where the two arcs
cross.
You now have a golden
isosceles triangle.
Estimate the size of
each angle.
Measure the three
angles of the triangle
43
$1$
36º
72º
$1$
72º
44
Golden triangle with one obtuse
and two acute angles.
45
Draw a line 10 cm long
Divide the length of the line by 1.618 ()
Write down your answer correct to 1
decimal place.
Your answer
should be 6.1804 …
which is 6.2 to 1
decimal place.
46
Set the compasses to 6.2 cm.
Put the point of the compasses on the
right hand side of the line and draw an
arc above the middle of the line.
47
Keep the compasses at the same
setting
Put the point on the left hand side of
the line and draw an arc to cross the
first arc.
6.1
48
Carefully join the ends of the line to the point
where the two arcs cross.
You now have a golden isosceles triangle.
Estimate the size of the three angles.
Measure the three angles of the triangle
49
108º
36º
36º
50
51
Quadrilaterals
Rectangle
Square
Rhombus
Parallelogram
Trapezium
Kite
52
6.472
Quadrilaterals made from two obtuse
angled golden triangles
36º
144º
144º
Parallelogram
36º
36º
108º
72º
216º
72º
72º
108º
Rhombus
36º
Arrowhead Kite
53
Quadrilaterals made from two acute
angled golden triangles
36º
108º
72º
72º
144º
144º
72º
108º
Parallelogram
72º
72º
144º
36º
Arrowhead Kite
Rhombus
54
55
Congruent shapes are exactly the same
shape and the same size.
56
Similar shapes are the same shape but
the sizes are different.
57
These Shreks are not similar or congruent
because the shapes have changed.
58
$1$
4
6.472
59
60
Start with a strip of paper
Carefully tie an knot in
the strip and flatten it.
Make sharp creases
on these folds
Untie the knot and straighten out the paper
61
Draw in lines to show the three creases.
What shapes do the creases make?
A parallelogram
Two trapeziums
62
Measure the lengths of the sides of the
parallelogram and a trapezium.
What do you notice about the lengths?
Can you find any golden numbers by
dividing the lengths of the sides?
63
Carefully tie the knot
again.
Fold back the spare paper here.
Make creases then cut along the creases.
What is the name of the
shape you have left?
64
65
A pentagram is a star made from five
equal straight lines
For over 5000 years the pentagram has
been used as a special symbol by
different groups of people.
66
Mesopotamia
The first known use of the pentagram was
found in writings from about 3000 BC in a
country called Mesopotamia which was
where Iraq is today.
67
To draw a pentagram start with a
regular pentagon.
68
Draw lines connecting one corner to
the two opposite corners.
69
Join the remaining corners in the
same way.
70
$1$
36º
72º
$1$
72º
71
36º
36º
108º
36º
36º
72º
36º
36º
72º 72º 108º
108º
72º
108º
72º
108º
108º
108º
72º
72º
36º
72º
72º
108º
36º
36º
36º
36º
72º
108º
108º
36º
36º
36º
36º
72
Measure the lines coloured red, green,
pink and blue to the nearest
millimetre.
73
Make a table showing the results of
your measurements.
Line
Length
(cm)
Red
Green
Pink
Blue
74
Without using a calculator work out
1.
Pink length + Blue length
2.
Pink length + Green length
Write down what you notice about
your answers.
75
Use a calculator to work out

1.
Red length
2.
Green length
3.
Pink length
Green length


Pink length
Blue length
Write down what you notice about
your answers.
76
Pentagrams inside pentagrams
77
Rhombus
78
Kite
79
Arrow Head Kite
80
Isosceles Trapezium
81
Trapezium
82
Golden Triangle 1
83
Golden Triangle 2
84
85
For hundreds of years artists have
used the golden number to help
with the design of pictures because
some people believe that this is
how to create the most beautiful
and pleasing picture.
86
In 1509, Luca Pacioli, an Italian monk
published a book about the golden number
and art. The book was illustrated by
Leonardo da Vinci.
87
Leonardo da Vinci
(1452 – 1519) was a
painter, engineer,
scientist and
mathematician.
Some people say
that he was the
most talented
person who has
ever lived. This is
his self portrait.
88
Leonardo da Vinci used the golden number to
help him to design his pictures. This picture
is called The Annunciation.
89
The Mona Lisa is
one of Leonardo da
Vinci’s most
famous paintings.
90
length of the red line  the length of the
white line is .
91
This painting is “Bathers at Asnieres” by the
French impressionist painter Georges Seurat
(1859 – 1891). He made use of the golden
number in many of his paintings.
92
The Dutch artist Piet Mondrian (1872-1944) Often
had golden rectangles in his paintings
Composition in
Red and Yellow
Composition in Red
Yellow and Blue 93
In 1955 Salvador Dali painted “The Sacrament
of the Last Supper” inside a golden rectangle94
95
Vitruvian man
created by
Leonardo da
Vinci around the
year 1487
96
97