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Greece, 16-21 october 2012
Comenius Project
2011- 2013
“The two highways of the life: maths and English”
The golden ratio
Introduction
What they have in common the following elements?
The disposition of a
sunflower seeds
The spiral of a shell
The shape of a
galaxy
The harmony of many works of arts and architecture
Introduction
Incredibly, the answer to this question is a simple
number called the golden number or the golden ratio
or… “φ” (phi).
The value of φ is
1.6180339887…
We can write this number in a
complete form as:
_
1 +√5
2
≈ 1.618033988…
Introduction
This number is unlimited and
aperiodic.
So it consists in endless numbers
that are not repeated in a
predictable way.
What does it mean?
How this strange number
can link the elements we
said before?
Where it comes from?
?!
Introduction
The golden ratio (φ) is an irrational number, and
probably, it is a discovery of the classical Greek
mathematicians. We can found it in “The elements of
geometry” of Euclid (300 b. C.).
The
symbol
φ
was
attributed to the number
only
in
the
twentieth
century
when
the
Norwegian
mathematician
Mark Barr proposed to link
this number to Phidias and
therefore to the Parthenon.
Portrait of Phidias, detail from “The
Apotheosis of Homer” (Ingres, 1827).
Definition
The golden ratio (phi) is represented as a line divided
into two segments a and b, such that the entire line is to
the longer a segment as the a segment is to the shorter
b segment.
_
φ=
1 +√5
2
Golden ratio
(a + b) : a = a : b
Go to demonstration
The golden rectangle
10 cm
In order to build a golden rectangle is sufficient that
the longer side is the result of the smaller side
multiplied for φ, in other words that the ratio between
width and height is equal to 1,6180...
16,18 cm
The golden rectangle
How to draw a golden rectangle?
1) We can start from a square
2) We divide it in two equal parts
3) We draw an arc using the diagonal of the half square
4) This will give us the right length of the second dimension
Ordinary objects
Do you know that many ordinary
designed with the golden ratio?
Credit cards
TV screen
badge
objects
were
SIM card
Why?
tape
The golden ratio has always been
considered a ratio capable to give great
harmony and beauty to the figures.
Among all its geometric applications the golden rectangle is
undoubtedly the most famous polygon.
Extraordinary objects
Many artists used in their works of architecture,
sculpture and picture the fascination of the golden ratio.
The Partenone (447-438 b. C.)
Extraordinary objects
The doriforo (450 b. C.) – Policleto
Extraordinary objects
La Gioconda (1503-1514) – Leonardo da Vinci
Extraordinary objects
L’uomo vitruviano (1490) – Leonardo da Vinci
1
0,618
Extraordinary objects
L’ultima cena (1494-1498) – Leonardo da Vinci
Extraordinary objects
La nascita di Venere (1482-1485) – Sandro Botticelli
Extraordinary objects
Composition (1921) - Piet Mondrian
Golden ratio and geometry
The number φ turns up frequently in geometry,
particularly in figures with pentagonal symmetry.
The length of a regular
pentagon's diagonal is φ
times its side. The triangle
ADB is named golden
triangle because the ratio
between AD and DB is φ.
Spirals and golden ratio
Starting from a golden rectangle is possible to construct
a golden spiral:
from a golden rectangle we subtract a square with a
length side equal to the height of the rectangle. A new
golden rectangle is built. Then, in the square, we draw
an arc as we show you in figure.
If we repeat this
operation, we can
construct an infinite
number of smaller
golden
rectangles
and a golden spiral.
Spirals and golden ratio in
nature
Different forms in nature follow the mathematical rules
of the golden spirals; as exemple: the petals of a rose
and the internal structure of a shell (Nautilus).
The Fibonacci sequence
Leonardo Pisano, known as Fibonacci, was born in Pisa
in 1170. Famous is its numerical sequence.
By definition, the first two numbers in the Fibonacci
sequence are 0 and 1, and each subsequent number is
the sum of the previous two.
0+1=1;
1+1=2;
1+2=3;
2+3=5;
3+5=8;
5+8=13;
8+13=21;
13+21=34;…
Fibonacci sequence and
golden ratio
What is the link between Fibonacci sequence and
golden ratio?
In a Fibonacci sequence, if we divide a term by its
previous we obtain a number that is closer to the
golden ratio moving towards the greatest terms of the
sequence.
1:1 =1
21 : 13
= 1.61538…
2: 1
34 : 21
= 1.6190…
3 : 2 = 1.5
55 : 34
= 1.6176…
5 : 3 = 1.666…
89 : 55
= 1.6181818…
8 : 5 = 1.6
144 : 89 = 1.617977…
13 : 8 =1.625
……………………………..
=2
Fibonacci sequence and
golden ratio in nature
In nature we can find several times the Fibonacci
sequence!
If we count the spirals in a “roman cauliflower” we can
find numbers of the Fibonacci sequence as 8 and 13.
Fibonacci sequence and
golden ratio in nature
Also in a sunflower can be counted spirals in a number
equal to the Fibonacci sequence, like 34 and 21.
Fibonacci sequence and
golden ratio in nature
… or in a pine cone: 8 and 13 spirals!
Conclusions
Mathematics is a science, but it is
also a key to understand many
aspects of nature and life.
Harmony and beauty, perceived by
our senses, are the result of
mathematics relationships.
The golden ratio shows how many
aspects of nature, art and history
of man can be deeply related.
References
Corbalàn Fernando, 2011. “La sezione aurea”. RBA Italia S.r.l. – Mondo
Matematico – ISSN 2039-1153
Lahoz-Beltra Rafael, 2011. “La matematica della vita”. RBA Italia S.r.l. –
Mondo Matematico – ISSN 2039-1153
Binimelis Bassa Maria Isabel, 2011. “Un nuovo modo di vedere il mondo”.
RBA Italia S.r.l. – Mondo Matematico – ISSN 2039-1153
Web sites:
http://en.wikipedia.org/wiki/Golden_ratio
http://www.isypedia.com/la-sequenza-di-fibonacci.html
http://www.macrolibrarsi.it/speciali/numeri-magici-in-natura.php
http://www.mi.sanu.ac.rs/vismath/jadrbookhtml/part42.html
Demonstration
In mathematics two quantities are in the golden ratio
(φ) if the ratio of the sum of the quantities to the larger
quantity is equal to the ratio of the larger quantity to the
smaller one.
_
√5
How you get to 1 +
2
?
If we said that (a+b) = 1
1:a=a:b
but b = (1-a) then
1 : a = a : (1-a)
for the fundamental property of proportions
1-a = a2
Demonstration
then
a2 + a – 1 = 0
back
and
a = (-1 + √5)/2;
because
b = 1-a
then
b = 1 - (-1 + √5)/2 = (3 - √5)/2
at the end, since φ = a/b
φ = (-1+√5)/2
(3 - √5)/2
= rationalizing =
_
1 +√5 = 1,6180339…
2