“The two highways of the life: maths and English” 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

We can write this number in a complete form as: 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: √5 _ 1 + 2 ≈ 1.618033988…

How this strange number can link the elements we said before? 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 Golden ratio (a + b) : a = a : b _ √5 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. Go to demonstration √5 _ 1 + 2 φ = Golden ratio (a + b) : a = a : b 6

The golden rectangle 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... 10 cm 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 8

Ordinary objects Do you know that many ordinary objects were designed with the golden ratio? Credit cards TV screen badge SIM card Why? The golden ratio has always been considered a ratio capable to give great harmony and beauty to the figures. tape Among all its geometric applications the golden rectangle is undoubtedly the most famous polygon. 9

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.) 10

Extraordinary objects The doriforo (450 b. C.) – Policleto 11

Extraordinary objects La Gioconda (1503-1514) – Leonardo da Vinci 12

Extraordinary objects L’uomo vitruviano (1490) – Leonardo da Vinci 1 0,618 13

Extraordinary objects L’ultima cena (1494-1498) – Leonardo da Vinci 14

Extraordinary objects La nascita di Venere (1482-1485) – Sandro Botticelli 15

Extraordinary objects Composition (1921) - Piet Mondrian 16

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. 17

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. 18

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). 19

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;… The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 203, 347, ... 20

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 = 2 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 …………………………….. 21

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. 22

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. 23

Fibonacci sequence and golden ratio in nature … or in a pine cone: 8 and 13 spirals! 24

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 _ 1 + 2 How you get to ? 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 27

Demonstration _ √5 then a2 + a – 1 = 0 and a = (-1 + √5)/2; back because b = 1-a b = 1 - (-1 + √5)/2 = (3 - √5)/2 at the end, since φ = a/b φ = (-1+√5)/2 = rationalizing = = 1,6180339… (3 - √5)/2 back √5 _ 1 + 2 28