1. THE GOLDEN RATIO GREEK PRESENTATION 3 rd MEETING – BONEN –GERMANY October 2009 COMENIUS PARTNERSHIPS

2. In the “Elements”, Euclid of Alexandria (ca. 300 BC) defines a proportion derived from a division of a line into what he calls its "extreme and mean ratio." Euclid's definition says: “A straight line is said to have been cut in extreme and mean ratio while, as the whole line is to the greater segment, so is the greater to the lesser.”

3. In other words, in the diagram below, point C divides the line in such a way that the ratio of AC to CB is equal to the ratio of AB to AC. Some basic algebra shows that in this case the ratio of AC to CB is equal to the irrational number 1.618….

4. C divides the line segment AB according to the Golden Ratio. In other words, we have to solve this simple 2 nd degree equation. If AB = 1 and AC = X then CB = 1-X and the proportion is:  x 2 + x – 1 = 0. And

5. = 0.618034.. Where only one root is accepted as positive number. So, CB = 1 – 0.618034 = 0.381966.. and if we divide AC/CB, the result is 1.618034…. The Geometric solution (using Rule and Compass only) is:

6. CAB A΄A΄Β΄C΄C΄ D D΄D΄ Ε΄ Ε ACC΄A΄ square, DD΄B΄C΄ square, EBD΄E΄ square, …..

7. This famous number was named, in honor of the name of the architect of Parthenon in Athens, Phidias, (ΦΕΙΔΙΑΣ) as Φ (Phi) number because this world masterpiece, Parthenon, was built by dimensions which have this strange number Φ.

10. A panoramic view of the ancient theatre of Epidaurus in Greece (4 th BC). The lower sector of seats consists of 34 tracks of seats, the upper one, of 21 tracks…. mm.. divide please these two numbers: 34/21 = … ? ;-)

11. This strange number Φ appears in many beings on earth. At first, in human bodies like the shape below:

13. Also appears in various plants, such as: a Sunflower

14. Or a pine cone

15. Or a sea being: Nautilus

16. Also, we can find this number not only in Architecture but in many famous paints. For example Leonardo Da Vinci (16 th AC) and his famous “Mona Lisa”

17. And now, another Leonardo… Leonardo Fibonacci and his rabbits. The original problem, that Leonardo Fibonacci in his famous book “Liber Abaci“, presented in the year 1202 was about how fast rabbits could breed in ideal circumstances.

18. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. At the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month. Well,

19. How many pairs will there be in one year? JANUARY FEBRUARY MARCH APRIL MAY.

20. The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ….. If we divide any two sequential numbers, we will find again this number Φ.

21. How strange and what an odd coincidence!!! Can we suppose that …perhaps… exists an “underground” link among all them in Nature?

22. WE THANK YOU MARIA CHARALAMPIDI GEORGE MAKANTASIS NIKOLAS PSYLAKIS