1. Fibonacci Sequence & Golden Ratio Monika Bała

2. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci Sequence and its properties Pascal’s triangle Pascal’s triangle Golden ratio and the number Golden ratio and the number φ Properties of the number Properties of the number φ Examples of occurrences of the number in real life Examples of occurrences of the number φ in real life

3. The Fibonacci Sequence In the XIII century an Italian mathematician Leonardo Fibonacci discovered a series of numbers that have very intresting properties. This is definitely one of the most interesting mathematical sequences. The Fibonacci Sequence is closely related with the Golden Ratio. Both of them are equally common in nature.

4. Definition and properties of the Fibonacci Sequence This is the kind of recurrent sequence in which every following term is equal to the sum of the two previous terms (except for the first two).

5. Sometimes this sequence is called the golden seqence because if we divide successive terms by the previous we get the value oscillating around the number. Sometimes this sequence is called the golden seqence because if we divide successive terms by the previous we get the value oscillating around the number φ. The greater terms of the series we divide by each other, the better approximation we have. However this value will never be equal to the golden number because The greater terms of the series we divide by each other, the better approximation we have. However this value will never be equal to the golden number because φ is irrational. 2 / 1 = 2.000000000000000000 3 / 2 = 1.500000000000000000 5 / 3 = 1.666666666666666670 8 / 5 = 1.600000000000000000 13 / 8 = 1.625000000000000000 21 / 13 = 1.615384615384615380 34 / 21 = 1.619047619047619050 55 / 34 = 1.617647058823529410 89 / 55 = 1.618181818181818180 144 / 89 = 1.617977528089887640 233 / 144 = 1.618055555555555560 377 / 233 = 1.618025751072961370 610 / 377 = 1.618037135278514590 987 / 610 = 1.618032786885245900 1597 / 987 = 1.618034447821681860 2584 / 1597 = 1.618033813400125230 4181 / 2584 = 1.618034055727554180 6765 / 4181 = 1.618033963166706530 10946 / 6765 = 1.618033998521803400

6. Pascal’s Triangle triangle is a triangular array of the binomial coefficients. Each number in the triangle is the sum of two numbers that are directly above it. The numbers in the rows correspond to the coefficients in the expansion of The Pascal’s triangle is a triangular array of the binomial coefficients. Each number in the triangle is the sum of two numbers that are directly above it. The numbers in the rows correspond to the coefficients in the expansion of (a+b) n. What the Fibonacci sequence and Pascal’s triangle have in common? So, calculating the sum of the elements in the diagonal columns we get the following numbers of the sequence.

7. Fibonacci Spiral except for te first two squares). This spiral is based on the squares of the lengths of the sides equal the following terms of the Fibonacci sequence (except for te first two squares).

8. The Golden Ratio of a line segment This is a division of a line segment into two parts in such a way that the ratio of the length of the longer part by the shorter is the same as the ratio of the length of the whole line segment by the longer part of the line segment. This ratio we denote by the Greek letter φ. This is exactly the golden number which is equal

9. The Golden number is an irrational number. Its approximation to 32 decimal places is the number: 1,61803398874989484820458683436564 Golden number

10. The point E divide in the golden ratio the line segment AB if: or Because is less than zero, so the only solution is:

11. Properties of the number φ: To get its square root we only need to add one: To get the inverse number we only need to subtract one:

12. The following fraction is equal to the golden number:

13. Golden triangle The Golden triangle is an isosceles triangle, which has an acute angle of measure 36° at the vertex and two acute angles of measure 72° at the base. The Golden triangle is an isosceles triangle, which has an acute angle of measure 36° at the vertex and two acute angles of measure 72° at the base. The ratio of the length of a side of a triangle by the length of the base of this triangle is the golden number.

14. Pentagram To draw a perfect pentagram we need to draw a regular pentagon and lead diagonals or extend his sides until the intersection. The interior angle of the pentagram is 36°. The whole pentagram has hidden in itself the golden ratio. 1. The ratio of the yellow segment by the blue segment = φ 2. The ratio of the blue segment by the green segment = φ 3. The ratio of the green segment by the red segment = φ

15. Golden proportions Leonardo da Vinci noted that if the human body is built proportionally then it is inscribed in a square and a circle. Such a square and a circle define a rectangle ABCD, which for a man of the correct proportions is gold, which is the height of a man to the length of the lower part of the body (from the navel down) is the golden number. (the ratio of the lower body to the upper is the golden number) Leonardo da Vinci noted that if the human body is built proportionally then it is inscribed in a square and a circle. Such a square and a circle define a rectangle ABCD, which for a man of the correct proportions is gold, which is the height of a man to the length of the lower part of the body (from the navel down) is the golden number. (the ratio of the lower body to the upper is the golden number)

17. PARTHENON- Athena’s temple on the Acropolis in Athens, built in the years 448-432 before Christ. Fronton of the temple was located in a rectangle in which the ratio of the sides is equal to the golden number. PARTHENON- Athena’s temple on the Acropolis in Athens, built in the years 448-432 before Christ. Fronton of the temple was located in a rectangle in which the ratio of the sides is equal to the golden number.

18. PYRAMIDS IN GIZA PYRAMIDS IN GIZA If we take a cross- section the Great Pyramid we get a right- angled triangle which is called the Triangle of Egypt. The ratio of the hypotenuse to the base is 1,61804 and differs from the golden number of only about 1 in fifth place after the decimal point. If we take a cross- section the Great Pyramid we get a right- angled triangle which is called the Triangle of Egypt. The ratio of the hypotenuse to the base is 1,61804 and differs from the golden number of only about 1 in fifth place after the decimal point.

19. BIBLIOGRAPHY: http://www.naucz31.republika.pl/ http://www.naucz31.republika.pl/ http://www.naucz31.republika.pl/ http://www.jakubas.pl/matematyka/1/zlota- liczba.htm http://www.jakubas.pl/matematyka/1/zlota- liczba.htm http://www.jakubas.pl/matematyka/1/zlota- liczba.htm http://www.jakubas.pl/matematyka/1/zlota- liczba.htm http://www.zobaczycmatematyke.krk.pl/003- Golonka-Kalwaria/index.html http://www.zobaczycmatematyke.krk.pl/003- Golonka-Kalwaria/index.html http://www.zobaczycmatematyke.krk.pl/003- Golonka-Kalwaria/index.html http://www.zobaczycmatematyke.krk.pl/003- Golonka-Kalwaria/index.html

20. Thank You for your attention!