Publiczne Gimnazjum im. Jana Pawła II w Stróży Polish Team GOLDEN RATIO IN MATHEMATICS Publiczne Gimnazjum im. Jana Pawła II w Stróży Polish Team

„Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel." Johannes Kepler, (1571-1630) In this presentation we are simply going to demonstrate the truth of these words …

φ = (a+b) : a = a : b WHAT’S THIS ? The golden ratio is the division of a line segment into two unequal parts such that the ratio of the whole to the larger part is the same as the ratio of the larger to the smaller φ = (a+b) : a = a : b

GOLDEN NUMBER  ( phi ) Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. It is solving the equation: precise value: decimal fraction 1,61803398874989484820…

continued fraction

UNIQUE PROPERTIES OF THE GOLDEN RATIO It is the only number which added to one is its own square … and subtracted from one is its own inverse. .

OTHER NAMES FOR THE GOLDEN RATIO golden section golden mean extreme and mean ratio golden number divine proportion (‘divina proportio’) divine section medial section golden cut mean of Phidias

In the book, he also introduced the so-called rabbits problem. Leonardo Fibonacci The gratest European mathematician of the Middle Ages. He was the first to introduce the Hindu - Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 0 . He wrote a book on how to do arithmetic in the decimal system, called "Liber abaci", completed in 1202. It describes the rules we are all now learn at elementary school for adding numbers, subtracting, multiplying and dividing. In the book, he also introduced the so-called rabbits problem.

FIBONACCI SEQUENCE Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 … The numbers in the Fibonacci sequence are called Fibonacci numbers

And now, something a bit more interesting  Each term in Fibonacci sequence is simply the sum of the two preceding terms 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 5 + 8 = 13 … F(1)=1 F(2)=2 F(n)= F(n-1) + F(n-2)

Want something more ? Every 3rd Fibonacci number is divisible by 2. Every 4th Fibonacci number is divisible by 3. Every 5th Fibonacci number is divisible by 5. Every 6th Fibonacci number is divisible by 8. Every 7th Fibonacci number is divisible by 13. Every 8th Fibonacci number is divisible by 21. Every 9th Fibonacci number is divisible by 34.

If you take the ratio of any number in the Fibonacci sequence to the next number, the ratio will approach the approximation 0.618. This is the reciprocal of Phi: 1 / 1.618 = 0.618. It means that the decimal integers of a number and its reciprocal is exactly the same. Fibonacci numbers are creating the numeral system. Every integral number can be introduced as the sum of Fibonacci numbers: 1 000 000= 832 040 +121 393+46 368+144+55 There are only two Fibonacci numbers which are squares: 1 and 144. 144 is number 12 in the series and its square root is 12. There are precisely two Fibonacci numbers which are cubes: 1 and 8.

SUMS OF FIBONACCI NUMBERS F1 + F2 + F3 + … + FN = FN+2 -1 1 + 1 = 2 3 - 1 1 + 1 + 2 = 4 5 - 1 1 + 1 + 2 + 3 = 7 8 - 1 1 + 1 + 2 + 3 + 5 = 12 13 - 1 1 + 1 + 2 + 3 + 5 + 8 = 20 21 - 1

SUMS OF SQUARES F12 + F22 + F32 + …+ Fn2 = Fn X FN+1 12 + 12 = 2 1 X 2 12 + 12 + 22 = 6 2 X 3 12 + 12 + 22 + 32 = 15 3 X 5 12 + 12 + 22 + 32 + 52 = 40 5 X 8 12 + 12 + 22 + 32 + 52 + 82 = 104 8 X 13

FIBONACCI SEQUENCE AND THE GOLDEN RATIO If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, , we will find the following series of numbers: 1 : 1 = 1,   2 : 1 = 2,   3 : 2 = 1,5,   5 : 3 = 1,666..., 8 : 5 = 1,6,   13 : 8 = 1,625,   21 : 13 = 1,61538... The ratio is settling down to the golden number

The Fibonacci Series is found in Pascal's Triangle. Pascal's Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1.  Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero. The numbers on diagonals of the triangle add to the Fibonacci series, as shown below.

Pascal's triangle has many unusual properties and a variety of uses: Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.) The horizontal rows represent powers of 11 (1, 11, 121, 1331, etc.) Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28... results in a perfect square (1, 4, 9, 16, etc.) It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row.  Do not count the 1's.) When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number

Some other examples of the golden section … The Greeks usually attributed discovery of the golden ratio to Pythagoras or his followers. One of the Pythagoreans' main symbols was the Pythagorean Pentacle, a pentacle inscribed within a pentagon. Pythagoras and his Pythagoreans venerated the pentacle and the golden ratio. Pythagoras viewed the pentacle as a symbol of mathematical and natural perfection. It is said that the pentagram had a secret significance and power to the Pythagoreans, and was used not only as a symbol of good health, but as a password or symbol of recognition amongst themselves.

The measure of the angle in each vertex of the pentagram is equal 36º The measure of the angle in each vertex of the pentagram is equal 36º. The sum of angles amounts 180 °. Each intersection of edges sections the edges in golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ. A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

An intersection of diagonals in the regular pentagon is dividing them according to the golden mean.

We consider a golden triangle with a top angle measuring 36° We consider a golden triangle with a top angle measuring 36°. Both base angles then measure 72°.. It is isosceles triangle, of which the attitude of the shoulder to the base is equal the golden number .

If we take the isoceles triangle that has the two base angles of 72° and we bisect one of the base angles, we should see that we get another Golden triangle that is similar to the first (Figure 1). If we continue in this fashion we should get a set of Whirling Triangles (Figure 2). Figure 3 Figure 1 Figure 2 Out of these Whirling Triangles, we are able to draw a logarithmic spiral that will converge at the intersection of the the two blue lines in Figure 3.

If we connect the vertices of the regular pentagon, we can get two different Golden Traingles. The blue triangle has its sides in the golden ratio with its base, and the red triangle has its base in the golden ratio with one of the sides.

If we inscribe a regular decagon in a circle, the ratio of a side of the decagon to the radius of the circle forms the golden ratio (Figure 1). Figure 2 Figure 1 If we divide a circle into two arcs in the proportion of the golden ratio, the central angle of the smaller arc marks off the Golden Angle, is 137.5° (Figure 2).

Look at the following rectangles: which of them seems to be the most naturally attractive rectangle? If you said the first one, then you are probably the type of person who likes everything to be symmetrical. Most people tend to think that the third rectangle is the most appealing If you were to measure each rectangle's length and width, and compare the ratio of length to width for each rectangle you would see the following: Rectangle one: ratio 1 : 1 Rectangle two: ratio 2 : 1 Rectangle Three: ratio 1.618 : 1 Have you figured out why the third rectangle is the most appealing? That's right - because the ratio of its length to its width is the Golden Ratio!

The rectangle is called a “golden rectangle” when its sides are in the ratio of the golden number. After drawing on it the square with the side equal of the long side of the rectangle a new, bigger golden rectangle is received

If we start with a square and add a square of the same size, we form a new rectangle. If we continue adding squares whose sides are the length of the longer side of the rectangle, the longer side is always a successive Fibonacci number. Eventually the large rectangle formed will look like a Golden Rectangle - the longer you continue, the closer it will be.

GOLDEN SPIRAL It is a similar thing as we did with the Golden Triangle. Successive points dividing a golden rectangle into squares (a set of Whirling Rectangles) lie on a logarithmic spiral. The length of the side of a larger square to the next smaller square is in the golden ratio.

We almost forgot about Fibonacci rabbits problem … that is why so many rabbits are coming in the world 

Rabbit Rules All pairs of rabbits consist of a male and female One pair of newborn rabbits is placed in hutch on January 1 When this pair is 2 months old they produce a pair of baby rabbits Every month afterwards they produce another pair All rabbits produce pairs in the same manner, and … rabbits don’t die  How many pairs of rabbits will there be 12 months later?

1st month Jedna para nowo narodzonych królików

2nd month Jedna para niezdolna jeszcze do rozmnażania

3rd month Jedna para urodziła już nowa parę

4th month Para rodzi kolejną parę, a ta urodzona miesiąc wcześniej jeszcze nie może się rozmnażać

5th month

6th month

etc … month adult babies total January 1 2 February 3 March 5 April 8 May 13 Juny 21 July 34 August 55 September 89 October 144 November 233 December 377

These are numbers of Fibonacci Sequence Received numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … These are numbers of Fibonacci Sequence

Thank you for your patience and attention 

made by: wonderful Polish students and their teachers 