Transcript golden number - Publiczne Gimnazjum im. Jana Pawła II w Stróży

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 …
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:
   1  0
2
5 1
2
precise value:

decimal fraction
1,61803398874989484820…
continued fraction
 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1....
UNIQUE PROPERTIES OF THE GOLDEN RATIO
It is the only number which added to one is its
own square …
   1
2
and subtracted from one is its own inverse.
.
1

  1
5 1


2
1
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
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
F(1)=1
F(2)=2
F(n)= F(n-1) + F(n-2)
5 + 8 = 13
…
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 + 1 2 = 2
1X2
12 + 12 + 22 = 6
2X3
12 + 12 + 22 + 32 = 15
3X5
12 + 12 + 22 + 32 + 52 = 40
5X8
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, F (n  1) , we will find the
F ( n)
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 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.
5 1
b
a
2
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 .
| AD |

| DB |
| BC |

| AB |
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
a

b
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
1.
All pairs of rabbits consist of a male and female
2.
One pair of newborn rabbits is placed in hutch on
January 1
3.
When this pair is 2 months old they produce a pair of
baby rabbits
4.
Every month afterwards they produce another pair
5.
All rabbits produce pairs in the same manner, and …
6.
rabbits don’t die 
How many pairs of rabbits will there be 12 months later?
1st month
2nd month
3rd month
4th month
5th month
6th month
etc …
month
adult
babies
total
January
1
1
2
February
2
1
3
March
3
2
5
April
5
3
8
May
8
5
13
Juny
13
8
21
July
21
13
34
August
34
21
55
September
55
34
89
October
89
55
144
November
144
89
233
December
233
144
377
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
