The document discusses the Fibonacci sequence and its properties. It defines the Fibonacci sequence as a sequence where each number is the sum of the previous two, starting with 0 and 1 or 1 and 1. It then provides examples of where the Fibonacci sequence and golden ratio appear in nature, such as spiral galaxies, DNA molecules, seed heads, tree branches, and fingers. It also discusses applications of the Fibonacci sequence in mathematics, pseudorandom number generation, and software development estimation techniques.

1. SHOHRAT OVEZOV

2. ▪ In mathematics, the Fibonacci numbers are the numbers in the
following integer sequence, called the Fibonacci sequence, and characterized
by the fact that every number after the first two is the sum of the two preceding
ones:
▪ 1,1,2,3,5,8,13,21,34,55,89,144,…

3. ▪ Often, especially in modern usage, the sequence is extended by one more
initial term:
▪ 0,1,2,3,5,8,13,21,34,55,89,144,…
▪ By definition, the first two numbers in the Fibonacci sequence are either 1 and
1, or 0 and 1, depending on the chosen starting point of the sequence, and each
subsequent number is the sum of the previous two.
▪ In mathematical terms, the sequence 𝐹𝑛 of Fibonacci numbers is defined by
the recurrence relation
▪ 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2

4. ▪ The Fibonacci sequence is named after Italian mathematician Leonardo of
Pisa, known as Fibonacci, His 1202 book Liber Abaci introduced the sequence
to Western European mathematics, although the sequence had been described
earlier as Virahanka numbers in Indian mathematics.
▪ Liber Abaci posed, and solved, a problem involving the growth of a population
of rabbits based on idealized assumptions. The solution, generation by
generation, was a sequence of numbers later known as Fibonacci numbers.
Although Fibonacci's Liber Abaci contains the earliest known description of
the sequence outside of India, the sequence had been noted by Indian
mathematicians as early as the sixth century.

5. ▪ The Fibonacci numbers occur in the
sums of "shallow" diagonals
in Pascal's triangle.
▪ 𝐹𝑛 = σ 𝑘=0
𝑛−1
2 𝑛−𝑘−1
𝑘
▪ These numbers also give the solution
to certain enumerative problems.

6. ▪The Golden ratio is a special number found by dividing a line into
two parts so that the longer part divided by the smaller part is also
equal to the whole length divided by the longer part.

8. ▪Seed heads: The seeds of a flower are often produced at the center
and migrate outward to fill the space. For example, sunflowers
follow this pattern.
▪Tree branches: The way tree branches form or split is an example
of the Fibonacci sequence. Root systems and algae exhibit this
formation pattern.
▪Fingers: The length of our fingers, each section from the tip of the
base to the wrist is larger than the preceding one by roughly the
ratio of phi.

9. ▪Spiral galaxies: The Milky Way has a number of spiral arms, each
of which has a logarithmic spiral of roughly 12 degrees. The shape
of the spiral is identical to the Golden spiral, and the Golden
rectangle can be drawn over any spiral galaxy.
▪DNA molecules: A DNA molecule measures 34 angstroms by 21
angstroms at each full cycle of the double helix spiral. In the
Fibonacci series, 34 and 21 are successive numbers.

10. ▪ Like every sequence defined by a linear recurrence with constant coefficients,
the Fibonacci numbers have a closed-form solution. The solution is,
▪ 𝐹𝑛 =
𝜑 𝑛−𝜓 𝑛
𝜑−𝜓
=
𝜑 𝑛−𝜓 𝑛
5
▪ where
▪ 𝜑 =
1+ 5
2
= 1.618033988749894…
▪İs the golden ratio, and
▪ 𝜓 =
1+ 5
2
= 1 − φ = −
1
𝜑
≈ −0,6180339887498 …

11. ▪Phi appears in the Solar System and the Universe, from the
distances between the planets, to the structure of Saturn's
rings to the shape of the Universe itself. New discoveries
are revealing it to exist in molecular structures and the
fundamentals of quantum matter and time.

12. When a light is placed on two glass layers that are in contact
with each other, a part of the light passes over, one part is
absorbed, and the other part is reflected. It is the 'multiple
reflection' event. The number of paths the glass follows in the
glass before it appears again depends on the number of
reflections the beam is exposed to. Ultimately, when we
determine the number of rays that emerge again, we know
that they fit the fibonacci numbers.

13. ▪ In January 2010 Science Daily and Physics World announced the discovery of
a hidden Phi symmetry in solid state matter. As explained in the Science Daily
article:
▪ “On the atomic scale particles do not behave as we know it in the macro-
atomic world. New properties emerge which are the result of an effect known
as the Heisenberg’s Uncertainty Principle. In order to study these nanoscale
quantum effects the researchers have focused on the magnetic material cobalt
niobate. It consists of linked magnetic atoms, which form chains just like a very
thin bar magnet, but only one atom wide and are a useful model for describing
ferromagnetism on the nanoscale in solid state matter.

14. ▪ When applying a magnetic field at right
angles to an aligned spin the magnetic
chain will transform into a new state
called quantum critical, which can be
thought of as a quantum version of a
fractal pattern. Prof. Alan Tennant, the
leader of the Berlin group, explains
“The system reaches a quantum
uncertain — or a Schrödinger cat state.
This is what we did in our experiments
with cobalt niobate. We have tuned the
system exactly in order to turn it
quantum critical.”

15. ▪ Stars that Pulsate to the Golden Ratio
▪ Phi and the Solar System
▪ İn Mathematical Physics:
▪ Ordering the complex numbers
▪ Primeness the Gaussian İntegers
▪ Bracket Functions

16. ▪Fibonacci numbers are used by some pseudorandom number
generators.
▪They are also used in planning poker, which is a step in estimating
in software development projects that use the Scrum (software
development) methodology.
▪The Fibonacci numbers are also an example of a complete
sequence. This means that every positive integer can be written as a
sum of Fibonacci numbers, where any one number is used once at
most.

23. ▪“ The most beautiful thing we can experience is the mysterious. It
is the source of all true art and science.’’
▪–Einstein, Albert (1879-1955), What I Believe.
▪“I received an excellent education in the methods of the nine
Indian numbers; the knowledge of these methods pleased me
more than anything else…’’
▪ – Fibonacci Leonardo of Pisa (1170-1245) Book of Calculation.

24. ▪ The Fibonacci Quarterly magazine
▪ https://www.goldennumber.net
▪ https://en.wikipedia.org
▪ https://phys.org