Ontogeny

During mitosis a living cell duplicates into two daughter cells. The amount of the genetic material in daughter cells is the same as in the mother cell. 

In cycle 1, there is a cell-creator: 1 A₀

In cycle2, our mother cell A₀ during the mitosis duplicates into two daughter cells:   2 A₁

So in cycle 3, the two mother cells, 2 A₁, duplicate into four daughter cells: 4 A₂

In cycle 4 the four mother cells, 4 A₂, during the mitosis duplicate into eight daughter cells: 8 A₃;

In cycle n, the 2^n-2 A_n-2 mother cells, duplicate into 2^n-1 daughter cells: 2^n-1 A_n-1 .

The number sequence which represents the cell division is a geometrical series: 

1,2,4,8,32,64,128,256,512

By the geometrical model of cell division, each cycle leads to the new cells of the same generation without surviving of the old cell. The cell A₀ becomes older by the law

t=n*T

where n is the number of the cycle and T is the period of the cycle. But, by the previous model the age of the cells is always T. That is in contradiction with  natural law and with the reproduction of the more - cellular organisms.

When we have reproduction of the more-cellular organisms the parents kept their generation and the children represent a new generation. The reproduction of the more- cellular organisms is the universal law of the cell division:

Cell division results into two cells of different age!
We recognise the mother cell and the daughter cell. 

1.So in cycle 2, our young cell becomes a mother for the first time and produces her first daughter cell: A₀ + A₁ 

2.In cycle 3, the mother cell A₀ reproduces into A₀ + A₁, as well as cell-daughter reproduces into A₁ + A₂ . Now, three generations are present: A₀ + 2 A₁ + A₂.

3. In cycle 4, the original mother cell produces another daughter cell. Two mother cells A₁ reproduce into 2 A₁ + 2 A₂. The mother cell A₂ also produces its own daughter cell. Now four generations are present:A₀ + 3 A₁ + 3 A₂ + A₃ ; 

4. In cycle 5, there are: A₀ + 4 A₁ + 6 A₂ + 4 A₃ + A₄; 

5. In cycle 6, there are: A₀ + 5 A₁ + 10 A₂ + 10 A₃ + 5 A₄ + A₅ ;

etc... 

The numbers of cells in each cycle produce the rows of the Pascal`s triangle:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1 

Pascal’s triangle is thus the Model of cell division. By adding diagonal numbers of the Pascal Triangle, the Fibonacci sequence can be obtained.

The Pascal Triangle is the ideal law of  cell division. All cells have  infinite division possibilities and the cells never die. However, that is not the case in Nature.

The Real Model of the cell division must assume that a cell dies after a few cycles of the division. 

In cycle 1, we have cell- mother A₀₀. There is 1 cell present.

In cycle 2 cell- mother produces cell - mother A₀₁, cell- first daughter A₁₀ and cell- second daughter A_-1 which doesn`t create a cell. There are 3 cells present.

In cycle 3, cell A₀₁ becomes a mother and produces her daughter : A₀₂ + A₁₀. Cell A₁₀ becomes a mother for the first time and produces its own daughter: A₁₁ + A₂₀. Cell- second daughter A_-1 doesn`t have its own production. There are :A₀₂+A₁₀+A₁₁+A₂₀ . There are 4 cells present.

In cycle 4 there are : A₀₃+A₁₁+A₂₀+ A₁₂+A₂₀+A₂₁+A₃₀ . The cell A₀₂ has no production and becomes A₀₃. There are 7 cells present.

In cycle 5 the cell A₀₃ passes away and the cell A₁₂ becomes A₁₃. There are: A₁₂+A₂₀+A₂₁+ A₃₀+A₁₃+A₂₁+A₃₀+ A₂₂+A₃₀+A₃₁+A₄₀. .There are 11 cells present.

In cycle 6 there are : A₁₃+A₂₁ +A₃₀+A₂₂+A₃₀+A₃₁+ A₄₀+A₂₂+A₃₀+A₃₁+ A₄₀+A₂₃+A₃₁+A₄₀+ A₃₂+A₄₀+A₄₁+ A₅₀. There are 18 cells present. 

In cycle 7 there are 29 cells and etc...

The number of cells in each cycle of cell division is : 

1,3,4,7,11,18,29,47,76,123,199,322

That is Lucas series of numbers. This series has the characteristic that each term is the sum of the two previous terms. The Lucas numbers Model of Cell Division is the Real Model of Cell Division. The connection between Cell Division, Pascal’s Triangle, Fibonacci numbers and Lucas numbers, is shown.