CELL DIVISION AND PASCAL TRIANGLE

Pascal triangle• Pascal's triangle is a number

triangle with numbers arrangedin staggered rows.

• Pascal triangle is the ideal law ofcell division

History• Named after Blaise Pascal, the official founder of this mathematical device.

• In Italy, Pascal's Triangle is actually known as Tartaglia's Triangle, namedafter Niccolo Fontana Tartaglia, a famous

• Befor pascal the numbers originated in the Hindu religion in India by omarkhyyam and it was also discovered by the Chinese in the 13th century.

Chinese version of pascal triangle

• The Chinese’s version of the Pascal’striangle was found in Chu Shi-Chieh'sbook "Ssu Yuan Yü Chien" (PreciousMirror of the Four Elements), written inAD 1303 which is more than 700 yearsago and also more than 300 years beforePascal discovered it. The book alsomentioned that the triangle was knownabout more than two centuries beforethat.

Property of Pascal triangle

• Sum of rows is the nth power of 2.

i.e. 2^n

Binomial expansion

• (𝑎 + 𝑏)2= 𝑎2 + 2𝑎𝑏 + 𝑏2

Fibonacci numbers

• The Fibonacci numbers can be found by adding up angles from certain ones to ones.

Cell division

• Cell division involves the distribution of identical genetic material, DNA, to two daughters cells.

There are two types of cell divison.

• Mitosis

• Meiosis

Mitosis

• Mitosis is a fundamental process for life.

• During mitosis, a cell duplicates all of its contents, including its chromosomes, and splits to form two identical daughter cells.

• the steps of mitosis are carefully controlled by a number of genes. When mitosis is not regulated correctly, health problems such as cancer can result.

REALTION OF PASCAL TRIANGLE WITH MITOSIS

Relation with nth power of 2

• 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 2n-2 An-2 mother cells, duplicate into 2n-1 daughter cells: 2n-1 An-1

.

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

1, 2, 4, 8, 16, 32, 64, 128, 256, 512……

We know that this type of sequence exist in Pascal triangle as we discussed above.

Relation with binomial expansion

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

• In cycle 2, 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.

• In cycle 3, 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 ;

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

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

• The number of cell in each cycle produces the rows of pascal triangle.

1A0

1A0 1A1

1A0 2A1 1A2

1A0 3 A1 3A2 1A3

1A0 4A1 6A2 4A3 1A4

1A0 5A1 10A2 10A3 5A4 1A5

Other examples of Pascal triangle

• Electronic configuration and second kind of Pascal triangle

• Architecture-lost in Pascal triangle

• Nature-Fibonacci numbers

Second kind of Pascal triangle

Electronic configuration and Second kind of Pascal triangle

Electronic configuration

• An electron configuration is a method of indicating the arrangement of electrons about a nucleus.

A typical electron configuration consists of numbers, letters and superscripts with the following format:

• A number indicates the energy level.( The number is called the principal quantum number.)

• A letter indicates the type of orbital: s,p,d,f...

• A superscript indicates the number of electrons in the orbital.

Relation• The maximum number of electrons is double square number. The square numbers

can be found in the second kind of triangle

11 2

1 3 21 4 5 2

1 5 9 7 21 6 14 16 9 2

1 7 20 30 25 11 21 8 27 50 55 36 13 2

1 9 35 77 105 91 49 15 2

Relation• Electronic shells actually have sublevels, i.e. s, p, d, f… number of orbitals in each

sublevels are 1, 3, 5, 7, 9,..... respectively.

11 2

1 3 21 4 5 2

1 5 9 7 21 6 14 16 9 2

1 7 20 30 25 11 21 8 27 50 55 36 13 2

1 9 35 77 105 91 49 15 2

Architecture

• Shanghai-based multidisciplinarydesign company super naturedesign has developed 'lost inpascal's triangle'.

• 100 triangular LED lights

• Xylophone triangles

Fibonacci numbers in nature

• The Fibonacci numbers play a significantrole in Nature. Many plants show theFibonacci numbers in the arrangementsof the leaves around their stems.

• One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers.

• E.g in grasses, rose, apple etc