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• Pascal's triangle is a number
triangle with numbers arranged
in staggered rows.
• Pascal triangle is the ideal law of
• Named after Blaise Pascal, the official founder of this mathematical device.
• In Italy, Pascal's Triangle is actually known as Tartaglia's Triangle, named
after Niccolo FontanaTartaglia, a famous
• Befor pascal the numbers originated in the Hindu religion in India by omar
khyyam and it was also discovered by the Chinese in the 13th century.
Chinese version of pascal triangle
• The Chinese’s version of the Pascal’s
triangle was found in Chu Shi-Chieh's
book "Ssu Yuan Yü Chien" (Precious
Mirror of the Four Elements), written in
AD 1303 which is more than 700 years
ago and also more than 300 years before
Pascal discovered it. The book also
mentioned that the triangle was known
about more than two centuries before
Property of Pascal triangle
• Sum of rows is the nth power of 2.
• The Fibonacci numbers can be found by adding up angles from certain ones to
• Cell division involves the distribution of identical genetic material, DNA, to two
There are two types of cell divison.
• 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.
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:
• 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 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
Electronic configuration and Second kind
of Pascal triangle
• 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
• A letter indicates the type of orbital: s,p,d,f...
• A superscript indicates the number of electrons in the orbital.
• The maximum number of electrons is double square number. The square numbers
can be found in the second kind of triangle
1 3 2
1 4 5 2
1 5 9 7 2
1 6 14 16 9 2
1 7 20 30 25 11 2
1 8 27 50 55 36 13 2
1 9 35 77 105 91 49 15 2
• Shanghai-based multidisciplinary
design company super nature
design has developed 'lost in
• 100 triangular LED lights
• Xylophone triangles
Fibonacci numbers in nature
• The Fibonacci numbers play a significant
role in Nature. Many plants show the
Fibonacci numbers in the arrangements
of 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