Pascal's Triangle was originally developed by the ancient
Chinese, but Blaise Pascal was the first person to discover all of the patterns
it contained.
At the tip of Pascal's Triangle is the number 1, which makes up the zeroth
row. The first row (1 & 1) contains two 1's, both formed by adding the two
numbers above them to the left and the right, in this case 1 and 0 (all numbers
outside the Triangle are 0's). Do the same to create the 2nd row: 0+1=1;
1+1=2; 1+0=1. And the third: 0+1=1;
1+2=3; 2+1=3; 1+0=1. In this
way, the rows of the triangle go on infinitly. A number in the triangle can
also be found by nCr (n Choose r) where n is the number of the row and r is the
element in that row. For example, in row 3, 1 is the zeroth element, 3 is
element number 1, the next three is the 2nd element, and the last 1 is the 3rd
element. The formula for nCr is:
n!
--------
r!(n-r)!
! means factorial, or the preceeding number multiplied by all the positive
integers that are smaller than the number. 5! = 5 × 4 × 3 × 2 × 1 = 120.
The sum of the numbers in any row is equal to 2 to the nth
power or 2n, when n is the number of the row. For example:
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20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
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Fibonnacci's Sequence can also be located in Pascal's Triangle. The sum of
the numbers in the consecutive rows shown in the diagram are the first numbers
of the Fibonnacci Sequence. The Sequence can also be formed in a more direct
way, very similar to the method used to form the Triangle, by adding two
consecutive numbers in the sequence to produce the next number. The creates the
sequence: 1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . . . . The Fibonnacci
Sequence can be found in the Golden Rectangle, the lengths of the segments of a
pentagram, and in nature, and it decribes a curve which can be found in string
instruments, such as the curve of a grand piano.
Connection to Serpinski's triangle.
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When all the odd numbers (numbers not divisible by 2) in
Pascal's Triangle are filled in (black) and the rest (the evens) are left
blank (white), the recursive Sierpinski Triangle fractal is revealed (see
figure at near right), showing yet another pattern in Pascal's Triangle.
Other interesting patterns are formed if the elements not divisible by other
numbers are filled, especially those indivisible by prime numbers.
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