The most famous mathematical work of Blaise Pascal is a treatise about the "arithmetical triangle" that consists of the binomial coefficients (Pascal's triangle), which has applications in the probability theory and has some surprising and entertaining features.
Martin Gardner wrote in his book "Mathematical novel": "Pascal's triangle is so simple that even a child can make it up. At the same time, it is has got unsearchable riches and binds together the various aspects of mathematics. Such unusual features make Pascal's triangle one of the most elegant schemes in all mathematics. "
The structure of Pascal's triangle: each number is the sum of the two numbers above it. All is elementary, but there are many wonders hidden there.
At the top of the triangle is number 1. The triangle can be continued indefinitely. It is symmetrical to a vertical axis passing through its top. Along the diagonals which are parallel to the sides of the triangle there are triangular numbers and their generalization.
Triangular numbers in the usual and familiar form show how many related circles may be arranged in a triangle - a classic example - the initial alignment of the balls in billiard. To the one coin, You can put two more coins – then can put three more - a total of six. Continuing to increase rows with a triangle shape we will get a row 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ..., that is what the second green line shows. This remarkable row in which each number is the sum of natural numbers (55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10), also contains a lot of well-known numbers 6 and 28 - perfect numbers, 36 - square number, 8 and 21 - Fibonacci numbers.
And the next green line (1, 5, 15, 35, ...) demonstrates an attempt to put some tetrahedron in four-dimensional space - a ball touches the four, and those, in turn, ten ... this is impossible in our world, only virtually.
To find the sum of numbers on any diagonal from start to necessary places just look at the number located at the bottom. For example, suppose that we want to calculate the sum of the natural numbers from 1 to 9. Look at the number 9 on a diagonal, we will see at the bottom on the left there is a number 45.
The amounts of numbers, standing along the less steeply dipping diagonals (on the figure marked by red lines) form the Fibonacci sequence. Pascal apparently had no idea that the Fibonacci numbers are hidden in his triangle. This fact was discovered only in the XIX century. The numbers standing on the horizontal rows of Pascal's triangle - are the binomial coefficients, coefficients of the expansion of (x + y) n in powers of x and y.
The number of possible combinations of n elements in m determined by the formula
Where n! = 1 * 2 * 3 * 4 * .... n so-called factorial of n.
Now, finally, we turn to the most interesting features of Pascal's triangle. Let’s replace every number in Pascal's triangle with a point. Moreover, the odd points we will paint with a contrasting color, and even - transparent or background color. The result would be unpredictable: Pascal's triangle is divided into smaller triangles, forming a graceful pattern. These patterns have got many surprises. At the top of Pascal's triangle, there is a triangle consisting of one - single point, then triangles containing 6, 28, 120, 496, ... of points go. Three of these numbers - 6, 28 and 496 - known as perfect, as each of them is equal to the sum of all its divisors other than the number itself. For example, a 6 = 1 + 2 + 3.
So you see that Pascal triangles could be very useful in some cases and nobody certainly could say that they are not interesting or beautiful and you even can program your own triangle. In Pascal's triangle there are a lot of other amazing things like the Reti sketch or Sierpinski triangle and a lot of others but today I tried to describe you the most interesting ones.
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With Love,
Kate
