May 23, 2024

The remarkable properties of Pascal’s triangle

However, the entries in Pascal’s triangle also follow some more obvious patterns: the diagonals of the triangle, which run parallel to the outer rows of the one, list the natural numbers, for example. The reason for this is that the numbers in the next row are always made up of the sum of the two values ​​above: because there is always one number on the outer edge, the sequence increases by one each step.

© Spectrum of Science / Manon Bischoff (details)

The pattern of numbers in Pascal’s triangle

The diagonals immediately adjacent to the values ​​1, 3, 6, 10, 15, etc. are also interesting. These are triangular numbers: the number of stones you need to be able to put an equilateral triangle out of. This episode appears because nThe third triangle number for the sum of 1 + 2 + … + n equivalent. This can be done according to Gaussian sum formula Through n(nCalculate +1) / 2 – and that’s just about the value B(n+1, 2), i.e. getting into n+ first row and second column of Pascal’s triangle.

triangular numbers | This is the number of points needed to build an equilateral triangle.

Another property related to triangles can be seen by coloring the odd and even entries of Pascal’s triangle differently. As François-Edouard Anatole Lucas proved in 1890, this creates a pattern that may sound familiar to some: it’s a fractal called the Sierpinski Triangle. This can be constructed by dividing an equilateral triangle into four smaller congruent triangles and removing the middle triangle. Repeating this process over and over with the three remaining smaller triangles creates a fractal pattern that can also be observed in Pascal’s triangle.

Sirepinski Triangle | Coloring the odd numbers in Pascal’s triangle black reveals a fractal pattern.

Other notable representatives of mathematics can also be found in Pascal’s triangle: for example the Fibonacci sequence. This infinitely long sequence of natural numbers is constructed by starting with the number 1, followed by another 1 and then making up the sum of the two previous elements of the sequence: 1, 1, 2 (= 1 + 1), 3 (= 1 + 2), 5 (= 2 + 3), 8 (= 3 + 5), 13 (= 5 + 8), 21 (= 8 + 13) and so on. This sequence also arises because of the cumulative structure of Pascal’s triangle: each number in a row results from the addition of the two numbers above it.

The Fibonacci sequence in the shell of pearl boats The spiral shell of pearl boats follows the numbers in the Fibonacci sequence.

Part of the reason the Fibonacci sequence is so popular is that the growth it describes in many people Nature’s processes can be observedFor example, the arrangement of the nuclei in a sunflower follows the Fibonacci series. In Pascal’s triangle you can find the individual terms of the sequence if you add up the inputs for all the short diagonals, as shown in the figure below:

Fibonacci sequence in Pascal’s triangle

Still not enough? Pascal’s triangle has many other interesting properties. For example, one also finds Mersenne primes in it. These are numbers in the form of 2n1 which has no divisors other than 1 and themselves, for example: 22−1 = 3, 23−1 = 7, 25−1 = 31, 27−1 = 127 and so on. The largest Mersenne head currently known is valuable n = 82589933 given. were through a Collaborative research project Found where volunteers donate their computing power to research Mersenne primes. And indeed, all the numbers that are in picture 2 appearn−1 also appears in Pascal’s triangle: for this you just have to search for all entries up to a particular row n Add. If you are about 25If you want to calculate −1, you have to add all the entries up to the fifth row. Since the sum of each row is the corresponding power of two, the result is: 20+21+22+23+24 = 31, which is Mersenne’s prime n = 5 corresponds.

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