A triangle that changed the world

November 15, 2022

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When you think of mathematics, you imagine a sea of numbers, formulae and relationships. This is, of course, as we know, quite often the case. However, mathematics hides quite interesting relationships you might not have thought of before. One of the most common “interesting relationships” in math is Euler’s identity: e^{i*\pi} + 1 = 0. Where two irrational numbers (real numbers that cannot be represented as a fraction) combined, they have one of the most elegant solutions in mathematics. When delving deep into the different types of  “interesting relationships” within mathematics, one stands out above the rest, mostly because of its simplicity in construction, application, and understanding. This is where we look at one of the most fun and increasingly interesting triangles that exist in mathematics to date: “Pascal’s Triangle”.

Constructing Pascal’s Triangle

​​While this triangle has quite a fascinating and extensive history dating back to 975, this article will mainly focus on its construction and multiple applications.

To construct this triangle, you must start at its peak. We will call this row n = 0 and place a 1 there. After that, we fill out the row below n = 1 by adding the number above it with the number to the left or right, treating the blanks as 0s. The following rows are filled following the same logic, which can be done an infinite number of times. This might seem quite tedious at first, but a better way of understanding the construction of the triangle is by looking at an example. 

Pascal’s Triangle

For ease of use and later purpose, each row will be numbered starting from the top n = 0  until the bottom of your constructed triangle, in the case above n = 7.

What can I do with this strange triangle?

If this is the first time you see this triangle, it might seem like a weird way of ordering numbers. However, this ordering is quite useful for many different applications in mathematics, such as combinatorics, computing powers, matrix exponentials and much more.

Combinatorics and polynomial expansions

When looking into combinatorics, one of the first things that pop up is the binomial coefficient expressed as m \choose k = \dfrac{m!}{k!(m - k)!}.  This is used for probability theory but also for polynomial expansions where it plays a big role in the Binomial Theorem expressed as:

(x + n)^m = \sum_{k = 0}^{m}{m \choose k} x^{k}y^{m - k}

This in itself already shows the importance of the equation m \choose k.  When expressing this equation for different values m and k you might notice a pattern. This pattern is exactly the same as the pattern shown in the rows of our triangle where m = n. Moreover, k represents the column counting from 0. So, for example, if you want to quickly compute the vale 4 \choose 3 instead of using the above equation, you can look at the triangle and go to the 4th (k starts at 0) entry of the row n = 4 and output that value, in this case, equalling 4.

4 \choose 3 = 4

This trick can also be used for the Binomial Theorem. For example, if one must come up with the polynomial (x + y)^5, one looks at the row n = 5 and then, using the theorem, results in 

1x^5 + 5x^4y+10x^3y^2 + 10x^2y^3 + 5xy^4 + 1y^5. If the pattern is recognized, the triangle aids you in coming up with these polynomial expansions in a fraction of the time it would take you to compute everything by hand.

This means that all equations that require the computation of m \choose k, such as the multiple-angle identities, the approximation of e, etc, can be computed with fewer intricacies.

Other patterns and applications

You might be satisfied with the above use, but the triangle’s usefulness does not stop there!

1 ) When adding up each row, you will find that this is precisely the solution to 2^n. For example, when looking at the row n = 5 and add up its values you will find that 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2^n.

Computation of 2^5

2) When treating each entry of a row as a decimal expansion, i.e. multiplying the numbers with 10^k where you start with 0, you might find a new pattern, namely that this will equate to the value of 11^n. For example, looking again at n = 5 and applying the above logic you get: 1*10^0 + 4*100^1 + 6*10^2 + 4*10^3 + 1*10^4 = 11^4

Computation of 11^4

3) Have you ever been interested in how many same-sized rectangles you need to build a 2-D triangle? Or how many same-sized blocks do you need to build a 3-D pyramid? Then look no further than the third diagonal line (1, 3, 6, 10, 15, etc.) of the triangle which gives us the triangular numbers. This is precisely the amount of rectangles you need to build a 2-D equilateral pyramid of a given height! Moreover, when looking at the fourth diagonal line (1, 4, 10, 20, 35, 56, etc.), it gives you exactly the number of blocks you need to build an equilateral pyramid.

The triangle possesses quite a few more interesting patterns, such as the creation of Sierpinski’s triangle, Fibonacci and computing the exponential of matrices. The mathematical world is, to this date, extracting and finding more patterns that are useful to its field. When first looking at the triangle you might have thought that there is not much to it, however, as seen above there is a lot to explore! Hopefully, this article was able to convince you of its use, and maybe someday, you might be able to find a new pattern residing within it.

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