Equation of the Day #3: Triangular Numbers

I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing!

The triangular numbers are the numbers of objects one can use to form a triangle.

Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)

Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with $n$ rows, this is equivalent to

$\displaystyle T_n = 1+2+3+ \ldots + n = \sum_{k=0}^n k$

This means that the triangular numbers are just sums from 1 to some number $n$ . This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let’s first add sequential triangular numbers together. If we add the first two triangular numbers together, we get $1 + 3 = 4$ . The next two triangular numbers are $3 + 6 = 9$ . The next pair is $6 + 10 = 16$ . Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.

(Image: Wikimedia Commons)

You can do this for any two sequential triangular numbers. This gives us the formula

$T_n + T_{n-1} = n^2$

We also know that two sequential triangular numbers differ by a new row, or $n$ . Using this information, we get that

$\begin{aligned} n^2 &= T_n + (T_n - n) \\ 2 T_n & = n^2 + n = n(n+1) \\ T_n &= \frac{n(n+1)}{2} = \begin{pmatrix} n+1 \\ 2 \end{pmatrix} \end{aligned}$

Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read “ $n$ plus one choose two.” It is defined as the number of ways to pick two objects out of a group of $n + 1$ objects.

For example, what is the $100^{\rm th}$ triangular number? Well, we just plug in $n = 100$ .

$T_{100} = \dfrac{(100)(101)}{2} = \dfrac{10100}{2} = 5050$

We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, “Well, that’s cool and all, but are there any applications of this?” Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called $n$ . This number can be any integer from 1 to $\infty$ . The energy corresponding to $n$ has $n$ angular momentum values $(l = 0, 1,\ldots, n-1)$ which the electron can possess, each of which has a total of $2l + 1$ orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given $n$ involves summing up all these possible orbitals, which takes on the form of a triangular number.

$\begin{aligned} \sum_{k=0}^{n-1}(2l +1) &= 2 \sum_{k=0}^{n-1} k + n \\ &= 2T_{n-1} + n \\ &= n(n-1) + n \\ &= n^2 \end{aligned}$

The end result of this calculation is that there are $n^2$ orbitals for a given $n$ , and two electrons can occupy each orbital; this leads to each row of the periodic table having $2\lceil (n+1)/2 \rceil^2$ elements in the $n^{\rm th}$ row, where $\lceil x \rceil$ is the ceiling function. (This complication is due to the Aufbau principle, which dictates how energy levels fill up.) They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by

$L^2 = \hbar^2 l(l+1) = 2 \hbar^2 T_l$

What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.

There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called “figurate numbers,” like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.

-A

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Equation of the Day #3: Triangular Numbers

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