**I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing! 8D**

The triangular numbers are the numbers of objects one can use to form an equilateral 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**

**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**

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

**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^{th} triangular number? Well, we just plug in *n*** = 100.** **T**_{100} = (100)(101)/2 = 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 infinity. The energy corresponding to ***n*** has ***n*** angular momentum values which the electron can possess, and each of these angular momentum quanta have 2***n*** - 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.**

**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⌈(***n***+1****)/2⌉**^{2}**elements in the ***n*^{th} row, where **⌈x****⌉ is the ceiling function.** 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

**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.