# MegaFavNumbers & the Monster

My favorite number over one million—and a surface-level dive into group theory

October 1, 2021

Just over a year ago, a group of math-based YouTube channels started the MegaFavNumbers project, making videos about their favorite numbers larger than one million and encouraging people to do their own. They argued that most people have favorite numbers only one or two digits long, while there are some larger numbers that deserve more attention, and I would have to agree. As such, I would like to bring back the project for a run on the *Talon*, and I encourage my fellow staff writers to write an article on their MegaFavNumber, as well as readers to choose their own.

My MegaFavNumber is considerably larger than a million and happens to be one I share with YouTuber and co-founder of the project 3Blue1Brown: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, the size of the monster. You may be thinking I chose this number for the cool name, and you would be right, but there’s also something perplexing about its meaning that just compels me. To explain that, though, we’ll need a crash course on group theory.

A group in mathematics, in its most basic terms, is the collection of all symmetries that an object has. For example, an equilateral triangle can be rotated 60, 120, and 180 degrees, or reflected across three axes, and look the same. This group is known as D_{3}. In less basic terms, talking about groups this way is like describing the number three with three literal objects; a group is a concept that can describe shapes, the same way numbers are concepts we use to count.

As suggested by the way the triangle’s group was named, groups (or rather, groups that are finite, excluding ones such as the symmetries of a circle) are categorized into families, 18 of them to be precise. These families include the ‘cyclic groups of prime order’ C, or the ways to rotate (not reflect) a polygon with a prime number of sides, the ‘alternating groups’ A, or the ways to arrange a collection of points, and a host of others that are harder to categorize. The monster, though? It doesn’t belong to any of them. The total collection of possible symmetries is itself not symmetric, having 26 outliers called the ‘sporadic groups’, the monster being the largest of them. The fact that these sporadic groups just stop abruptly is what makes the monster so intriguing to me, especially since right now we have no good explanation as to why they’re there.

Perhaps fittingly with how absurd these groups are, the mathematicians discovering them did have their fun naming them. For instance, the second-largest sporadic group after the monster is called the baby monster, having a size of 4,154,781,481,226,426,191,177,580,544,000,000. There are also a set of outliers to the sporadic groups, six of them, which are called the ‘pariahs’, while the rest are called the ‘happy family’. The connection that these groups have to other fields of mathematics (that even I, the resident math nerd, am struggling to understand) have been dubbed ‘moonshine’, a reference to what mathematician John Conway called the first of these connections, as in it was a crazy idea. He wasn’t wrong, the sporadic groups are indeed crazy, but not only have they proved to be useful, they beckon you to wonder what the structure of the universe is like to have them exist, and to have them cut short.