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Monster Group
Such a crazy thing. M, the monster group, is the largest of the 26 sporadic groups. The sporadic groups are all the finite simple groups not members of the four infinite families of finite simple groups. The monster group has all 15 supersingular primes as divisors of its order.
What's supersingular? So glad you asked.
Blame it on Mathober Day 13: sporadic group
Number Tournament: NEGATIVE ONE vs THE ORDER OF THE MONSTER GROUP
[link to all polls]
-1 (negative one)
seed: 33 (15 nominations)
previous opponent: omega
class: negative unit
definition: less than nothing, but only slightly
|M| (the order of the Monster group)
seed: 49 (9 nominations)
previous opponent: [final vote count]
class: group theory
definition: the incomprehensible answer to an extremely foundational question
which is the best number?
negative one
the order of the Monster group
Was in the mood to draw the monster friends together being badass. Been a while since I did anything with Meteora or Mariposa since I've kinda been on my HT Bendy Gravity Falls three-way. Wanted to draw them all together as adults + teens.
Jack belongs to @lovelylivelyv
@heartsong1994 @deathfangirl9 @hotelt-resurrection @geekgirles @speedygivermusic @howling-nightmare
@anicemyth you're about to make us soooo autistic about this you don't even know bless you
so group theory: this is a very rudimentary explanation but in mathematics a group describes all the ways a structure can be symmetric aka every operation or transformation that can be done on a structure while still it remains unchanged. for example a symmetry of an equilateral triangle would be mirror flipping it on its axis or rotating it 120 or 240 degrees. the untransformed state of the structure (e.g. rotating said triangle by 0 or 360 degrees) is also counted as one of the total symmetries within the group. There's a lot of detail in the defining of this regarding the arithmetical and algebraic behavior of groups that the resources I'm going to add at the end will surely do a better job of than I could.
A group of these symmetries can be broken up into "building blocks" similarly to how an integer can be broken down into its prime factors. These building blocks of groups are known as simple groups. putting aside the fact that there are infinite simple groups bc let's not even go there - the monster group is one of the finite simple groups.
through an incredible mathematical undertaking it has been proven that we have discovered all the possible finite simple groups that can exist. they fall into categories based on their properties and this categorization is depicted in something that looks a lot like a periodic table of elements:
in the colored columns are the 18 assorted categories of group - cyclic, alternating, etc, but at the bottom the 2 rows in light green show the sporadic groups which are 26 groups that do not fall in any of the above categories. at the bottom right is the monster group.
the reason why this is crazy - the numbers listed at the bottom of each box there are the total number of symmetries contained within the group. for an equilateral triangle like I mentioned above you get 6 symmetries including both rotational and reflectional symmetries and including the baseline state of the triangle without any transformation having been done on it.
The monster group? Contains about. 8 x 10^53 symmetries. That is
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
symmetries. what the fuck. both massive and specific. if that triangle with 6 symmetries is 2 dimensional - with this many symmetries how big must this monstrous object be?
196,883 dimensions.
in addition to that the monster group actually contains (including itself) 20 of those 26 sporadic groups. (Fun fact those groups contained within the monster have been dubbed the Happy Family with the 6 outliers being named the Pariahs lmao). it's notable also bc it is very difficult to represent it concisely compared to other finite simple groups including the rest of the sporadics.
so it's just this.... thing. that is out there. we know what it is, we know its incredibly specific parameters, but of course we don't know WHY it's there or WHY those are the numbers you arrive at (if thats even a reasonable question to ask), it looks very arbitrary but it is ultimately a fundamental mathematical entity regardless of how inelegant it may seem, the universe is an interesting place
this weird abstract yet very specific structure has connections to other fields of mathematics - it has a connection to modular functions as described by the monstrous moonshine conjecture. yes it's actually called that and it is waaay above my paygrade but this somehow connects to a 24-dimensional variant of string theory (note I absolutely hate string theory for unrelated reasons but the mathematics of it is very interesting) in some way.
in short there exists an incredibly high dimensional object with an obscene number of symmetries that can can be used in tandem with something from a seemingly totally unrelated area of mathematics (the modular j-function) to describe a physics theory. ?????????? they called it moonshine bc they thought it was an absolutely batshit thing to even consider but apparently it works
that is my best attempt at explaining this so here are some resources I really recommend:
Researchers are on the trail of a mysterious connection between number theory, algebra and string theory.
additionally I'd like to just plug John Conway as a whole here he's in the first video linked talking about his work regarding the monster group and the moonshine conjecture. you can find him on the channel speaking on other topics including the game of life which is an unrelated but very interesting cellular automaton that is available free online to be played with. his group theory work is what stands out to me though, he sadly passed of covid a few years back at an old age but he is one of my favorite mathematicians of all time not only because of his work but also because he just seems like a chill fucking guy
my fanciful conclusion is like. this Thing evokes in my mind images of angels or eldritch horrors or what have you. vast and incomprehensible it dwells in a space so complex it defies any human understanding beyond the mathematics used to describe it. it is beautiful and unthinkable and perhaps i want to kiss it. the end
(If anyone with a better mathematical background than us which is not at all a high bar to set wishes to add to this please do!)
#834
Always be mindful; The jaws of death have more reach Than they appear to.
They did the mash. The (math) Monster Mash!
https://wolfr.am/oWFHRcXZ
Number Tournament: SEVENTEEN vs THE ORDER OF THE MONSTER GROUP
17 (seventeen)
seed: 16 (35 nominations)
class: prime number
definition: one greater than 2^2^2
|M| (the order of the Monster group)
seed: 49 (9 nominations)
class: group theory
definition: the size of the largest sporadic simple group, equal to 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. the Monster group itself is the symmetry group of a 196,883-dimensional object.
which is the best number?
seventeen
the order of the monster group
