Since leaving university, I have tried to find any excuse touse the mathematics that I learned there in everyday life, even if a lecturerdid tell us that it would be difficult. It always makes me think back to when Iwould use maths as procrastination from maths during exam season.
A set is defined to be a collection of well defined distinctobjects, often being of numbers, for example:
{0,1,2,3}.
You can have a set containing anything though. Anotherexample would be the set containing the years that Liverpool have won thePremier League, ∅.This is the empty set, and is a subset of every possible set. Now let’sconsider a set S. The power set of S, written P(S), is a set itself containingevery possible subset of S. If you found that sentence harder to unscramblethan an egg I do not blame you, so it is best that I give an example: If
S= {1,2,3},
P(S)= {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }.
I used to think as a power set as every possiblecrossover between the elements in the set, as if they were films to be made,but we’ll come back to this later.
When I was younger I had no objection to superheromovies. Being a child who was easily entertained and subsequently thought thatevery film I saw was the greatest thing ever, I quite obviously enjoyed them.There came a point though where they just began to annoy me, with theirsomewhat copy and paste storylines and somebody defying all probability yetagain. There are a few exceptions to the rule of course, The Dark Knight andDeadpool to name a couple, but on the whole, they are usually not for me. I thought, at what point will they run out? A quick Google search when I was meant to berevising for my final exams found that there are around 947 of them (although Icopy and pasted the superheroes list straight from the marvel website, deletingany duplicates and it came out closer to 1250, so we’ll just play it safehere and stick to 947).
Scrolling through this giant list made for a goodlaugh. Some superhero names are pretty cool, but then again you put two coolwords together and hey presto! Any 10-year-old could nail it: Black Panther,Hawkeye, Steel Serpent and my favourite... Mathemaniac. Not sure whatMathemaniac’s superpower could be though, maybe he can total the price of hisweekly shop before he gets to the checkout or something. The cool names areshort lived though. There are 10 superheroes or villains that start with ‘Mr.’and 8 that start with ‘Captain’. Then there are some that must have been thought up at 5-to-5 on aFriday with names like; Bloke, which accurately describes him I suppose but thenaming here is about as lazy as Garfield and Homer Simpson holidaying in TheMaldives together; Giant Man, which suggests he may be a giant man, but who canreally be sure, and; Hindsight Lad, I am not making this up. I can only imaginethis is some guy walking around in a cape who keeps telling people he meets atcheckouts or in locker rooms that he wishes he travelled more or that hebet on Leicester City to win the Premier League. But we are just gettingstarted. Those are just the lazy names. All of these names only strengthened my annoyance towards mostsuperheroes as I continued this list. There are;Spider-Ham, a joke in The Simpsons Movie and not surprisingly, as a comic too;Xavin, what is he Xavin, his face? 3-D man, or as I like to refer to him, Man.Then the final 3 sound like a Pokémon evolution line, being Namar, Namora andNamorita.
Insults aside, let us have the set S being the setcontaining the approximate 947 marvel superheroes. You can call the setanything, but ‘S’ is easy as it stands for superhero, although the wordsuperfluous is a better fit.
S= {Ant-Man, Banshee, Bloke, Hindsight Lad, Iron Man…}
When you have a set of elements, you can see thatall the elements in its power set can be divided into how many elements eachset has. To easier visualise this, consult the examples below. Assume we havesome set X of 4 or more elements. The number of subsets of X that containthemselves 2 elements can be found with the formula below. This formula doesnot change, no matter how big X is.
We can repeat this to find the number of subsets ofsize 3, 4, 5, up to any number you want. When you add up all of thesequantities, they sum to 2nelements in the power set of X, or in our example, S. As we are looking atfilm crossovers though, we will minus 1 as the empty set ∅ would be a film containing nosuperheroes, but this will make no noticeable difference to our outcome, as 2nin the case of S, is going to get more out of hand than a wet bar of soap.
Now we see that there are just under half a million 2-charactercrossovers between all marvel characters, which let’s assume you can watch 8films a day, would total a time of over 153 years to watch them all. 3 and 4-charactercrossovers increase each time up until 473 and back down again in the shape ofa standard deviation correlation, so just how many can we get?
Ihave been watching a lot of maths and science based, very interesting videos onYouTube recently on a channel called ‘Vsauce’. Here is the video, entitled ‘How Many Things Are There?’, which caught my attention most.
Inthe video, Michael approximates the possible number of thoughts that ouruniverse could have before a googol (10100) years has past and thereis no usable energy left in the universe. First he takes the approximate mass of theuniverse, 3.4 x 1060kg and the fastest possible computing limit due tothe speed of light and the uncertainty principle (called Bremermann’s limit) 1.36 x 1050bitssec-1 kg-1. Using these he simulates making the universe into a universe sized supercomputer that estimates that for the 3.154 x 10116seconds that the universe has left, there are 1.458 x 10227 thoughtsthat could be thought. This is assuming that each thought takes about 1sentence, or 800 bits worth of information. This is an interesting result whenit comes to the amount of superhero movies there could ‘possibly’ be…
So now we just need to plug in the numbers. 2n-1 gives the number of crossover films for a group of superheroes of size n, which wefind to be in the case of n=947 is
5.80866x 10281,
making them literally unthinkable. Thanks for reading.
Luke Bennett
