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What is Group Theory?

In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubik’s cube–which we’ll simplify to a 2x2 square for now– or anything, so long as they follow 4 specific rules, or axioms.

Axiom 1: All group operations must be closed, or restricted, to only group elements. So in our square, for any operation you do—like turn it one way or the other—you’ll still wind up with an element of the group. Or for integers, if we add 3 and 2, that gives us 1—4 and 5 aren’t members of the group, so we roll around back to 0, similar to how 2 hours past 11 is 1 o’clock.

Axiom 2: If we regroup the order of the elements in an operation, we get the same result. In other words, if we turn our square right two times, then right once, that’s the same as once, then twice. Or for numbers, 1+(1+1) is the same as (1+1)+1.

Axiom 3: For every operation, there’s an element of our ground called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is 0. Not very exciting.

Axiom 4:  Every group element has an element called its inverse, also in the group. When the two are brought together using group’s addition operation, they result in the identity element, 0. So they can be thought of as cancelling each other out. Here 3 and 1 are each other’s inverses, while 2 and 0 are their own worst enemies.

So that’s all well and good, but what’s the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, let’s expand our square back into a full-fledged Rubik’s cube. This is still a group that satisfies all of our axioms, though now with considerably more elements, and more operations—we can turn each row and column of each face.

Each position is called a permutation, and the more elements a group has, the more possible permutations there are. A Rubik’s cube has more than 43 quintillion permutations, so trying to solve it randomly isn’t going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, that’s exactly what most solvers do, even using a group theory notation indicating turns.

From the TED-Ed Lesson Group theory 101: How to play a Rubik’s Cube like a piano - Michael Staff

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partypupper  asked:

hello! :] i really like your blog and wanted to ask something! i'm sure you've been asked this before and in this case just idk tell me which post to read and all that but!! could you recommend any stories about wolves? like stories about realistic wolves, anthropomorphic wolves (like The Plague Dogs or The Lion King or Balto - basically 'talking' animals) or even werewolves maybe? i don't know a lot of those and would reaaally like to find some stories to read... thank you!! :>

Hi, maybe my followers can help out? You can leave a comment on this message if you know anything! :)

EDIT: messages I received on this topic: 

raakxhyr said: Child of the Wolves and Night Fang are pretty good :)

pezwolf said: White Fang, and Call of the Wild, both by Jack London. For non Fiction I recommend The Wolf by L. David Mech

stormyautumnday said: Some wolf POV books for your anon, from an animal fiction loving library worker: Wolves of the beyond by lasky, runt by bauer, the sight and fell by clement-Davies, wolves of time by horwood, wolf: the journey home by Bowen, wolf chronically by Hearst, the hunt for elsewhere by vine, midnight’s sun by kileworth, the messenger by gartland, jackal and wolf by shixi

ssecretsdontsleep said: Wereworld is also a good book series with a werewolf in it!! For the anon that asked