# solve

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

Animation by Shixie

A simple illustration to how the brain reacts and behaves towards stimulus and how it can be rewired and trained towards further development using the same behavior.

Riddle

You leave me a hundred times a day, and seldom think about it. But if you do something illegal and leave me behind I will betray you.
Who am I?

As time goes on, you’ll understand. What lasts, lasts; what doesn’t, doesn’t. Time solves most things. And what time can’t solve, you have to solve yourself.
—  Haruki Murakami, Dance dance dance

“And the mystery was solved-”

BULLSHIT, NO IT ISNT YOU STILL HAVE HALF AN EPISODE LEFT

Nothing will solve more problems than the women exerting her balanced leadership and guidance. — Bryant McGill

☆ some dapper baphomet

[art commissions / portfolio @ god-bird.com]
do not republish/repost my artwork or remove my comments.

—  Attachments, Rainbow Rowell (via womanbythesea)

Choose to not worry at all, freeing your creative mind and spirit to solve your problems instead. — Bryant McGill

Love triangle

‘Me watching another move where the heroine has to choose one man’

Me: Problems poly could solve!

Life is not a problem to be solved, but a reality to be experienced
—  Søren Kierkegaard

Whatever the problem, be part of the solution. Don’t just sit around raising questions and pointing out obstacles. We’ve all worked with that person. That person is a drag.

- Tina Fey