Rubik cube: cohomology and homotopy

I want to a new series of articles describing the solution of the Rubik’s cube as a method to learn modern, advanced math.

I can safely assume you can arrange all the same color cells in one face by yourself, see the image below.

Now we need to solve the edge (first of all the middle cell) between two contiguous faces.

That’s the boundary function of a cochain!

We simply look for topological invariants!

In our case we have a non abelian Lie group action: we move right face and top face clockwise, then in the same order counter-clockwise and we land on a different phase-point.

But it’s the key to the solution. We have displaced a corner of the already solve face in order to get the correct middle edge. We will leverage the degree of freedom of our gauged math to recover our temporarily displaced cell. Indeed the top rotation is a degree of freedom at this stage.

In conclusion we should be left with 2 of the 3 levels done and only the top face still to do!

Great result for today.

Stay tuned for next articles…

Summary of the second layer

What have we done so far? We’ve solved the second layer of the Rubik’s cude as a 2-category with a 2-bracket displacement and a 2-bracket replacement.

10-middle-layer-edge-left-algorithm

First bracket: 2bracket_1

Second bracket: 2bracket_2.JPG

Third layer

Now we move from a 2-category to a 3-category and topologically speaking from the middle cell of the four lateral edges to the four cross-cell of the upper face.

12-yellow-cross-rubiks-cube-solution

where the above transformation is a 3-bracket indeed: 3bracket.JPG

Conclusion

You can follow the complete solution in one of the many online solvers: link here.

The algebraic view is that you are escalating a category tower, as in homotopy type system consisting of cochain complexes.

Appendix

You can find an alternative math explanation from MIT.
In fact they speak about a non commutative (= non abeian) group, they mention the commutator (= bracket), 2-cycle and 3-cycle (more or less the 2- and 3-category and the cochain) and the Lagrange’s Theorem (but one can give a homological argument as I’ve done above, thinking about exact sequences and equivalence classes of group extensions and so on…)

Resources

The best explanation of homology: “Computing homology groups | Algebraic Topology | NJ Wildberger

and the best explanation of homotopy: “Lie Groups and Lie Algebras: Lesson 35 – The Fundamental Group

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