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How to solve a Rubik's Cube in five seconds

DavCrav Silver badge

Something like this: a group is a collection of symmetries of an object. Any object will do, as long as it has constituent parts that possess symmetry. An example of a group is the group of Rubik's cube: the symmetries here are all moves, which might not look like symmetries because we swap the colours around, but if we ignore the colours then they are symmetries, and the colours are simply there to show you that you are doing a symmetry. Solving Rubik's cube is equivalent to the following: given a symmetry of Rubik's cube, write it as a sequence of "easy" symmetries, i.e., quarter turns of the slices.

This is an example of a finite group, where there are finitely many symmetries. Of course, there are objects with infinitely many symmetries, such as a disk. This has a rotation of any angle, and a reflection through any line passing through the centre. Another example of an infinite group is the real numbers, with addition being the way of combining objects. Here there is also a notion of closeness, in that two numbers are 'close' if their difference is 'small'. Of course, close and small are relative terms, and the appropriate mathematical concept to encapsulate this is a topology. A topology on a set, such as the real numbers, is a collection of subsets of it, called 'open sets', and they have to satisfy three basic properties: the empty set, the set with nothing in it, is open; the intersection of two open sets, so everything in both of them, is also open; the union of any number of open sets, so everything that's in any of them, is open.

For the real numbers, the open sets are collections of open intervals (a,b), which means all numbers between a and b, but not a and b themselves. Two numbers are 'close' if they are in lots of open sets together, in some sense.

If we have pairs of real numbers then we can put a topology on this, the 'product topology', which says that a set is open if it is the product of open sets in each variable, and then throwing in more open sets for this to be a topology. (Notice that, given any set of open sets, this can be a topology by including unions and intersections, so we can do this.)

There was no reason to choose just pairs of real numbers, we could have chosen n^2 real numbers: then we can arrange these numbers as n by n arrays, and this gives us a topology on all matrices. The determinant map is 'continuous', meaning that if we take an open subset of the real numbers, U, then all matrices with determinant in U form an open subset of the set of matrices.

With this topology, we can talk about matrices being close to one another, more or less their co-ordinates are close in the real numbers. If X is a group of invertible matrices, then it is closed if all other matrices form an open set, and closed groups of matrices are Lie groups.

That's some more words, but it might not be any better.

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