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##### Physics, Astronomy & Cosmology / Re: The Hopf gauge

« Last post by**varsigma**on

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**Today**at 01:35:22This is much like talking about something you've studied and you're trying to explain it during a darts game, or down at the bowling alley. But you assume people can generally understand numbers, and what an ordering is.

The nice thing about the poset, which was published to announce the "God's number" for the 2x2x2, but seems to have been ignored since, is that it is a set of chains and antichains, Dilworth's theorem applies, but (isn't there always a but?) there is an extra set of chains and they are 'quadratic', I'm tempted to call them cochains, but that's going to take a bit of proving.

These are all generated from the 'real' chains. There is an easy way to see through it all with restrictions and inductions, on the group action. This can represent the poset as just say, the Kleene closure of a set of generators, with the required restriction.

To illustrate what I mean, consider the diagonal line in ZxZ, from (0,0) to (11,11). This has the following tuple of numbers mapped to it: (1,6,24,96,384,1416,4788,14260,36780,69960,43984,96). This has an ascending and then descending order, and, if you factor the numbers there are some surprisingly large primes. The quadratic 'group action' divides each number by two (except the 1), generating a new chain from that point, and there are six chains. The maximum antichain length is six and Dilworth's theorem holds.

The primes have a different kind of partial order though. This is one reason it's almost impossible to solve a scrambled cube in less than the width of the poset graph of G.

Time for a cuppa. I've decided to shelve ribbon graphs, it's a bit messy in more ways than one.

The nice thing about the poset, which was published to announce the "God's number" for the 2x2x2, but seems to have been ignored since, is that it is a set of chains and antichains, Dilworth's theorem applies, but (isn't there always a but?) there is an extra set of chains and they are 'quadratic', I'm tempted to call them cochains, but that's going to take a bit of proving.

These are all generated from the 'real' chains. There is an easy way to see through it all with restrictions and inductions, on the group action. This can represent the poset as just say, the Kleene closure of a set of generators, with the required restriction.

To illustrate what I mean, consider the diagonal line in ZxZ, from (0,0) to (11,11). This has the following tuple of numbers mapped to it: (1,6,24,96,384,1416,4788,14260,36780,69960,43984,96). This has an ascending and then descending order, and, if you factor the numbers there are some surprisingly large primes. The quadratic 'group action' divides each number by two (except the 1), generating a new chain from that point, and there are six chains. The maximum antichain length is six and Dilworth's theorem holds.

The primes have a different kind of partial order though. This is one reason it's almost impossible to solve a scrambled cube in less than the width of the poset graph of G.

Time for a cuppa. I've decided to shelve ribbon graphs, it's a bit messy in more ways than one.