[Zer0]

& what do these Logicians think/say about a 1x1x1 model?

[varsigma (OP)]

I'm one of the logicians who might say that a 1x1x1 cube is just the trivial case for an induction on nxnxn.
And that's only in three dimensions.

If you propose that a single, solid cube can be colored such that the colors can be 'printed' onto a plane surface, you have an algebra of face turns, and a color polynomial which tiles a section of the Euclidean plane. Moreover, there is a set of nets for a cube and the algebra above would partition this set, given some restrictions such as paths that must visit each face only once.

Mostly since a single cube is a geometric object, with symmetries. Coloring is then a way to coordinate the algebra, or be able to distinguish the symmetries. Complex vectors are just a handy way to linearise everything.

For instance, in electronics you analyse circuits in terms of a frequency response, a curve. Using complex frequencies gets you an additive domain, and simplification. You then recover a real response from the analysis. This is very common in engineering, even in transmission line design.

[varsigma (OP)]

This 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.

[varsigma (OP)]

So here is what I'm doing I believe. I have a partially ordered set, but it has a total order.

I'm trying to complete a symmetric chain decomposition on this set. Moreover, because the metric graph of this set is embedded in at least Z x Z, although that means the totals in each metric are overlaid or overlapping unions of disjoint equivalence classes, there are Z-modules up to Z₁₂ x Z₁₅. Because these are effectively structures that let you contract the paths to loops in a graph, and link together the 2-tori to define the paths which each chain is the union of, this must be a fibration of the 3-sphere.

The Turing/Von Neumann version is, this is an algebra which defines a partition function, and this function essentially counts the different ways to encode certain information, why three dimensions is the minimum for this to make sense, and why measurement--decoding--means breaking a symmetry.

This symmetry breaking is easy to see if you have a cube that isn't colored, then color one face. There is an invariant: a 0-colored cube is equivalent to a cube with the same color on every face. So coloring only one face breaks this, such that there is a unique face, and a unique encoding under the group action.

[varsigma (OP)]

And I'm reasonably sure now that I can describe this poset graph as a Boolean lattice, since by inspection the chains are multiplied at each part of the whole partition.  There is 1, then 1 + 1, then 1 + 2 + 1, 1 + 6 + 1, . . . ,as there is always 1 lower and 1 upper chain, with the multiple chains in the interior.

The first minimal chain is just the first abstract tile, a 1-dimensional line embedded in Z₂ x Z₃. It represents the cover of the partition at (0,0), the identity permutation. This singleton can be contracted to a loop on the point (1,1). This is ok for various reasons, and the simplest reason I can cook up is that the "solution" can be defined as "less than or equal to one move" from this permutation. So an algorithm is free to halt when it's at (1,1) or (1,2).

And by restricting the generators of the poset to two instead of all three, the sizes of each equivalence class are reduced to 2ⁿ, for some n. This is an example of a restriction (2 of 3 generators) which reduces the cardinality of a set. Then the dual or adjoint induction is the Boolean lattice, with a ring algebra which should then give some rules of composition for the tiles. There are 54 in total, arranged as parts of chains and antichains.

One other thing that's possible if you remove or forget the identity permutation from the poset, then you have 7! x 3⁷ - 1 total permutations, which factors to 17 x 216127. So the two prime factors also have something to say about the complexity of this problem, and the limits on packing "information" into three dimensions.

Something along those lines. Also how the algorithmic aspects of finding a solution, for  a given permutation, will always need to employ reduction and restriction, such that an inductive "path" exists.

Dilworth's theorem and its dual allow the decomposition of a poset into chains and antichains with a relation between them which is derived from Greene's theorem. It's quite an elegant way to analyse everything about the structure of the poset graph, but it doesn't give you the total, saturated poset because the B_n lattice is a subset of the total lattice.

[varsigma (OP)]

I think I can say a few things about the modular structure of this graph.

In that, a free additive module is just the idea of repeated addition, a sequence like "+ 1 + 1 + 1 . . .". This, in modular addition, say mod 2, will mean you have the sequence 0 + 1 -> 1 + 1 -> 0 + 1 -> . . .
This means you can define a loop, you can identify the two points, {0,1} and make the distance between them + 1, as the loop so this represents the action of addition on both points, as a single point "{0,1}". So it maps a loop, representing addition modulo 2, to a set of points. This works for {0,1,2, . . .,n}.

So then you have 0 + 1 + 1 + . . . n times, = 0 (mod n). Yay. That's because 0 is the kernel of a function that maps to a homomorphism as the image of the next , or is it the previous? I need to figure that out, function (the homomorphism between additive groups) in the chain.

A Turing machine that models this is one that reads a 0 and writes a 1, or reads a 1 and writes a 0 ( adds 1 mod 2). So there are two read/write loops on a single state, and no halt condition.

[varsigma (OP)]

I've got  a nice result now for the chain-antichain decomposition. The key is to use a restricted--reduced--set of generators for the poset. Then each chain is just a string of alternating letters, these are accepted by a finite-state machine which constructs the poset graph according to string length and how many single repeats exist.

The graph is finite, so all strings which repeat are repeatedly--iteratively--mapped to this set of chains/antichains according to the structure of each string of characters.

For instance the string URURUR... will map repeatedly to the chain from (0,0) t0 (11,11). Why this is the limit depends on the symmetry of the group, or if you will, how the symmetries, extended into strings of length k (a group algebra), grade the set of permutations. Repeats like URRUR... will "lift" the string vertically, or alternatively I can push this into a third dimension in the graph, because It's my graph. I own this mother.

Because I have Z-modules, I have flat tori and these can embed in a topological space where the tori are linked together, but none of the connected paths in the graph are linked because they are all on the surface.
However there is a way to nest everything, and I don't know that it means a whole lot except in a topological, graph-theoretical context. It doesn't make the algebra any more useful or understandable, it's better as a nice linear map in at least two integer dimensions.

So the poset graph is embedded in a bundle of flat tori, which are linked together in an abstract space, i.e. are Hopf-connected, but G the cube group is not. Apparently.

ed apologies for that ownership claim. I own perhaps some of the ideas, not original really, but I don't own this group or a set of numbers. Those have been around for a bit longer than me, possibly even than the universe of matter and reality itself. Philosophical considerations aside, of course.

[varsigma (OP)]

There are various techniques or methods that let you count permutations in a set, such as cycle decomposition and Stirling numbers, or polynomials of various types.

However, I can't help but notice that with a reduced set of generators and strings of compositions of two letters, counting is also much simpler.

Since a string like UR, is also a pair of group elements, there are inverses. So you also have U'R', U'R, UR', or a multiplicity of four. You only need UR x 4 because this is known about combinations, and by trivial induction that RU gives you four more. Then you extend this to 3-letter and 4-letter strings to derive some closed form expression or generating function.

[varsigma (OP)]

It seems that the initial excitement about Erno Rubik's new puzzle, has seen the development, evolution if you will, of problems in the optimization or scheduling classes, or machine learning, in which, like chess or card games, the Rubik's cube and similar 3-dimensional puzzles are a good fit.

There are three main disciplines in which these kinds of puzzles appear, pure mathematics, or group algebras, vector spaces and modules, even as a good tutorial example of what they are. The second two disciplines are related, one is machine learning and optimization of solving algorithms in machina without human knowledge, the other is developing algorithms ad humana for humans to solve these puzzles efficiently, perhaps to study how well different people learn such things. Maybe the military can use it (just sort of kidding, there)

After all, demonstrating you can solve "the cube" is still a kind of test of your intelligence, and doing it quickly says you know some good algorithms.

[varsigma (OP)]

Ok, I have to include some actual stuff about what the big fuss is, why Penrose says much the same thing as, well a few people I suppose. The Hopf fibration does seem to be fundamental. Perhaps because the universe is just weird (except strips of paper are ordinary).

So from the Journal of Physics:

Hopf and the qubit?two-level quantum systems

The occurrence of the Hopf fibration in two-level quantum systems?systems that can be described using a two-dimensional complex Hilbert space, also nowadays known as qubits?was mentioned, e.g., by Penrose [36], and is described more extensively in [48]; so we can be brief and somewhat schematic about it here.

We take the two-dimensional Hilbert space just to be
with its standard Hermitian inner product ≔

. The states of the system are given by density matrices, i.e.,
Hopf and mechanics?the harmonic oscillator

It may surprise some that the simple two-dimensional isotropic classical harmonic oscillator provides an example of the occurrence of the Hopf fibration in physics. But indeed, its phase flow, restricted to a non-trivial energy shell, Hopf-fibers the latter! To see this easily, we scale everything such that the mass and the angular frequency of the oscillator become 1; the Hamiltonian then reads
whence the equations of motion

Employing the usual
Hopf and General Relativity?Taub-NUT space

Around 1950, G?del and Taub independently started the investigation of spatially homogeneous cosmological models in General Relativity. The ones constructed by them are space-times
having line elements of the form where the ωk are essentially left-invariant 1-forms on a three-dimensional Lie group and the product of forms in ds2 is symmetric tensor multiplication as usual. More precisely, , where

is some interval for τ, to be determined during the process of
Hopf and twistors?Robinson congruences

The concept of twistor can, according to its inventor, Penrose, be introduced in a number of ways, and all conformally invariant field laws in flat space-time can be reformulated in terms of twistors, generating new ways of looking at such laws; this has already been very fruitful in the past in many cases.

Clearly, this is not the place to give any of the different ways of approaching or using twistors. We rather pick out one way of representing a twistor space-time-geometrically and highlight
Hopf and Wignerism?helicity representations

In all examples discussed so far, the bundle, or total, space
was of immediate importance and thus in the forefront of the formalism; the base space

, on the other hand, did figure in the qubit case and is used in technical treatments of NUT space that work with the space of Killing trajectories. In the two examples to follow, it will be the other way round: it is the base space that is in the foreground, while the bundle space appears as derived, e.g., via a patching construction. The
Hopf and magnetic monopoles

This example of our collection is probably the first one that caught the attention of a larger number of physicists. As announced, the Hopf fibration comes here from a bottom-up construction.

In his 1931 article ?Quantised Singularities in the Electromagnetic Field? (magnetic monopoles), Dirac [19] starts out looking for a generalization of wave mechanics in which the wave function does not have a definite value for its phase difference between two points. In the 1970s, it was realized [50] that
Hopf and the Dirac equation9

The two-level quantum systems of Section 2 may be considered, in particular, as describing the spin states of a non-relativistic spin-

particle?i.e., one ignores its spatial degrees of freedom. If the latter is not done, one must write spinorial wave functions, subject to appropriate wave equations. In the relativistic regime, the relevant wave equation is the Dirac equation. We shall point out here that the Hopf fibration has significance also in this context. Most directly this comes about
Acknowledgements

I am indebted at least to the following persons for directly or indirectly educating me on matters included in this article: R. Beig, I. Bengtsson, P. Nurowski, R. Penrose, H. Rumpf, W. Simon, and A. Trautman. In particular, important information came from participants of a meeting in honor of E.T. Newman held in June 2001 at the Banach Center, Warsaw; I thank the organizers of that meeting for inviting me there to present a preliminary version of this article, for hospitality, and
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[varsigma (OP)]

Some comments on the way terminology has to evolve, and new mathematical kinds of objects are always subject to some overloading of meaning.

For instance, torsion is well understood in physics as the application of torque, in mathematics (group theory) it has a different meaning. Also, it seems to be down to whether or not polynomials exist in some mathematical theory which claims to represent reality i.e. the reality of entangled particles, the reality of measurements, etc.
The reality of a simple relation between sets and statistics that gives "reality" Bell's inequalities.
And a Nobel for proving, mathematically, that the universe is non-local. (??)

So the leading edge of all this Hopf algebra stuff, is a quantizable theory, this apparently means braid groups and abstract cobordisms between embedded links in R x R³. It has to preserve certain invariants, whatever theory emerges, and there is uncertainty about what these are, exactly.

[varsigma (OP)]

Time for a tentative solution to the question what is the Hopf gauge?

It's a thing in a theory that gives the theory a place to start counting something else. Roughly.

Why graphs, ribbon graphs, are a way to represent this theory is buried in graph theory and homology and homotopy classes. Knots, and stuff like surgery of the kind that changes a symmetry, generally a topological one. It's all totally abstract, but, the idea is to have the right ansatz so you can plug in real particles with real paths; real external fields, and so on.

You get knot polynomials which represent real Lagrangian/Hamiltonian flows. And you get the need to start using new terminology. I can recommend a look at the Princeton Companion to Mathematics (google that), it's refreshingly honest.

I mean it doesn't just use the big words, it tells you what they mean and gives you concrete examples--it should explain better than I can what a gauge theory is.