[varsigma (OP)]

I suspected something about the "thickened" K₂ graph I made by cutting a twisted, connected strip of paper.
The recipe, or the algorithm to make this bad boy, is to start near an edge (a local one!) and cut in a narrow-ish spiral, so you go through the centre and towards the "other" edge. Edges are local when you cut near them with scissors, but you have a global view too. The edges in the resulting looped thingus, are also where you cut near, if that's what you do.

I can keep cutting along either loop, separating it from the nice flat edge. Eventually paper and scissors won't support this. But mathematics will. If I have a topological strip I can keep cutting and make increasingly smaller loops. The paper gets skinnier and the loops will get more like little circles, eventually the twists will vanish when the strips are one-dimensional. But they will stay linked together.
And they will be really small, like smaller than atoms small.

p.s. I think I've learned how to play my little plastic toy into being a reduced algebra, with icosahedral symmetry and it seems to have the chops. It totally shreds S₅ because there are 4 copies on the thing. Because of the face-coloring, if you're strict about the map, it must be a missing red one. This is the reduced copy because it isn't in the set of remaining red triangles. Reducing a permutable set in mathematics is a matter of constructing a partition of the number, in this case 5, and using Young diagrams, you look for a parabolic subgroup. Yawn. This puzzle does it for you

It's a nice example of a polytope, a thing which is several things in the same place, or with the same status. A graph which is permutable, has orientable elements, etc. You orient the faces of the graph. You color the edges of a hole that wanders around the icosahedrally-sliced 2-sphere, mapping 3 of 4 colors (a restriction on the set and the color map) to its edges. Total Picasso, a rasguero of modules on a free monoidal category.

It needs something, though

[Eternal Student]

Hi.

    Thank you @Origin and @paul cotter .    You're probably right about the magnetic vector potential being the best candidate to explain how your toroidal transformer works.     
    It's also noteworthy that although Maxwells equations do work when you consider relativistic changes in the co-ordinate frame,  the E fields and B fields will be different and it's a little tricky to write equations that would show what the E fields and B fields would be in a different frame. For example, there is no 4-vector where the usual 3 component E field vector would be its spatial components.
      It's far easier to construct a 4-vector potential where the usual magnetic vector potential A will be the 3 space co-ordinates and, almost as if it couldn't just be coincidence, the scalar potential Φ for the Electric field turns out to be exactly what we needed as the time component to build this 4-vector.  We can transform this 4-vector potential easily and, if we wish, extract information about the E fields and B fields in each frame from that.   The potentials for the fields and not the fields themselves seem to be the thing that will combine to make a proper spacetime 4-vector.
     Without spending too much time going over the details (which are in several textbooks anyway), this suggests that the electromagentic potential is the natural physical or physically meaningful quantity when you consider Electromagnetism in a frame independent way.  The 4-vector electromagnetic potential will change only precisely as you would expect a vector to change when you change co-ordinates or reference frames.  The actual E and B fields are not so well behaved, they will change in strange ways when you change co-ordinates.  If we paraphrase this further:  The electromagnetic 4-vector potential is a quantity that really does seem to describe a physical thing, a thing that we can describe and talk about in a frame independent way.    By comparison, the actual E and B fields are completely frame dependent and may have no physical significance outside of the frame where they were measured.   So (in my opinion) it probably shouldn't be all that surprising that what may be happening in the potential field is going to matter and possibly matter far more than what appears to be happening with the E or B field.

    We seem to be side-tracking the original thread, so I'll stop.

Best Wishes.

[varsigma (OP)]

What it needs is less freedom.

That's ironic because the design of the puzzle means only one of the triangular faces can move at a time. The movement of a face or a tile into the hole exchanges the positions of the hole and the tile. But the tile has an oriented figure, a number, on it so the orientation rotates on the tile, whereas the hole doesn't have a number or a color.

The color map on the edges of the hole is a three-character combination of 4 x colors, so it's a word in a monoid and it's a free one; only some words are permutations.  Perhaps that's why it lacks a fixed arrangement and you have to decide what to do to "solve" it. I showed that it's a four-color map and you can arrange at most three sets of five tiles or triangular faces, so the 'pentad' shares a single vertex. The remaining fourth color can't be mapped like that so it's distributed over two separate regions, at most a tetrad plus a single tile.
 
Nonetheless, the thing is still a good pedagogical model that demonstrates group actions on sets, combinations have a subset of permutations, directions can be permuted. It's nice in a mathematical way but a bit useless as a puzzle.

Or you could drink lots of coffee say.  Yeah, nah.

p.s. watching Monica Vazirani again and thinking over her advice to her grad students, about categorification and how decategorifation gets you into the Hom sets (category of homomorphisms), it's about choosing a single number and partitioning it.

The prime numbers are like single points along a line, with composite numbers in between them. A prime number has a set of divisors with two elements, 1 and the prime are the only divisors of a number which is prime.

So exclusion: 1 is the trivial case--every number is divisible by 1. So using this, we have the single divisor remaining. Vazirani shows how Young's lattice of partition shapes S, can be used to derive useful stuff. You have have to watch, listen, and hopefully understand what's being presented. But what you want to do with a prime p is glue it to a boundary, maybe the real line, and this is just 1 ⊕ p, two points on the boundary, the  1 is the distance between p and the number 1 less than p. So easy it makes your eyes water. Or is that onions, again . . .

(I have abused the notation above, usually you see "otimes" sometimes "boxtimes" for the operation that adds another point to the boundary. I used oplus which is usually taken to mean a direct sum.)

But you can put every prime into a single cell of a Young's diagram and take the direct sum, using the usual ordering of numbers to get that shape, by moving up and down the lattice of shapes. It's cool stuff, but it's only rock 'n roll.

So you want a row of primes and a partition of numbers (compositions of cells with singletons--prime numbers--in them) and you can partition everything in as many dimensions as you need.

I suggest if you are interested, you google the nice lady's name on Youtoob. There is a series of lectures on Combinatorics and group representation theory. She is a rock star of advanced algebra.

[varsigma (OP)]

So here I am dithering somewhat, about various possibly related subjects, ribbon graphs and the Hopf gauge, algebraic graphs as free or restricted monoids, and 3d permutation puzzles.

At my age I should be playing cards with people my age. But here I am playing with toys. Well maybe as we get older we learn how to play with toys the right way. I like to see If I can take something apart and put it back together.

Another little beauty of a problem I was meant to find some kind of computational solution--a mock up if you will--for a switching problem. This is, if you have n inputs and to start with, the same number of outputs how do you connect a request for a connection to an output, so it doesn't block the network? In a thing we were handed a diagram of, there is a set of switches that do the routing of such a request from an input to a given output, called a Benes network.

But I know now that it's a graph of a construction, of fixed and free elements from B₂, the braid group on two letters. You compose these vertically and horizontally, in an ambient space, this is also in graph theory.

To fix braid elements, you restrict their orbits. A way to think about what fixing a braid group element means, is take a switch apart that switches between the two, which is how many there are in B₂. The identity isn't fixed if you can switch it to a transposition. To fix it you take the two strands in the switch you took apart, and say it's a direct sum of the single strand in B₁, and glue it back into the graph. You do the same with the fixed crossing but just say it's one crossing. Now your fixed set has a pair of elements with 0 and 1 crossing. The switches permute, the fixed elements don't.
The action on the switching elements is free, and Abelian because you can change the switch with addition modulo 2, addition is Abelian, and, for a pair or a bank of two switches, the group action is isomorphic to the Klein 4-group, There's the basis for your free module.

But all this is so you can have a lot of inputs and outputs and a way to connect them. I was told our efforts should look at the difference between routing one, and routing many in parallel,  parallel processing was the name of the course, at the time. There is a communications protocol you need to have, in a communications network.

One approach is to locate the graph of your switching network in a Hotel, that can connect any room to any other room. It's for the guests, this one. So the Hotel Benes, has a desk clerk who routes room A to room B, he knows which number room is which, and which guest is staying, which rooms are empty. He's the desk clerk.

So he might have hired a phone switchboard operator, who knows which switches to flip to get the guest in room A connected to room B. In the parallel version there are several operators, so they compete for access to the computational resource, the switches and their settings.

It gets complicated.

[varsigma (OP)]

Yes, complexity. We enjoy being able to solve complicated puzzles. There are many videos on the internet of many people doing this, explaining how to simplify the problem by doing in in parts. You partition the solution.

That is, whatever internal algebra one uses to apply whatever algorithms you cook up, you solve part of such a puzzle, then figure out how to solve another part, so that parts add together, A partition of algebras.

You want a free one and a fixed one. You realize the fixed one is one you will need to be able to partition again, then "re-fix". Solving a Rubik's cube means essentially that if you solve one layer, you need to be able to unsolve and resolve, freely, this layer. You take a measure (the solved bit) and distribute it so it will go back together, under some algebraic operation.

[varsigma (OP)]

I've been thinking on my spherical toy, the group action and where the vector space I need to write down the right stuff all is.

But I've thought about free and restricted group actions before, We probably all have.
So this thing has design and engineering restrictions, The puzzle is a solution, but it restricts the action because to have any in the permutation space, it has to be designed with a removable piece.

This frees up three pieces around the hole, and they can exchange places with it. You don't seem to have cycles and the inverse operation acts as the identity. So you get to add or remove one permutation from the set, but it's a big set.

So it walks through the graph, and it's a monoid in an affine algebra: no path starts or ends anywhere. There is no initial or final permutation and no home base for this hole in the surface.

Perhaps I should pay more attention.
oops I forgot about the cycle on orientations, or the direction the number is pointing, on a face when you repeat the exchange. a 3-cycle. I should have said you can't compose any cyclic permutations on the positions, because the hole is the cab, the triangles are passengers.

[Zer0]

At my age I should be playing cards with people my age. But here I am playing with toys. Well maybe as we get older we learn how to play with toys the right way. I like to see If I can take something apart and put it back together.

Reminds me of the Damm Rubik's Cube!

i tried n i tried n i tried...
& gave up!

Then i decided to Cheat.
I pulled all the blocks apart.
Rearranged them in their orderly pattern.
& then put em all back together.
(yep! brute force)

Now it is laid to rest in the showcase, where guests time n again marvel at my accomplishment, while only i who Truly knows how victory was achieved, on the inside hang my head in Shame!
: (

ps - i wanted to pass it of to someone else, but knowing it's worth, did not.

*Note -
A Rubik's Cube might be the Worst gift for someone with a.d.h.d./o.c.p.d.

[paul cotter]

I managed the rubik cube ONCE, but could never repeat it.

[varsigma (OP)]

I was once told, along with the rest of the class, that some people have difficulty with abstraction.

Well, maybe. I like to think that isn't really true, it's possibly more the case that people have difficulty thinking about a thing that requires a bit of focus. Some people pull engines and other bits of car apart, and put them back together.

Some people write software and have to stay focussed on the problem and finding solutions. Some people keep computers and software working in a useful manner. Some people design communications networks. I like to think those are skills I might have already, if I just look for them hard enough. I like to think it's about letting go of how important you or someone else thinks it is, and just do it because maybe you can. Maybe you should stop listening to people who tell you it's hard to do.

[varsigma (OP)]

Ok, now that I've bootstrapped my concerns about how important it all is--to whom, and in what context--I can take that Hillary step and forget about the step I just made.

Abandon physics for a while, stop thinking about geometry. This is advice Lenny Kauffman gives in his notes about knots and links. Stop thinking about how complex a knot looks, and check out the locally defined symmetries and abstract those.

As to a solution for the icosahedrally-sliced puzzle:-  I have recognized that I can choose an initial color map. This would be where a 4-color map fails to be 4 disjoint regions, you only get 3 and the 4th is two regions. I have managed to get the puzzle there, but then the numbers can be pointing anywhere (one of three directions), although it's possible to arrange these, the puzzle is tedious to use. But mathematics gets you there without that kind of fuss..

If I choose this particular permutation of the set of colored tiles, then remove the red piece which is located by the color mapping, so it lies at a centre of the three colored regions which are disjoint but arranged in lots of 5 tiles, then it has a permutation on its edges. There is exactly one such permutation on the icosahedron, all the other tiles are surrounded by a combination which is not a permutation. Lovely isn't she?

oops again there are in fact three more permutations around the red triangles in the disjoint red tiling, there because of the 3-cycle (i would guess). Ok so the inital state has 4 and no more than 4 permutations, the same as the number of regions which are disjointly colored, So now I need to be able to count up the operations (compositions of tile swaps around the hole) so that I return to the ah, mountain.

I have to embrace the tIger first (that's tai chi stuff, btw).

[varsigma (OP)]

I began thinking recently about what this might have to say about another kind of physics where the toys are states of matter, condensates and superconducting solids.

Well, I think you need to turn the diagram monoids into a switching fabric. How you define a switch then depends on what you need to do in the classical sense, vs what you would like to do in the quantum-exchange sense.

Switching a classical switch is affine if you don't care about whether it's off or on--there is no start.
Hence a string of 0 characters with 1 here and there (i.e. sprinkled in the 0s) will switch a switch off or on, it's the Boolean XOR product of two inputs, which inputs? One has to be monitoring--read only--the state of the switch. I surmise this is a necessarily continuous measurement. The other input monitors a string of 0s, waiting for a 1.

In the quantum domain the monitoring is quite different to the classical version. This could be exemplified by the braid groups that are being used in theories. A braid on two strands is different to a switch that cross two strands, because the switch only needs left over right, or right over left, not both.

The switch 'factors' the braid group on 2 strands, and so it's a permutation switch. You would have to know which way the switches were crossed to have any chance of constructing a braid, and then all you could do with it is unbraid it.

That could say something about braid groups and why they're popular with the physicists.

[varsigma (OP)]

Of course, the ambient isotopy! (excuse me while I slap my forehead)

The puzzle I have speaks to both the concept, and the realization of a switch that switches or permutes positions.

But the action requires ambient "holes" that sum to the area of the hole itself. During the swap, there are three holes, the ambient space has to allow compact support for the function (a rotation around one of two vertices); there are four edges supporting the rotation, and three triangular holes. So the hole is physically partitioned into three smaller triangular shapes, as part of the classical restrictions. Ok, that could be interesting.

Ambient isotopy is where deforming what is embedded is equivalent to deforming the ambient space. This puzzle says how you do it with triangular tiles on a sphere.

[varsigma (OP)]

If you want to know what a monoid is, it's a simple (!) thing to understand.

I encountered them in language theory, where it's the concatenation of letters or characters from a set.
Then all the finite strings are in a set--the length 1, length 2, . . . strings form a set which is closed.
 obviously length 1 strings are the alphabet itself.

When you walk around, your steps are strings in an alphabet of steps. Walking is the concatenation of steps, ordered by a relation between the two halves of your local symmetry. Monoids are everywhere.
And, your strings of steps are all accepted by a finite state machine,  Logic is what you say it is, pretty much.

I need to correct something I said before about the twisted strip I call a graph with two loops.  I realised that if I continue to cut along the strip, I'm only separating the middle from the loop which doesn't contract, it stays the same size except for getting thinner. That is actually a bit more interesting because now, I want to know what I did so the length of either loop is invariant, under this slicing part of it away. It's a partition that gives you a longer and longer flat but tapering ribbon. If I continue slicing it I'll approach a circle attached to a line, and the circle will have a fixed length.
And it's a graph with half-edges (if I say it is).

Moreover, it's a Mobius 'looped ribbon with a twist', and it's sliced or cut in a spiral. So it visits a set of points on a line across the strip in an ordered fashion. It does that with a straight cut too, but the order is different, An algebra of cutting through a set of points along a line emerges. Time for a bit of a rest though.

[Eternal Student]

Hi.

   Could I ask:     What input (if any) do you require from any readers of this post?

It's an interesting collection of thoughts you've written down but there aren't any questions or anything that suggests you are seeking any discussion.   I'm probably not the only reader who is wondering what it is that we are supposed to do.
    We can read what you have written - but that may not help or make any difference.   What are you hoping to do with, or get from, this forum post?   What can we do that may be helpful?

Best Wishes.

[varsigma (OP)]

Hi.

   Could I ask:     What input (if any) do you require from any readers of this post?

It's an interesting collection of thoughts you've written down but there aren't any questions or anything that suggests you are seeking any discussion.   I'm probably not the only reader who is wondering what it is that we are supposed to do.
    We can read what you have written - but that may not help or make any difference.   What are you hoping to do with, or get from, this forum post?   What can we do that may be helpful?

Best Wishes.

Ok. My apologies.

Going back to the opening page of this thread, it was kind of orbiting the idea that the Hopf fibration, hence the Hopf gauge and link, are about "our reality".

So I know now that the Hopf gauge is about ribbon graphs, which are not ordinary graphs, and I still don't really know about the reality side of it except for some embeddings (of spheres into spheres with more dimensions), one of these is the 3-sphere, which our real space is an example of.

So this 3-sphere is packed with circles. You can pack an ordinary circle into a circle without constructing a circle. Or rather, you map a circle to itself such that the map has another circle "in between" the first a double cover that links a circle, folded up and twisted, around another circle so they are linked together.

So to do that with a strip of paper, whereas the first can be achieved with a loop of wire, a twist action and a fold action to generate a central circle, you have to twist the entire strip by cutting it and gluing it back reverse-wise.

Both maps have a circle down their 'median' section that is linked to the one on the edge. Bingo, that's a homomorphism. So  bit of geometry of folding, cutting and gluing, with a twist gets you the same kind of algebra over a fixed set of points.

In some sense, you give a circle some "room" for another circle, and, with a loop of wire folded up a certain way (i.e. tensored) you give a circle along the middle of a surface room, and there it is. You don't have to draw it or cut it out.

Now what you want is a permutation of a fixed set of points;

[varsigma (OP)]

I feel a bit like I'm playing my bagpipes in the garage here, but I guess I'm a garage kind of guy.

I've fixed cars, both inside and outside of a garage, I've played a few musical instruments, but not drums, inside a garage, and I've done a few DIY experiments. Now it's garage math.

But you are doing math playing the bagpipes, and here's the story.

If you forget about most of reality except for the sounds a bagpipe makes when someone plays a tune, there is an ambient background, the drones. These are like fixed points in a diagram and the only permutation is restricted to the identity, over all the drone sounds.

Then the melody permutes sequences of notes, monophonically. When a note changes in time, it has two ways for the change, up or down a scale. I smell math, right there. Music is mathematically symmetric, and there are many ways to invent an algebra using a basis for some vector space. Melodies can be mono- or polyphonic, and there is a time 'signature', which most instrument players tend to improvise with.

Obviously it has a rich structure, as they say.

[varsigma (OP)]

And I've been trying to figure out how "rich" the space of a twisted loop of paper is, as a model of some spacetime.

How is it related to the notion of a ribbon graph (fairly obvious, but there's more) and why is it a bunch of linked circles, embedded in this twisted sheet of material? (math note, an embedding is an immersion, with a dual submersion)

A gauge theory is about trying to make a global symmetry into a local one. This is exactly what making a Mobius strip or twisting up a loop of wire so it matches the shape of the edge, is the action you need to abstract.

To 'break' the global symmetry in your theory, you make it a local symmetry. To 'restore' the global symmetry, you introduce a force, a vector space. The theory then follows. After a bit of thinking about linked circles and visiting some vertices in a graph.

[varsigma (OP)]

This is still me trying to relate what Kauffman is saying about his version of the Jones polynomial, and a Hopf algebra.

I am considering that it is connected to the way a loop can be the boundary of a surface, like how a loop of wire can support a film of soapy water. Which is a surface that minimises tension everywhere. It also frames the loop, according to  Kauffman, you can extend the surface beyond the boundary and embed it, in a space with ambient isotopy. You thicken the 1-dimensional boundary.

You can also contract the whole surface to a small ribbon across the closed loop, like contracting  soap film, but with some topological algebra that lets you do it, an equivalence relation say.

So if you change the shape of the boundary by twisting and folding it over itself, and then let the surface expand, it finds a "path of minimal tension", and looks different to the initial surface which was inside the flat loop, now its the surface of a twisted ribbon, and there's a link between the circle around it's center and the edge of the loop. It's a way to prove that you can wrap a circle up this way and it's at least one pair of linked circles, in 3-space.

[varsigma (OP)]

If nothing else I've learned recently that the abstract algebra approach means you have a few things to understand.

I think I've managed to do this using 3d permutation puzzles and shown coincidentally why they're good things to think about if you study the subject.

Why is a Rubik's cube restricted, or any permutation puzzle that involves rotations, over slices or however you choose to define the physics, it's one of the restrictions you should consider.
And there is another thing to the algebra you get to see, since the internal slices don't get a colouring. The colour map is independent of the rotational functionality of these things. As algebraic objects, there is a certain amount of freedom.

But whatever you do, it will be an induction, because you want there to be a restriction. You want a certain "colour invariance" to be part of the algebra. A natural way to colour a graph is to colour the faces. You need to derive (via induction on the number of faces) how many colours are needed to uniquely distinguish or identify each element in the set being permuted.

Since you are acting on this colouring, and on the rotations on each or any face, you are the "G ex machina", but of course you abstract everything to cycles and a group acting on the set. What is or isn't transitive depends on the colour map, and you get to say what that is.

[varsigma (OP)]

Ok, so what I'm thinking on is, this model I want to have of a result that tabulates or counts the number of permutations of the smallest cube puzzle--the 2x2x2 model--which was I think a program written before there were pentiums, or even 486 cpu boxes around.

The author, I forget who, is a mathematician and I think had a hand in the Cube Explorer software, explains his approach as using group theory to avoid a brute-force attack, because . . . memory and cpu resources were restricted. These days, most laptops could probably handle the brute-force counting method because the smaller puzzle has a mere 3.7 million and change, permutations in Perm{X}.

So what I'm doing is looking backwards, and that's one of the key ideas with group theory--the action and what is being acted on--which means I can look at an induction on n and n!. The restrictions start out being pretty obvious, but there are some rocks to avoid, such as, part of the action involves rotating from one 4-cycle to another, which is a rotation of the whole cube, and this is factored out when you, the conveyor of the action, rotate some part of it in a local sense, i.e. with the other part fixed.

The quarter turn metric is a thing that counts rotations as a cycle on 4 vertices, in the G-set {X}, but a 4-cycle is a permutation of length 3, it has three crossings in the permutation diagram. So the restrictions get you to the start line, with three generators for S₈ and an inclusion of three generators for S₄, since a 4-cycle includes a 2-cycle. The three 2-cycles are three inversions on three faces (centres of rotation!) and generate the product of three copies of S₄, as 180^o rotations, and these conjugate with the quarter turns.

The conjugation is just a point in the partition of Perm{X}. Notice how a concise notation can describe all the details . . .
This is where the quarter turn meets the half turn (an inversion, recall), and everything else can be rolled up and tucked away, then it's on to the next section where a pair of generators is the restriction (i.e.3 choose 2, and act once), and there are more points that intersect like the first one does.

Other computational or algorithmic observation, it took a while for the logicians to prove that the next size of puzzle in this domain, the 3x3x3, has a limit of 20 for the face turns. The limit for the 2x2x2 is 11, which is 9 less. Why that is is  a question the algebra behind both puzzles can answer.