Post by: varsigma on 15/01/2024 09:01:43
This guy is some mathematician, a professor, and he said the Hopf fibration is our reality; I assume he meant something like the universe is a principal Hopf bundle.
?
Post by: varsigma on 16/01/2024 09:20:15
Apparently, apart from topology and homotopy theory, it's important in Newtonian rigid-body mechanics (or classical phase space) and in quantum information theory.
Say what.
Post by: alancalverd on 16/01/2024 10:08:45
So the question is what does a Hopf model of the universe predict?
Post by: varsigma on 16/01/2024 17:47:05
Important to maintain the distinction between what something is, and how it can be modelled.
So the question is what does a Hopf model of the universe predict?
I don't know that there is a model so much as a way to connect existing theories to a Hopf gauge.
I'm reading a paper that gets there with quaternions. That makes sense to me because they are a division ring with four dimensions.
One of a handful of such division rings. I don't think you get any further than the octonions (I could be wrong about that).
On the other hand, can I say the quaternions are a universal algebra in a literal sense. I feel I need some healthy scepticism.
Post by: varsigma on 16/01/2024 21:24:07
All the number systems that have normed division algebras (vector space with a "euclidean" norm) are even-dimensional, beyond one dimension (arguably this one-dimensional algebra "falls out" of the others, maybe when we simplify things).
There's the uni-onions, or real numbers with the usual algebra.
The bini-onions or complex numbers.
The quattri-onions and the octi-onions.
n-onion rings.
Post by: varsigma on 17/01/2024 17:20:12
It turns out the original thought-provoking vid is by Eric Weinstein.
And he has a theory of almost everything. He got chucked out of Harvard, no less.
Nonetheless, there are exactly four normed division algebras and exactly four ways to fibrate the n-sphere with n-1 spheres. Maybe that means something but Mr Weinstein appears to be quite sure that the octonions are not well-understood. I can go with that; what the hell are they? Why are they there? Are they any use for anything or just mathematical?
This from Wikipedia
Quote
In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres:
S0 ↪ S1 → S1 ,
S1 ↪ S3 → S2 ,
S1 ↪ S7 → S4 ,
S7 ↪ S15 → S8 .
By Adams's theorem such fibrations can occur only in these dimensions.
Post by: varsigma on 18/01/2024 19:01:12
I've heard they encode rotations in 'ordinary' 3-space. This must mean that the Rubik's cube and indeed any 3-dimensional puzzle that has rotating parts is "quaternionic".
I don't know how much, but I do know that any finite string of "face" rotations has a cyclic order (to find out what the order is, repeat the string until you return to where you started). I guess that makes the 3-d puzzles an example of cyclic quaternions or somesuch. But I have to bear in mind there's a color map, I can change it. Graph coloring is a whole other branch, and graphs on the surface of a sliced 3-ball are interesting because there are quaternionic elements.
I have also wondered about the mathematical consequences of slicing a 3-ball, and whether that means you get access to a foliation or fibration of the 3-sphere. I have a partition of the complete graph for a 2x2x2 "face-colored" cube and I understand it to some extent--it's the quotient of the group by an equivalence relation--you make all the 'moves' on a single face equivalent.
This has a nice algorithmic context. All the states of the cube which are one move from the solution are in a solved state (say, by inspection). For the 2x2x2 this is exactly 9 states; it's easy to work out why there are that many, or you can ask the cube.
Post by: evan_au on 19/01/2024 11:40:22
Quote from: varsigma
I've heard (quaternions) encode rotations in 'ordinary' 3-space.That is correct.
Computer graphics geeks try to get the highest possible number of polygon transformations per second.
- You can do transform the position of a 3D point (a vector) to another point (eg rotation, translation, magnification, etc) by matrix multiplication. You could do it with 3x3 matrices, but I understand that they used 4x4 matrices to allow magnification (and it has nicer symmetries??).
- You can also do it by representing 3D points (a vector) and multiply it by a vector of quaternions
- Many years ago (long before GPUs), I saw a careful comparison reporting (IIRC) that the matrix multiplication method required slightly fewer multiply-and-add operations than octonions (after careful optimisation).
- Of course, the massive hardware parallelism of today's GPUs may have rendered all that careful hand-tuning obsolete.
The classic definition of quaternions was: i2 = j2 =k2 = ijk= -1
- However, ijk = -jik, ie the operations do not permute.
- This is also true in matrix arithmetic: Change the order, and you usually change the answer
- This is also true in the physical world
- this is not so for real or imaginary numbers, which do permute
If you were following directions in a building, two paths might be:
- Turn right, forward 100m, Turn left, forward 100m, catch the lift up 100m.
- Turn left, forward 100m, Turn right, forward 100m, catch the lift up 100m.
- ie Change the sequence, and the endpoints are not the same...
See: https://en.wikipedia.org/wiki/Quaternion#Quaternions_and_three-dimensional_geometry
...and the following section comparing quaternion arithmetic to matrix multiplication.
Post by: varsigma on 19/01/2024 20:08:48
The classic definition of quaternions was: i2 = j2 =k2 = ijk= -1
- However, ijk = -jik, ie the operations do not permute.
- This is also true in matrix arithmetic: Change the order, and you usually change the answer
- This is also true in the physical world
- this is not so for real or imaginary numbers, which do permute
More generally, you have 'scalar' multiplication; a + bi + cj + dk. Usually the coefficients are real. The thing with rotating things like rigid cubes, is that yes, it depends how you define rotations and generally these will not commute.
Consider the following heuristic. A cube is sitting on a plane surface and gravity determines that it will have one face against this surface; there are however two other ways to 'glue' the cube or fix part of it w.r.t the plane surface.
You can fix an edge or a vertex. Fixing an edge is a natural way to think about a formal way to define rotations from face to face of a cube and show they form "words' in some kind of algebra (actually sort of a tiling).
How or why you connect quaternions (under what motivation) is the question I'd like to answer. Do you need to? I already know you don't. Would it be a nice way to write some kind of program? Maybe.
The clues at the scene of the crime here, include that list of n-sphere fibrations in post #5.
The first one says complex numbers exist because: the circle embeds in 2 dimensions and has a fibration (Huh?)
The next one says something similar about the quaternions (the 3-sphere embeds in 4 dimensions).
Then the octonions exist because the 7-sphere needs 8 dimensions.
The sedenions are there because there's a fibration of the 15-sphere.
So yeah, where are the reals? they're a subset of each embedding. The sedenions do not form a division algebra; I guess we can consider these guys as the doorkeepers of division. Then why is division such a special operation?
What the hell happened?
p.s. I do have another clue, possibly connecting the quaternions or at least projective spaces, to rotations of cubes.
The original 3x3x3 puzzle is a set of individual cubes, but each is 'restricted;, firstly by being part of a larger cube, secondly by having a restricted coloring--some elements have two, some have three, there are six centres with only one face that can be meaningfully colored.
Anyhoo, the operational space is that of fractional linear transformation(s). Taken together, FLTs and the quaternions appear to be uneasy partners. What am I missing here?
Post by: varsigma on 20/01/2024 16:50:45
S0 ↪ S1 → S1 ,
S1 ↪ S3 → S2 ,
S3 ↪ S7 → S4 ,
S7 ↪ S15 → S8 .
That's better. The fibre is the leftmost term. it's "over" each point in the total space which is the middle term. The rightmost term is the base space. These are the only sphere fibrations with spheres.
Post by: varsigma on 21/01/2024 01:27:10
First, I notice that the central point (0,0) "of" the unit circle isn't in the set of points defined by x2 + y2 = 1. I think of this as the central symmetry of a circle, which can be "explored" by a radius line (with unit length) and an angle, plus a definition of rotation of the radius line.
But the projection of a point (x,y) on the unit circle (I use "on" although it's "in" a set, because the circle is a manifold). onto the x axis is a symmetry you explore by fixing a point on the circle, not the centre.
This fixing of something is the step Hamilton needed to get his quaternions working. It's what gets the complex plane going in the above, you fix a point and rotate a line.
Post by: Eternal Student on 21/01/2024 06:30:56
It's not entirely clear if you ( @varsigma ) want any replies or discussion. Maybe you do. It seems to be an interesting collection of ideas that you are considering.
Some way back you said this:
they (the octonions) are a division ring with four dimensions.....One of a handful of such division rings. I don't think you get any further than the octonions (I could be wrong about that).You seem to be taking division algebras over the Reals. It's not clear which additional properties (such as associativity, commutativity and dimensionality over R) you wish to have.
Frobenius' Theorem classifies the finite dimensional associative division algebras over the Reals. There are only 3:
(i) The Reals.
(ii) The Complex Numbers.
(iii) The Quaternions.
The Quaternions are not commutative, while R and C are. So if you wish to have commutativity then there are only 2 possible finite dimensional division algebras over the Reals.
The Octonions would not be included until you relax the associativity demand. The Octonions satisfy only a weaker form of associtivity usually called alternativity.
Mathematicians Kervaire and Milnor are credited with demonstrating that, even if you drop the demand for asscociativity and commutativity completely and just demand finite dimensionality then that dimension must be 1,2,4 or 8 only. Further work demonstrated that all finite dimensional division algebras over R would have to be isomorphic to one of these:
R , C , H (the quaternions), or O (the octonions) and there are no others.
There are still some properties that you may choose to drop or alter. If you drop the demand for finite dimensionality then ..... I know of few theorems that place a lot restrictions on what you can get.
You may also decide that there was nothing very special about R and you could have taken some other field as the scalar field for the algebra. It is known that if you choose a field K that is neither algebraically closed or Real closed the dimensionality of a finite dimensional division algebra over K is not limited to just 1,2,4 or 8. Specifically, there is an infinite set of different (non isomorphic) finite dimensional division algebras over K.
- - - - - - - - - - -
what the hell are (the octonions)? Why are they there? Are they any use for anything or just mathematical?All mathematics is valuable and worth having. I may be a little biased in making that statement.
As it happens there are some applications for octonions in physics: Wikipedia lists some..... https://en.wikipedia.org/wiki/Octonion#Applications - not many of these are demonstrably practical or have yielded anything that could be used, tested and verified except some applications in robotics and machine learning.
- - - - - - - - -
This must mean that the Rubik's cube and indeed any 3-dimensional puzzle that has rotating parts is "quaternionic".I'm sure most mathematicians will be pleased that you use Mathematics to describe real world objects and phenomena and not the other way around. For example, you don't say "the quaternions have properties that mimic rotations of a 3-D puzzle". However, the quaternions (and any mathematical object) may be abstract and it's a bit of a leap to assume the world (or universe) could only follow some pattern or behaviour that exists in some algebraic structure - but maybe that is how it is, I don't know. It's more common to imagine that the world does what it does, there is then some mathematical structure that can be constructed which will represent that behaviour.
------------
I guess we can consider these guys (sedenions) as the doorkeepers of division. Then why is division such a special operation?In many ways it isn't division that is special. The simplest algebraic structure is a Group, it has only one binary operation and you can use a + symbol and addition notation to represent it. However, you can just as easily use a x sign, or a dot . and multiplicative notation to represent it. Rings are a more complicated algebraic structure where you have two binary operations and we naturally choose to use addition notation for one of these and multiplication notation for the other. We put a greater restriction on the properties and behaviour under one binary operation and the Ring is actually an abelian Group just under that operation. By convention, we mark this first binary operation with + and talk about it as if it was addition. There are less restrictions placed on the other binary operation and the Ring is not a group under the other binary operation because various things were not demanded. The use of additive and multiplicative notation is just convenient. It is arbitrary which of these operations you might consider to be analagous to addition in the Real numbers and which one is analagous to multiplication. Indeeed it's un-necessary to make any analogy to operations on ordinary numbers, they are just "the first" and "the second" binary operation acting on some set of elements.
The various theorems about division algebras you've referred to are not marking "division" as being special. They are just limiting the behaviour of the structure under "the second binary operation" (whatever that may be).
Best Wishes.
Post by: varsigma on 21/01/2024 19:43:49
The various theorems about division algebras you've referred to are not marking "division" as being special. They are just limiting the behaviour of the structure under "the second binary operation" (whatever that may be).As with any exploration of mathematical structure, it's hard to not go off on some tangent, and it's hard to stay on course since you aren't sure if you're exploring in the right place.
Division is "special" in the sense it's restricted to certain places. That's what I'm exploring, the why is that true question.
If you "lift" the restriction of closure then things change.
Mathematics gives you different kinds of freedom, but this comes at the cost of having to fix something.
To me, a fundamental part of the logic involved is this comparison of differences. Fundamentally any mathematical structure is about what is fixed and what isn't. Or at least, that's as good a place as any to start.
But this is about the Hopf fibration of the 3-sphere and, that there are other sphere fibrations; there does seem to be some kind of connection to division rings. Or, you know, there might be a coincidence; it might not be saying anything.
But we have a professor who is convinced enough that this approach (Hopf algebras) is the answer (what was the question?) that he got booted from the Harvard faculty. Maybe it's all shirt and no trousers.
Post by: varsigma on 21/01/2024 21:05:42
If you "lift" the restriction of closure then things change.See, that could be a cosmological understatement.
The questions appear to include some about closure in the observable universe. One is, how many ways can you fill 3-space with circles? How come that's interesting, and how do you do it so every pair of circles is linked together? How many dimensions are required to link two circles? (I know the answer to that one, but, so do you guys, right?)
And, notice how this "simple" idea, linking circles, leads to a lot more; why though? Is it related to the complexity of the number of ways (i.e. is a combinatorial 'problem').
Post by: varsigma on 21/01/2024 21:35:28
Can I ask if anyone here is into graph theory? What's the significance of thickening edges in graphs? Is it that now, you can introduce twist?
Can anyone explain ciliated ribbon graphs in a few words? I'm plowing through arxiv and haven't found anything straightforward as yet.
Post by: Eternal Student on 22/01/2024 06:50:42
Can I ask if anyone here is into graph theory?I did study some discrete mathematics but didn't specialise in it.
What's the significance of thickening edges in graphs?You may be merging two things into one bundle of terms here.
1. The thickness of a graph, G. This describes the minimum number of planar graphs required {Gi} such that the graph G can be described as the union
G = Ui Gi.
A planar graph is one you can draw just on a flat sheet of paper and no edge would need to cross over another edge.
The vocabulary is evidently based on imagining that you may need to have many layers (stacked one on top of the other in 3-D space) where each layer will look like a graph that doesn't need any edges to cross over. With more layers required, the graph is thicker. For some graphs with an infinite set of vertices, you may need more than 1 extra dimension in which to stack the layers but the term "thickness" has stuck. You can have the same vertex appear in multiple planar graphs Gi. So there simply is no need to imagine weaving some edge down through the layers to reach a vertex that was only in a lower layer, if it's needed in some other layer you can also have it there. So a standard 3-D stack of layers will always be sufficient.
2. A Ribbon Graph which is sometimes called a Fat Graph. Here the edges are shown with some width rather than just as lines. These edges are then often called "edge rectangles" although they need not be drawn with straight sides like rectangles in Euclidean geometry. Topologically and for most purposes in Graph theory, an edge connecting vertices can be shown as a straight one or a curve, it matters not.
As you probably know, a Graph is just an object you can define as a set. It is a set of sets. You have a set of vertices, and a set of edges. The set of edges is itself a set of sets - it's a structure that effectively describes an ordered pair (x,y) where x, y are vertices (which could be the same vertex). A pictorial representation of this is generally more usefull and we draw the vertices as points, with the edge (x,y) shown as a line connecting vertex x and y. Once you appreciate that a graph is simply a set theoretic object, it's apparent that how you draw it is almost irrelevant, the graph is not the picture, the picture is just a way of representing the graph. It also makes formal sense out of the definition given just above for the thickness of a graph: The graph, G is just a set and it certainly can be written as a union of sub-sets.
The Wikipedia page describing Ribbon Graphs is currently quite poor. As already mentioned, a graph is formally just a set theoretic object and a new picture has not changed it at all. Wikipedia currently describes Ribbon Graphs only in this way - as a representation you could use.
A ribbon graph can also be defined as a new object, a graph with some additional structure. (Wikipedia will probably get around to this eventually). It's a set whose elements are all of the things just described plus some relation that describe whether an edge will twist. Under this definition a ribbon graph is different to a (ordinary) graph even if all the edges are untwisted. In an ordinary graph "twist" is simply not a structural property that existed but in a Ribbon graph all edges have state of being twisted or untwisted.
The current applications of Ribbon Graphs are often limited to having them as a representation, a picture of the graph. It seems to be interesting to know how a graph could be drawn on various different types of surfaces. Wikipedia takes a few lines to describe how useful Ribbon Graphs are for visualisations of embeddings. That, I can only assume, may be true.
I don't know a lot more. The Wikipedia pages should exceed what I could tell you (although some of it does seem quite poor to me, as mentioned)
https://en.wikipedia.org/wiki/Ribbon_graph this isn't a lot of information but following all the links off from that page gets you a bit more and there are some good old fashioned references (mostly books) listed at the end.
I just enjoyed talking about sets for a moment. All mathematical objects can be considered to be built from simple sets, they can be a mathematicians fundamental objects much as an atom would be to a scientist.
- - - - - - - -
You asked about the "significance of a thickening edge in a graph" and that may be an awkward blend of terms: It could be an edge whose removal would reduce the thickness of the graph. You probably didn't want to know that. I'll assume you just wanted to know why ribbon graphs have edges with some width.
If you've seen a diagram of a ribbon graph with edges getting thicker, they were either trying to show a twist in the edge or else it's an artifact (presumably a desirable artifiact) of how the representation was constructed. Wikipedia mentions ribbon graphs where the edge width would be determined as an ε-neighbourhood of points along that edge. The representation starts to look like a collection of long thin ribbons when ε is suitably small. So you need to start by recognising that the metric defined on the surface on which the edge is drawn would always allocate a metric distance of ε to the width of the edge. If that edge is getting wider on the diagram, then the way space is being represented on that diagram differs from the metric distance. So, for example, they may be showing a representation of the graph on a surface where there is some preferred way to space out the points in that surface (maybe using a Euclidean metric) but this preferred way is not the same way that the given metric would allocate distance. In some regions the given ε metric distance is very different to the diagrams allocation of distance and the edge becomes much wider or much thinner.
I hope that makes some sense and I'm sorry I couldn't tell you much more.
Best Wishes.
Post by: varsigma on 22/01/2024 11:41:37
Did you know a Mobius strip isn't just a non-orientable surface? It's a thing you can think of as a ribbon graph-- a loop on a single vertex. The ribbon is twisted, but the edge of the ribbon is a circle, but so is a continuous straight line down the middle of the strip. That's two distinct circles, the strip is the map from one circle to the other.
Now this strip can be a minimal surface--a catenoid, When you unfold the surface by pulling the edges apart, the surface spontaneously reconfigures. You can model this with wire and soap films.. Another thing you can do is cut the strip along the meridian (surgery!). This is a way to prove the circle on the edge of the minimal surface is linked to the meridional circle
Post by: varsigma on 22/01/2024 21:14:14
Can you figure out why cutting a Mobius strip in half by cutting once (i.e. rotating the scissors through one complete circle) results in a larger, but single strip, whereas cutting the strip by rotating the scissors by two complete circles results in two linked but unequal ribbons?
How is that related to the mathematical and physical fact the strip with one twist has only one side? You can prove that there is only one by drawing a colored stripe down the centre--to complete this colored stripe you need to rotate the pen twice around the strip (or, the torus).
Since the twisted (Mobius) strip can be embedded in a solid torus with the edge on the surface (on the 2-torus); there we more or less have it--the edge of the Mobius strip is a torus knot with one crossing.
According to nLab, this double cover of SO(2) has a map to the Hopf fibration (as you can see, a torus can be filled with twisted strips, and a twisted strip is a 'bundle' of linked circles.
(https://upload.wikimedia.org/wikipedia/commons/b/b9/Hopf_Fibration.png)
Can you spot the "almost a 2-torus" in the middle of the upper image (of the 3-sphere): it's yellow and green. This is the bundle over the lower "line of latitude" on the 2-sphere. The equator or any latitude circle on the 2-sphere "fibrates" to a torus of linked circles on the 3-sphere (or, in its projection back to three dimensions).
Fibrations, submersions, that stagger the mind . . .
Post by: varsigma on 23/01/2024 03:39:11
More things to think about, or more things I am thinking about.
A pair of linked circles in three dimensions can't be constructed without opening one of them.
How many dimensions do you need so this is like, I don't know, say, cutting along a ribbon?
And, another thing is, any two of the circles in that colored picture of nested, incomplete tori, is linked in the same way you get two linked ribbons when you cut a twisted strip open a certain way.
You need to "go around" at least twice to get there; you could go more than that, or wind your cutting circle around more than twice. If you cut it open by joining the ends of the cut together after one winding, you get a single ribbon but if you repeat this, i.e. go around again the same way, bingo there she is, the link from nowhere.
Post by: varsigma on 24/01/2024 01:29:09
The Hopf gauge is a group; a gauge group is sometimes called a structure group.
A ciliated ribbon graph is a special kind of graph which is an upgrade if you will, from lattice gauge theory (a lattice is a graph too, but generally lattices have a periodic structure).
Mathematicians like to be careful about things like graphs. If you change something like, you decide to use ribbons instead of lines, and oriented disks instead of vertices, you have to introduce operations on a graph that do these things.
So you get an operational algebra for your graph. The Hopf gauge is part of the way you "decorate" the graph with things like arrows or with reference points that determine an order on the edges at a vertex. Generally 'gauge' means reference. A gauge is like a basepoint, something is fixed and something is known about its value (or its order) . . .
I challenge anyone to put that on a bumper sticker or a T-shirt.
Post by: Eternal Student on 24/01/2024 01:47:41
Can you figure out why cutting a Mobius strip in half by cutting once (i.e. rotating the scissors through one complete circle) results in a larger, but single strip, whereas cutting the strip by rotating the scissors by two complete circles results in two linked but unequal ribbons?That one I can just about get my head around.
Here's a video that can help to visualise how and why things are happening.
[Approx. 4 minutes, no spoken words, just some text and animations, it all becomes very easy to understand]
You ( @varsigma ) may not need it but anyone else trying to follow might appreciate some stuff being shown and explained in the good old fashioned 3-D world.
Best Wishes.
Post by: varsigma on 24/01/2024 04:23:34
That one I can just about get my head around.The image (I haven't watched the video yet) illustrates what happens if you cut down the middle of the strip.
One reason you get a longer thinner strip is, you cut down the centre of rotation for the twist (recall to make a Mobius strip you tape the ends together after you flip one of them over).
I've made at home about 6 twisted strips and cut them different ways. I have found a way to generate a ribbon graph with 5 twists in it. It's a complete K2 with a twisted loop on each vertex. One loop has 1 the other has 4 twists. The loops are linked. Weird or what?
It's because I didn't complete the cutting procedure. I'm investigating how many ways there are to cut a strip with one twist so it does or doesn't come apart. Of course there are then all the ways I can cut open a strip with more than one twist in it.
Graph theory, don't get me started.
Post by: varsigma on 24/01/2024 06:18:47
The image posted by Eternal Student shows a Mobius strip and a cut strip. Can you see that the one on the right is the one on the left when it's glued back together, but as presented has more twists in it? How many are there and how does it affect what happens when you cut this one down the middle?
Post by: Eternal Student on 24/01/2024 11:01:13
I haven't watched the video yet
I've made at home about 6 twisted strips and cut them different ways.That's fine, you're under no obligation to watch a video. Many of us don't watch videos that get posted on a forum thread.
It's mainly there so that if anyone else was trying to follow they will have something tangible. It's a 4 minute video, I would recommend it and it's a lot faster than trying to make your own strip and cutting it in real life.
The image .... illustrates what happens if you cut down the middle...Warning, a thumbnail image of a You Tube video may be different for different people. I think it depends on whether they have ever started to watch any of it - there may be some temporary files already in their browser cache.
Best Wishes.
Post by: varsigma on 24/01/2024 14:31:29
That's fine, you're under no obligation to watch a video. Many of us don't watch videos that get posted on a forum thread.But, in real life you can choose to do things with strips of paper and scissors, some tape etc, that a video doesn't.
It's mainly there so that if anyone else was trying to follow they will have something tangible. It's a 4 minute video, I would recommend it and it's a lot faster than trying to make your own strip and cutting it in real life.
I did watch the video, it reminded me of something I did recently, but more often and in different ways.
To me, it's remarkable that you can easily make something which immediately gets you thinking (it worked for me) about paths along the strip or its edge.
It has one edge, although locally it has two. It has one side but locally, etc.
It's a thing that generally, I find is more interesting if I draw lines and don't cut along them.
Moreover a strip of paper can be an edge in a ribbon graph.
Post by: varsigma on 24/01/2024 16:21:30
A path in or through a graph visits both vertices and edges. This is already a way to partition different kinds of paths in graphs.
Graphs are free objects because they don't have to have directions (one way signs) along edges. If they do, then paths are restricted. The order you move along edges or visit vertices is now more important.
Ribbon graphs encode an order for path walks between sets of vertices. When you upgrade an edge from a one dimensional line to a ribbon, by "thickening" it, it has two edges, if you orient it by drawing a directed edge around it or a circular arrow, that's a restriction which imposes an order for the edges, pathwise.
I'm plowing through a long paper from arxiv which is mostly equations, but there are some diagrams. The message that gives me is there can be an awful lot of equations in a graph.
Well, ok, the Rubik's puzzles are all graphs as well as polytopes. There are books, full of notation, that people have written, and probably a lot that haven't about what these things are, mathematically or algebraically.
Give me an algebraic object and I'll show you a graph . . .
Post by: varsigma on 31/01/2024 06:21:56
I've realised that making Mobius strips or twisted loops of paper in general gets you into knot theory.
But, you need a sort of algebra of cutting along lines which are embedded circles.
You need a rotation algebra in that many dimensions--three is the minimum needed to make a knot.
So if I wanted to explain what it's about to a young person, I'd say there's a 4-dimensional algebra of rotations, hiding inside a Rubik's cube. It's hard to see it because it needs to fix a 4th dimension and rotate everything around it.
Unfortunately this universe has a time dimension and you can't fix it and rotate around it, because it keeps vanishing at a point. You probably can't do that in a universe with more dimensions, because motion depends on that last detail about time disappearing, all the time. But with more spatial dimensions and time, you can have motion of objects through other objects or themselves; you can turn a sphere inside out.
Post by: varsigma on 11/02/2024 06:38:10
I've been walked through a proof of the real and the complex plane being isomorphi√√c. Both are vector spaces, but how does that tell you anything about "imaginary" numbers like i?
I've seen a method that gives you a geometric view of imaginary roots of a polynomial. For example a parabola is either concave up or down. In the first case, if it has a positive y intercept it won't have any real roots. The method reflects the parabola through the y intercept, and rotates it into another dimension out of the plane (where the roots are).
Likewise, the square root of -1 can be said to "live" halfway between 1 and -1, in a rotated plane. And I think the easiest way to see the isomorphism is because of the way you can represent a complex number as a 2x2 matrix. This matrix is the sum of two 2x2 matrices, the real and the imaginary part. The real part is a fixed plane (times I, the identity rotation), the imaginary part is a rotated plane (times a positive rotation of 90 deg.)
That's what I see when I note that the imaginary part is a rotation matrix.
Post by: varsigma on 05/03/2024 08:21:29
My best heuristic for this idea is the Aharonov-Bohm effect.
What this is is a measurement of a shift in the interference peak--the central peak shifts or splits in two because of a phase change in the fermion matter field, The phase is of the amplitudes for each fermion's probability of being you know, a this or a that, or both or neither. It doesn't matter, in terms of paths for particles which are fermions.
The matter field's amplitude is given at any point by this wavefunction of probability (not actually the "physical" thingy!).
So long story short, the magnetic potential, shielded from the electron beams going around it in a coherent split matter-wave, gauges the phase of the field, of . . . amplitudes.
The shift in phase, from the gauge field, breaks a symmetry. This symmetry-breaking is a measurement, they are one and the same thing.
In the Hopf gauge, the algebra breaks a symmetry at vertices, where the order of the edges is gauged by a ciliation.
I'm looking at algebras which are about braids and knots, they don't need ribbon graphs but I have seen Kauffman do this for knot polynomials using his bracket notation.
This stuff is all active, on the drawing board right now kind of stuff. Knots, ye gods.
Post by: Origin on 05/03/2024 16:21:27
the central peak shifts or splits in two because of a phase change in the fermion matter fieldWhat is a "matter field"?
What is the phase change you are referring to?
So long story short, the magnetic potential, shielded from the electron beamsWhat electron beam? Where did that come from? How do you shield a magnetic potential?
Post by: Kryptid on 05/03/2024 16:35:08
Post by: varsigma on 05/03/2024 21:01:07
Is this a new theory?Nope.
I recommend you look up the Aharonov-Bohm experiment. A beam of particles with a definite velocity is a matter-field. So is a lump of matter at rest.
A diffraction grating with two slits is a matter-field, a moving field of fermions interacts with (the geometry of) a fixed field of matter. Bingo, you measure a pattern of interference by breaking some symmetry. This symmetry can be the global phase of the fermion field. It can be that thing because you can't ever measure or detect it.
If that sounds a bit weird, yes it does.
Post by: Origin on 05/03/2024 23:56:43
If that sounds a bit weird, yes it does.Could you answer my questions from reply #29?
Thanks.
Post by: varsigma on 06/03/2024 09:39:58
About removing circles from a Mobius strip, which you can do with an untwisted one, in which case it's a cylinder.
Where does it go? To the same place you can get it back from, i.e. anywhere you can glue back the edges that weren't there before you used scissors to cut out "a circle worth" of twists.
More to this, is the fact that a circle is a 1-torus, so pedantically you remove a twisted strip with the scissors, but it's embedded (immersed) in a 1-torus--approximately since paper has a small thickness. When you glue paper together along edges, it fails to correspond to the mathematical version.
So this cutting and gluing with a strip that is closed and connected, ok, maybe it needs to be compact if you want there to be bijective maps, but yada yada, it speaks to what a circle is, or what anything mathematical like the number of twists is.
If you can increase this number by cutting the strip, then it follows that removing a circle acts on this number. What is a number? You can partition a number, you can categorify a number.
A number is not just a number.
Post by: Origin on 06/03/2024 12:27:13
I'm not a performing seal, mate.I realize that. You made some statements that I didn't understand so I asked for some clarity. I thought that was the point of a discussion forum. If you don't want to answer that's fine with me.
Post by: Eternal Student on 06/03/2024 20:41:21
You made some statements that I didn't understand so I asked for some clarity.I didn't originate the posts but this is what I think was meant:
What is a "matter field"?In QFT (Quantum Field Theory) it is any field for which oscillations in that field or energy in that field could be considered as a particle of matter. Usually we're thinking of fermions - typical matter particles like an electron, a quark or a composite particle like a proton.
I'm not entirely sure that @varsigma really needed to consider QFT for what was being discussed. The Aharonov-Bohm effect is explainable or can be described with just a non-relativistic, plain vanilla version of quantum mechanics.
What is the phase change you are referring to?In the standard experiment, it would be a shift in what is called the complex phase of the wave function. Basic idea: Wave functions have a value that is a complex number, they have real and imaginary parts. So a wave function might be something that looks like this:
(https://readingpenrose.files.wordpress.com/2015/01/circular-polarization-circularly-polarized-light_right-handed-animation-305x190-255colors.gif)
If you look at the place where the wave function is being measured by the... disc shaped speedometer looking thing with the big arrow spinning around.... you'll see the magnitude of this wave function never changes. However, the portion of it that is in the imaginary axis (say running in the up-down direction of the page) compared to the portion of it that is the real axis (say running in the direction that looks like it's coming out of the page or going into the page) does change. Sometimes all of the magnitude of the value is in the imaginary axis, sometimes all of it is alog the real axis. Overall the complex valued wave function seems to spin or rotate when you view the bare complex value of the wave function. The modulus, or overall size of the wave function is what is usually important, the modulus squared is proportional to the probability of finding the particle at this place, so in this diagram that never changes and the particle is as likely to be found at place X now as it will be likely to be found there in 20 seconds or tomorrow. The fact that the complex value of the wave function is rotating in the complex plane doesn't matter in the slightest. Indeed for almost all purposes, the phase or exact way in which the magnitude of the wave function is spread between the real and imaginary axis doesn't affect any physical observation you could make. All we usually care about is the overall magnitude of the value and the complex phase of the wave function was, historically, thought to be essentially irrelevant and something that was only there as an artifiact of the mathematics.
However, the individual oscillations in the real and imaginary axis WILL matter if you brought two quantum mechanical wave functions together, i.e. made one wave packet approach another wave packet and merge. On merging, you just add the wave functions much as you would for any other sort of wave mechanics. However, it is not the modulus or magnitude of the two wave packets that you would add, you add the direct or bare complex value of the two wave functions. If the two wave packets had their rotations in the complex plane synchronised or "in phase", then the real part adds to the real part, imaginary part will add to the imaginary part, all of it gets made bigger and you end up with one really big valued wave function that still spins or rotates around the Complex plane as before (it's just got a bigger magnitude or longer spinning arrow on the speedometer).
On the other hand if the two wave packets that merge had their rotations in the complex plane precisely 180 degrees, or one half a cycle, out of phase, then when you add them you end up with the oscillation tending to cancel out: Whenever one was going big in the positive direction along the real axis, the other is going big in the negative direction of the real axis etc. When you add the two wave packets together, you get only destructive interference and 0 oscillation all together. Hence 0 probability of finding the particle(s) in this place where the wave packets merged but were so badly out of phase.
What electron beam? Where did that come from?What the Aharanov-Bohm experiment did was to take a beam of electrons and split it into two. Pass one of those groups of electrons around the left side of a barrier and the other electrons around the right side of the barrier. While travelling, the complex phase of any beam doesn't matter at all. Now re-combine the two beams after they get around the barrier and let the beam hit a screen that will glow on contact with an electron.
If the two split beams remained in phase, then you get a certain pattern on the screen. If the two beams went out of phase then you'll get a different pattern on the screen due to all the destructive interference as discussed just above.
Here's a diagram (the magnetic field will be explained next):
Aharanov Bohm.jpg (64.35 kB . 978x663 - viewed 180 times)How do you shield a magnetic potential?Well you don't. What you can shield is an ordinary magnetic field (the thing that should exert force on a moving charge). The potential, it seems, you can do nothing about. What was done in the Aharabov-Bohm experiment was to use a special sort of barrier that they sent electrons around. The simplest one to use is just a very long coil. As you may know, inside a coil of wire carrying an electric current, there is a very strong magnetic field. Outside of the coil the magnetic field is really small and effectively negligble at most distances. The magnetic field wasn't really "shielded", although that could have been done, but it was just easier to use a coil and assume the magnetic field exists only inside of it and not around the outside of it.
The thinking of the time was that since there wasn't any magnetic field outside the coil and in the space where the electrons were being passed, the behaviour of the split beams should be much the same regardless of which side of the coil they were passed around. In particular, the complex phase of their wave functions were in phase before they were split and they ought to just stay in phase. So, on being recombined at the other side of the coil, you'll get only the usual pattern on the screen.
As you may have guessed... you don't.... the complex phase of the beams does seem to be different depending on which side of the coil they were passed around and you end up with a pattern on the screen that suggests destructive interference has occurred when the two beams were recombined.
For quite a while this was quite a serious issue: There was no magnetic field where the electrons were travelling, what on earth was affecting one beam differently to the other? Turn off the coil and everything is back as it should be... good old pattern 1 on the screen when you recombine the electrons. Somehow the magnetic field that existed only inside the coil seemed to be influencing the electrons that were only ever outside of the coil. Sure, if some charged particle was moving and it went through a magentic field you'd expect to see some effect but you need to realise that the magnetic field is precisely 0 valued outside of the coil, there is no magnetic force of any sort to be felt where the electrons were travelling.
Let's use gravity as an example because that's usually easier for us to appreciate. Let's assume I can put two equal sized planets next to planet earth, one on the left and one on the right, equal distance from the earth, everything nice and symmetric. You would have a new gravitational force pulling to the left but also a new one pulling right, so overall no net force. You would carry on with your life as usual and expect to notice no change in anything whether these pair of planets were there or not. What the Aharanov-Bohm experiment suggests is that... actually there may be a difference. Although there is no net gravitational force acting on you, you have certainly changed your gravitational potential. There are loads of things for which we still think "potential" is fairly arbitrary and all we need is some arbitrary reference point we ca call 0 potential. In electronics, the earth line is reference 0 volts and everything works provided the +5 Volt line is at a potential 5 Volts higher. We don't try to guess what the actual potential of the earth line was and we assume it doesn't matter. If we took the box of electronics to another planet, drove an earth pylon into the ground there and used their "earth" as our electronic earth, we would still expect the box of electronics to work.
The Aharanov-Bohm experiment suggests that for magnetic potential (and possibly other potentials) things are not as we once thought. It made a difference when the coil was turned on because the magnetic potential of the space the electrons were in had changed, even if there was actually 0 magnetic field (no magnetic force if you prefer) in the space. It's maybe not that interesting these days where we flippantly add another field whenever required - but back in the day it was important to imagine that magnetism (and possibly every other force) may demand that two fields existed throughout all of space. There is a magnetic field, which can exert a force on moving particles but there is also a magnetic potential field. More importantly, the potential field may actually have physical effects rather than just being a mathematical tool. This alone is important, it means that maybe knowing that the +5V line in your box of electronics is 5 volts higher than the earth line is NOT enough, it may actually matter what the absolute potential at the earth line really is.
Unlike the electric or gravitational fields, the magnetic potential field cannot be described just as a scalar field, we require a vector valued field for magnetic potential (i.e. it has both a magnitude and also a direction). This was another reason why the notion of a magnetic potential was contested or considered to be just a mathematical artifact. An explanation for the Aharanhov-Bohm experiment is not attainable if the magnetic potential is just a scalar field. There is a perfect symmetry around the coil in the experiment , an electron passing on an equivalent path on either side of the coil should have been put into all the same potentials for all the same amounts of time, so even if it did affect the complex phase of their wave functions, they should be affected equally. On re-combining the beams at the other side of the coil, you'd still expect to have both beams remain synchronised or in phase - and get a perfectly standard pattern for constructive superposition being shown on the screen.
If we allow the magnetic potential to be vector valued instead of just a scalar, then we can break this symmetry. The potential field has direction in addition to numerical value and there can be a net circulation of the vector field around the coil: We imagine that electrons passing around one side can be travelling against the direction of the magnetic vector field, the ones travelling around the other side would be travelling in the same direction as the magentic potential vector.
That's probably all I know about the Aharanov-Bohm effect, I hope it's of some use to you.
Best Wishes.
Post by: varsigma on 06/03/2024 21:51:22
It was an engineering problem setting up the experiment. An electron beam in a SEM can be focused on a small wire which is the beam-splitter. This split beam has a "coherence factor", it doesn't stay coherent for very long. the beam is recombined by a second wire. the two wires have opposite charges, to split the beam the first must have a -ve electric potential, to recombine the two partial beams the second must be +ve.
The paths of these two split beams aren't affected, the phase of the matter-wave is. Electrons see this geometry of a magnetic potential acting on their phase--it's a parallel transport of a phase vector along a surface, which is flat in our classical experimental frame, but 'rolled up' into a truncated cone in the phase-space. That is a description of the effect in terms of a geometry--is there really a cone-shaped potential erm, quantum waveguide? Yes, the electrons say. We just can't ever see the thing directly.
And my conjecture is that a measurement--the change in the pattern--is a symmetry-breaking "operation". There is an action on a symmetry group. Which one? Well it's likely living in SU(2) x U(1) somewhere.
Post by: varsigma on 06/03/2024 22:23:33
This is the unit for the module you have when you take a set of permutable objects and embed it in a field of numbers, generally you want them to have some other symmetry, a shape or a color map maybe. Geometry need not be a consideration here because your module is abstractly a string of points along a boundary. An ambient space.
The Rubik's cube and 3-d puzzles in that class or permutable sets, comes with an identity permutation--easy to define as just " any face has one and only one color on (identically!) all of its squares"
--hang on the missus needs a cuppa, back in a ticky boo.
Ok so the deal is, I can choose any permutable set so I can choose one of these, made for that purpose, mathematical toys
Screenshot from 2024-03-07 11-28-37.png (299.65 kB . 479x488 - viewed 116 times)Each is in a permutation of the set, this is the sum of orbits of all the pieces such that they leave the set unchanged. There are a lot of orbits to sum over, but you don't have to do that. Each puzzle in the posted image is in this state and you can prove that there are permutations that change the positions as you do them, but leave the puzzle unchanged when you finish. The sum includes the group action on positions, there are only eight for the 2 x 2 x 2 puzzle in the photo. So each piece has the same size orbit, which is 8.
You need to say that this number is "one orbit", and we are done (at least, we are done when you write down this mathematical proof).
But you can just say it without the proof, if you're a professor of mathematics and know that fact about orbits and stabilizers, because you know you're talking about one of the symmetric groups Sn. So she might mention you have to include the Sn-equivariance "over orbits".
You really are saying that the number which is given by orbit sums, is "one number" it's a categorification of N0. which is the 0-category of natural number objects.
Whew
Post by: Origin on 06/03/2024 22:31:50
That's probably all I know about the Aharanov-Bohm effect, I hope it's of some use to you.Outstanding explanation, thank you!
Post by: paul cotter on 07/03/2024 17:43:16
Post by: varsigma on 08/03/2024 08:40:44
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
Post by: Eternal Student on 08/03/2024 17:28:57
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.
Post by: varsigma on 08/03/2024 17:35:47
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.
Post by: varsigma on 10/03/2024 18:13:17
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.
Post by: varsigma on 10/03/2024 19:00:13
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.
Post by: varsigma on 11/03/2024 12:57:05
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.
Post by: Zer0 on 11/03/2024 17:17:04
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.
Post by: paul cotter on 11/03/2024 22:11:16
Post by: varsigma on 11/03/2024 22:17:25
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.
Post by: varsigma on 11/03/2024 23:28:19
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).
Post by: varsigma on 12/03/2024 22:56:11
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.
Post by: varsigma on 12/03/2024 23:28:46
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.
Post by: varsigma on 14/03/2024 21:26:28
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.
Post by: Eternal Student on 15/03/2024 01:33:11
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.
Post by: varsigma on 17/03/2024 03:43:53
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;
Post by: varsigma on 19/03/2024 09:18:34
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.
Post by: varsigma on 19/03/2024 17:37:28
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.
Post by: varsigma on 20/03/2024 21:58:06
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.
Post by: varsigma on 27/03/2024 23:07:53
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.
Post by: varsigma on 01/04/2024 01:39:39
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 180o 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.
Post by: Zer0 on 03/04/2024 17:47:59
Post by: varsigma on 10/04/2024 21:41:35
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.
Post by: varsigma on 17/04/2024 01:35:22
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.
Post by: varsigma on 20/04/2024 10:33:33
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.
Post by: varsigma on 22/04/2024 00:41:29
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 Bn lattice is a subset of the total lattice.
Post by: varsigma on 23/04/2024 23:19:22
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.
Post by: varsigma on 30/04/2024 22:42:11
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.