[varsigma (OP)]

I heard someone say something thought-provoking on a Youtube vid about the Hopf fibration.

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.

?

[varsigma (OP)]

This mathematical object, a fibration of the 3-sphere (embedded in four dimensions), can be visualized, or parts of it can, through modern graphics animations, and in particular the use of coloring.

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.

[alancalverd]

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?

[varsigma (OP)]

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.

[varsigma (OP)]

The onions.

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.

[varsigma (OP)]

Ok, well.

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

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.

[varsigma (OP)]

The quaternions might or not might be universal.

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.

[evan_au]

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.

[varsigma (OP)]

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?

[varsigma (OP)]

I hate when you edit something and don't notice an error.

    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.

[varsigma (OP)]

I can expound on the first line in that list.

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.

[Eternal Student]

Hi.

    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.

[varsigma (OP)]

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.

[varsigma (OP)]

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').

[varsigma (OP)]

Be advised:

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.

[Eternal Student]

Hi.

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  {G_i} such that the graph G can be described as the union 
          G  =  U_i G_i.
    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 G_i.   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.

[varsigma (OP)]

Ok. Graphs. Ya, I did some graph theory, we did minors, planar and toric embeddings, yada yada.

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

[varsigma (OP)]

The surgery involves a bit of thinking however.

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.


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 . . .

[varsigma (OP)]

If any of this is making you cry, that might be the onions.

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.

[varsigma (OP)]

Where does the Hopf gauge come into all this?

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.