Talk:Hopf fibration

message: I've been looking for an explanation of why the circles of the Hopf fibration become linked.  This is a request for someone more knowledgeable to fill in this missing information - Gauge 17:51, 2 Apr 2005 (UTC)
replies: text: On a 2-sphere (globe), if you go off in any direction and keep going straight you eventually arrive back at your starting point. Same with the 3-sphere, except you are no longer restricted to a plane, you can go off in any 3-dimensional direction.  For the 2-D case all great circles intersect. You can avoid this for the 3-D case. Step away from your initial starting point and go off in a new direction.  You want to pick this direction so that you don't intersect the previous geodesic.  To this end you have to give your new direction a little "skew" so that your new starting direction is not exactly parallel, and out of the plane, to your old direction. This avoidance of intersection causes the two loops/geodesics to spiral around each other and/or interlock.  For the Hopf fibration the farther away you are from the initial starting point, the more "skew" you add.  When you are 90 degrees away, you add 90 degrees of skew.  This is the most extreme case and you have two interlocking rings passing through the midpoint of the other ring. With this construction you can parameterize the whole 3-sphere, with no two rings ever touching each other. Cloudswrest (talk) 19:58, 31 October 2012 (UTC)
thread_title: Untitled

message: This is a somewhat flaky question, but ...  I'm wondering if there's a "natural" metric associated with a Hopf fibration. The "natural" metric on CP^n is the Fubini-Study metric, which is identical to the ordinary metric on the two-sphere for CP^1.  I can certainly pullback the metric on S^2 to define a metric on S^3, but I'm wondering how "natural" this really is, if it has any interesting non-intuitive or enligtening properties.  
For example, if I envision S^3 as the EUcliden space R^3 that we live in, with an extra point at infinity, then the Hopf fibration fills this space with non-intersection circles (as illstrated by the "keyring fibration" photo). Each circle has a center ... what is the density of the distribution of the centers of these circles in R^3, (assuming a uniform density on S^2)?  Are the centers of these circles always confined to a plane? What is the distribution on the plane? Uniform? Gaussian? Each circle defines a direction (the normal to the plane containing the circle). What is the distribution of these directions? linas 16:23, 26 June 2006 (UTC)
replies: text: When I think of a metric on S^3 using the Hopf fibration, I think of the Berger spheres.  That article needs clean-up, by the way.  --Horoball 22:12, 7 October 2007 (UTC)
thread_title:  natural metric? 

message: Why is this article at "Hopf bundle" instead of "Hopf fibration" anyway?  --Horoball 17:39, 9 October 2007 (UTC)
replies: text: I have been asking myself the same question since I saw it, so I've now moved the page. Geometry guy 17:45, 9 October 2007 (UTC)
thread_title:  naming 

message: I found some notes I made while studying how to fill space with nested toruses made of Villarceau circles. For the benefit of others who may wish to experiment:
; Latitude and longitude
: On S2, take θ∈[0,π] as latitude and φ∈[0,2π] as longitude on the unit sphere. Then
::\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi \\ \sin \theta \sin \phi \\ -\cos \theta \end{bmatrix} . 
; Projected parametric circle
: When S3 is stereographically projected to R3 from (0,0,0,−1), the parametric circle (parameter t) for the fiber of (θ,φ) is
::\begin{bmatrix}x\\y\\z\end{bmatrix} =
 \begin{bmatrix}
  \sin \theta \sin (t-\phi) \\
  \sin \theta \cos (t-\phi ) \\
  (1 - \cos \theta) \sin t
\end{bmatrix} \frac{1}{(1 - \cos \theta) \cos t + \sqrt{2(1 - \cos \theta)}} . 
; Circle data
: The projected circle for the fiber of (θ,0) has radius, center, and plane equation
::\begin{align}
 r &= \frac{\sqrt{2(1 - \cos \theta)}}{\sin \theta} , \\
 \begin{bmatrix}x\\y\\z\end{bmatrix} &= \begin{bmatrix} 0 \\ -\tan \tfrac{\theta}{2} \\ 0 \end{bmatrix} , \\
 0 &= x (1 - \cos \theta) + z \sin \theta . 
\end{align}
Since φ merely rotates the circles around the z axis, the general center and plane are easily obtained. In the limiting cases for θ, the torus degenerates to the unit circle in the xy plane when θ = 0 and to the z axis when θ = π. --KSmrqT 10:24, 11 October 2007 (UTC)
thread_title:  Figure data 

message: The sentence "discovered by Hopf in 1931" is what's in question.
In "Such silver currents,..." the biography of W.K. Clifford, it says Clifford discovered the "Hopf" fibration, and that Hopf was more scrupulous than anyone in giving credit to Clifford. Rather than trying to dig up 100-year-old references, does anyone here know more details of this? Perhaps KSmrq, who might be referring not just to conceptual importance of the locally trivial aspect, but also to some historical importance as well. Is the fibration nature of this example due to Hopf?  —Preceding unsigned comment added by 137.146.194.173 (talk) 19:27, 12 June 2009 (UTC)
replies: text: The fibration of the 3-sphere is expressed by the exponential map applied to the vector subspace of quaternions. As such, it was known to William Rowan Hamilton and is expressed in § 548 of his Lectures on Quaternions, Royal Irish Academy, 1853 (see page 555). He writes, "the logarithm of a versor of a quaternion ... is the product of axis and angle." The axis corresponds to a point on the 2-sphere, and the angle may unwind as a fiber.Rgdboer (talk) 21:29, 10 November 2013 (UTC)
thread_title: Hopf or Clifford?

message: Example from the fluid mechanics looks strange: from equations
:
p(x,y,z) = -A^2B \left(a^2+x^2+y^2+z^2\right)^{-3}  ,  \rho(x,y,z) = 3B\left(a^2+x^2+y^2+z^2\right)^{-1}
it follows that 
: \frac{\rho^3}{p}=\frac{3^3 B^3}{-A^2 B}=-27 B^2/A^2
id est,
:p=\frac{-A^2}{27 B^2} ~ \rho^3
To me, such a relation looks strange, is this valid for some realistic fluid? 
Shouldn't  $A$ and $B$ be real constant? 
Shouldn't $p$ and $\rho$ be positive?
Shouldn't pressure $p$ increase with increase of density $\rho$?   dima (talk) 04:07, 2 September 2012 (UTC)
thread_title:  Fluid mechanics 

message: The recent addition of the 600-cell, "The 600-cell partitions into 20 rings of 30 tetrahedra each." needs some clarification.  The tetrahedron does not have opposing parallel faces, so there is no way you can stack these "end-to-end" in a great circle.  It necessarily has geodesic curvature. At best you can make a closed chain that has some helicity, whos axis would be a great circle. Still very interesting. Cloudswrest (talk) 23:24, 2 October 2012 (UTC)
replies: text: I see this path and while is appears technically true that the 600-cell partitions into 30-tetrahedra rings, these rings are qualitatively different from the rings in the other three polytopes. In the other three the rings are regular. Each cell is equivalent and centered on the great circle. In the 600-cell partition the 30 tetrahedra in a ring are grouped into 10 sets of 3 tetrahedra, spiralling around, and tangent to, the 10 vertex great circle (said great cirle is the dual of the 10-dodecahedron, 120-cell great circles). The 30 tetrahedra in a ring are not all eqivalent to each other. Cloudswrest (talk) 20:23, 11 October 2012 (UTC)

text: This section was deleted for lacking references. I had been adding some graphics. I copied the section to User:Tomruen/Regular polychoric rings. Tom Ruen (talk) 04:01, 15 November 2014 (UTC)

text: I started the "Discrete Examples" section because of the 120-cell.  I consider it a perfect example of the Hopf fibration in a different context. A "physical" example.  People might not "get it" when given equations, or theory, or even a continuum picture, but seeing the 120-cell example might provide an "aha moment"!  You can SEE it in the Todesco Youtube video of the 120-cell. The other face-to-face cases quickly became obvious.  Then somebody mentioned, without any references, the BC helices in the 600-cell, and all the tet based polytope fibrations fell into place also.  For the most part math articles have been a pretty safe subject to edit as it's objective, self documenting, except for very esoteric stuff, and people who don't know anything about it are usually uninterested, unlike articles on subjects like say, Martin Luther King, or Hitler, or date rape, where all the social justice warriors and polemicists come out to play. When Eppstein first complained about references over a year ago I did a web search.   There are bits and pieces on various web sites and blogs, including some on John Baez's blog, but I could not find any coherent full coverage of the subject, which in any case is pretty obvious to interested parties.  But I do know that making snide and sarcastic remarks on the rather competent and prolific work of a Wikipedia math illustrator is over the top. Cloudswrest (talk) 02:00, 17 November 2014 (UTC)
thread_title:  Discrete examples 

message: What do "regular" and "quasi-regular" mean in this context? —Tamfang (talk) 19:18, 12 October 2012 (UTC)
replies: text: I was using "regular" as in the sense of a "regular polygon", since the other mentioned polytopes do form regular polygonal chains.  The tetrahedrons form helical chains so the chain of cords thru their centers cannot be a regular polygon.  Although the chain chords through the center axis of the helix looks to be a regular 30-gon.  So perhaps it can be better phrased. Cloudswrest (talk) 19:39, 12 October 2012 (UTC)
replies: text:  Let me back up,  I remember now what I was talking about.  In the Euclidean case the per cell helical pitch is an irrational fraction of the circle.  In the 600-cell case the chain obviously closes in 30 cells and regularly repeats itself. Cloudswrest (talk) 19:50, 12 October 2012 (UTC)
thread_title:  the Boerdijk-Coxeter helix 

message: I think the article is excellent but that there is still room for improvement.
Two of the illustrations are, I think, misleading:  
a) One is the beautiful multicolor picture of the Hopf fibration S3 → S2 at the upper right, of a snail-shellish surface that's a union of Hopf fibres.  It is certainly true that one can find such a snail-shellish surface that is the union of Hopf fibres.  But a much simpler way of describing this Hopf fibration is used in the text: a union of tori, all having the same core circle, and each filled (foliated) with a family of circles of Villarceau (which are all congruent to the common core circle.
Also, the caption of this illustration reads in part: "The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball."  I don't know how this can be done as stated other than in a very distorted manner. 
Rather, after *removing* one of the fibres in S3 one can arrange that the image be R3 minus the z-axis, and *this* can be compressed to a maximal open solid torus of revolution that shows clearly the "concentric" tori and their circles of Villarceau.
b) The illustration of the circles of Villarceau is certainly accurate. But it is also extremely misleading in the context of the Hopf fibration S3 → S2, since it shows the two circles intersecting!  Of course, this never happens in a fibre bundle.  This illustration is perfectly appropriate for the article on circles of Villarceau.  But here it is not. It would be much better if a picture showed clearly how a family of (disjoint) circles of Villarceau can fill up a torus of revolution.
c) In the text, neither the section titled "Geometry and applications" nor any other section seems to mention anything about the radius of the base space in a Hopf fibration. If the Hopf fibration has as total space a sphere S2k-1 (k = 1,2,3, or 4) of radius = 1, with the usual action of the unit (reals, complexes, quaternions, or octonions) or in other words, the usual way to define P1, CP1, HP1, or OP1, respectively, as the quotient of this action.
The radius of the base spheres in the Hopf fibrations can be seen to be = 1/2.  This ought to be stated.Daqu (talk) 21:31, 29 August 2013 (UTC)
thread_title:  Some things need improvement 

message: It says "... a 3-sphere (a hypersphere in four-dimensional space)..".
Is that non-generic description within brackets really necessary? -  Subh83 (talk | contribs) 07:01, 2 January 2014 (UTC)
replies: text: I think the n-sphere terminology can be confusing — it's not obvious whether the n refers to the dimension of the sphere itself (as it does) or of the space in which it is embedded (in the standard embedding). The parenthetical helps avoid confusion. On the other hand, it presupposes that the hypersphere has its standard embedding, which is unnecessary here (the fibration does not use an embedding).—David Eppstein (talk) 08:27, 2 January 2014 (UTC)

text: I agree with David, like a 3-polytope is a polyhedron in 3-space, so when I first saw 3-sphere, it confused me. Tom Ruen (talk) 20:26, 2 January 2014 (UTC)
thread_title:  introductory sentence 

message: Where can I find the image of a stereographic projection of S0-S1-S1? 67.243.159.27 (talk) 12:31, 24 February 2014 (UTC)
thread_title:  Stereographic projection 

message: Should there be a mention of Bowers' polytwisters (and polytwirlers/polywhirlers)? — Preceding unsigned comment added by 98.207.169.109 (talk) Sep 1, 2017
: I'm not aware of any sources we can reference. Tom Ruen (talk) 05:05, 17 September 2017 (UTC)
thread_title:  Polytwisters? 

message: I started working on a project(Github), which represents rotations with Euler Axis, Axis-Angle, or angle-angle-angle sort of representation.  All rotation methods such as matrices and quaternions have a common base that is angle-axis.
These early demos... https://d3x0r.github.io/STFRPhysics/3d/index4.html (hopf fibration generation, generating a single fiber with analog steps instead of a discrete number of separate rings)  https://d3x0r.github.io/STFRPhysics/3d/index.html  This really shows just a single loop, which is the rotation coordinates of a base rotation, rotated around the 'yaw' of that point.
The hopf fibration generator though is simply a few loops that takes 3 rotation axis and iterates the angles that are applied for each.  There's a outer loop of 0-2pi, and inner loops from -2pi to 2pi for a number of turns around...  The settings are - a number of turns - which breaks up the second loop into an integer fraction... and then applied with all the appropriate fractional steps.
I had initially done it in discrete loops, so there was a for count of turns, for 0-4pi around each turn, rotate Q0 by Q1 and rotate that result around the axis Q3.  I don't find any discontinuities.... so I'm skeptical of MUCH content of this whole article.  Generally the resulting coordinates generally form a toroid; mind you though it's plotted in 3D, there's nothing linear about this, it's all about rotations, and the evolution from one point in time to another by compositing three rotations.
This is basically the loop code... (A,B,C)*T = R0, R1 and R2 are other unit vector axii of rotations which are used.  'freeSpin' is application of the composite rotation Rodrigues' Rotation Formula, which rotates an axis and angle around another axis and angle.
  const lnQ0 = new lnQuat(  0, T*A/lATC, T*B/lATC, T*C/lATC ).update();
  for( let nTotal = 0; nTotal < steps; nTotal++ ) {
     fibre = nTotal * ( 4*Math.PI ) / ( steps );
     const fiberPart =((fibre + 1*Math.PI)/(Math.PI*2) %(1/subSteps));
     const t = (Math.PI*4)* subSteps*(fiberPart) - (Math.PI*2);
     
     const lnQ = new lnQuat( lnQ0 ) // copy lnQ0 before 'spinning'
                    	.freeSpin( fibre, {x:R1.x,y:R1.y,z:R1.z} )
                        .freeSpin( t, {x:R2.x, y:R2.y, z:R2.z} );
     // plot each point connected to each prior point
     // each point is scaled angle*(axis) where axis is a unit vector.
Edit: Initially I was doing it more like fibers to make individual rings, so rather than rotating the rotation, I was just iterating the axis, and then rotating that in the end; when doing it as a single path like above it's not actually correct to step the axis like that; however, for more of a Hopf Fibration like the youtube video with interlinked rings (non chaotic).  but then the result betrays that half of a cycle is interlaced with the other half.
But I don't really know anything about abstract topologies; and maybe what they think is discontinuous is actually apparently continuous to the layman?
21:06, 2 April 2021 (UTC)
thread_title:  So much abstraction.... 

message: The section here needs some serious work. It claims "The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space." - except that p is defined as a map from S^3 to S^2. I know that p descends along the quotient map S^3 -> CP^1, as it is invariant under multiplication on the domain by unit complex numbers, so in principle one could instead specify a map on CP^1 using homogeneous coordinates coming from S^3, and then claim that that is a diffeomorphism. Albeit it's not obvious this map is either injective or surjective, nor what its inverse is. 121.45.89.81 (talk) 23:33, 19 September 2021 (UTC)
thread_title:  Complex projective line section