Talk:Hopf fibration

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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)

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)

natural metric?

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)

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)

Dumbing down

While I sympathize with the aim of beginning articles with accessible language, the claim that "the Hopf bundle (or Hopf fibration) … is a partition of a 3-dimensional hypersphere into circles" misrepresents the essential mathematics.

Yes, a fiber bundle has fibers, but the topological relationship between the base space and the total space through the projection map is what makes it important. In particular, if we look at the inverse image of a neighborhood in the base, that portion of the bundle looks like a product of the neighborhood and the fiber space. This "local product space" structure is what allows us to do, say, path lifting.

Better pictures of the Hopf bundle suggest this topology by showing nested tori, not just circles. Some of the earliest computer graphics instances are the work of Thomas Banchoff, whose "flat torus" is the inverse image of a circle of S². And he shows circle geometry, not just topology, because the image uses stereographic projection from S³ to R³. A visualization of the entire bundle, not just one torus, can be found at the Hopf Topology Archive. Follow the link from the main page to see an image using colors on both S³ and S², and other strategems, to reveal structure. (It is also found in the SIGGRAPH 94 Art and Design Slide Set, and in Graphics Gems IV.) In this one the circles are only topological, but are confined to a finite ball.

Image showing fibers and corresponding points on the two-sphere

The (pre-existing) keyrings "model" in the picture leading the article is as unhelpful as the "partition" prose, though it has other appeal. (The better images I mentioned cannot be used because of copyright.) Also, I'm afraid the "One topological model" sentence is a move in the wrong direction, especially for the lay reader, for whom it will be gibberish.

So, care to try again? --KSmrq^T 22:20, 8 October 2007 (UTC)

I largely agree, but haven't fixed this yet. Meanwhile, KSmrq, I know you are a whizz with SVGs. Do you think you could produce one? Homotopy groups of spheres also really needs a better lead image. Geometry guy 22:34, 8 October 2007 (UTC)

I have done some work on images of the Hopf fibration, inspired partly by those mentioned above. I've attached one to this page which is very similar. Does anyone object to replacing the 'keyring model' with this one? Nilesj (talk) 19:12, 31 October 2012 (UTC)

I'm happy to have my prose dissected and improved. I'd like to defend the "one topological model" part, though: the point was to describe the bundle in terms lay readers might hope to be able to visualize (circles in the one-point completion of 3d) rather than leaving them with the impression that as an object involving abstract 3-manifolds it is unvisualizable. Similarly, while the local product structure is essential to the mathematical content, I don't think it's essential to a lead that gives lay readers some idea of what this is about. —David Eppstein 23:34, 8 October 2007 (UTC)

Now it's your turn to critique or improve! I made a number of revisions to the intro, with mixed results. The first paragraph says more and says it better, I hope. The "one topological model" portion really didn't work for me there, so I moved it and rewrote it. I'm not thrilled with diving into notation and technicalities in the second paragraph. It happened because my imagination failed me: how do we describe the local product structure colloquially? A product space is an easy idea, but not one the lay reader knows. And a local product? Argh. I could do it in a paragraph, but not in a sentence. So I moved up some material that was already there, and improvised. The implications need expanding, but by then I was tired of losing the wrestling match. While I was at it, I switched the references to {{citation}} form so I could get the automatic links from {{harv}}, and expanded them a little.

As for an image: While tinkering with Villarceau circles several months ago I began playing with some 3D renderings, just to show the nested torus idea of stereographic projection, using transparency. If I'm doing the Hopf fibration, I want S² in the picture as well. No way would I tackle this in SVG! PostScript maybe (see Casselman); but even that would be quite the challenge. With modern graphics cards, the really cool approach would be interactive 3D graphics, but Wikipedia doesn't support that. (The chemists must really chafe.) --KSmrq^T 12:47, 10 October 2007 (UTC)

naming

Why is this article at "Hopf bundle" instead of "Hopf fibration" anyway? --Horoball 17:39, 9 October 2007 (UTC)

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)

Figure data

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

${\begin{aligned}r&={\frac {\sqrt {2(1-\cos \theta )}}{\sin \theta }},\\{\begin{bmatrix}x\\y\\z\end{bmatrix}}&={\begin{bmatrix}0\\-\tan {\tfrac {\theta }{2}}\\0\end{bmatrix}},\\0&=x(1-\cos \theta )+z\sin \theta .\end{aligned}}$

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. --KSmrq^T 10:24, 11 October 2007 (UTC)

Hopf or Clifford?

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)

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)

Hamilton fibration

In 1853 W. R. Hamilton published Lectures on Quaternions, a topic he had developed over the period of a decade in Dublin. As an astronomer, he required spherical trigonometry to do calculations on the celestial sphere. A quaternion q = a + bi + cj + dk has squared norm $a^{2}+b^{2}+c^{2}+d^{2}$. Hamilton called the tensor of q the square root of this quantity. Dividing q by its tensor gives Uq, the versor of q. The versors populate the 3-sphere S³ in quaternions. Hamilton called a the scalar of q, Sq, and bi +cj + dk the vector of q, Vq, so q = Sq + Vq. Thus, as Hamilton was coining the term vector, the original vector space was VH = {bi + cj + dk : b, c, d in R }. The intersection of VH with S³ is a 2-sphere called the right versors. The right versors are imaginary units in Hamilton's algebra. If s is a right versor then Euler's formula holds with s inserted for i : $\exp(\theta s)=\cos \theta +s\sin \theta .$ Thus θ parametrizes a circle bundle over the base space of right versors, for total space S³. Here the "lines" are directed great circle arcs passing through 1 in H.

Rather than "Invented by Hopf in 1931", the article should read "In 1931 Hopf revived the Hamilton fibration of S³ into versors". The article section "Geometric interpretation using quaternions" refers to versors without crediting Hamilton, who wrote 78 years before Hopf.

According to Norman Steenrod (1951) Topology of Fiber Bundles, there are "various fibrings of spheres discovered by Hopf" (page 105) where Steenrod cites Hopf's 1935 paper that includes fibration of S⁷. Thus, the article is incorrect claiming the S³ fibration of 1931 is "Hopf’s invention". Steenrod also mentions quaternions on page 37 where the bundle is identified. The oldest citation in his bibliography (113 references) is Linear Algebras (1914) by Leonard Dickson (no versors there). At the time Steenrod wrote, Hopf's first bundle article was just twenty years old, and Hamilton's Lectures 98 years old. Whether through ignorance or neglect, the versor fibers are a historic fact that clarifies the content of this article. A source is needed to patch these ideas together, perhaps Lyons (2003) Mathematics Magazine, which has been inaccessible so far. — Rgdboer (talk) 21:57, 7 September 2024 (UTC)

Fluid mechanics

Example from the fluid mechanics looks strange: from equations

$p(x,y,z)=-A^{2}B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-3},\rho (x,y,z)=3B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-1}$

it follows that

${\frac {\rho ^{3}}{p}}={\frac {3^{3}B^{3}}{-A^{2}B}}=-27B^{2}/A^{2}$

id est,

$p={\frac {-A^{2}}{27B^{2}}}~\rho ^{3}$

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)

Discrete examples

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)

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)

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)

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)

the Boerdijk-Coxeter helix

“ The 600-cell also partitions into 20 rings of 30 tetrahedra each in a very interesting, quasi-regular chain called the Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes regular. ”

What do "regular" and "quasi-regular" mean in this context? —Tamfang (talk) 19:18, 12 October 2012 (UTC)

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)

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)

Some things need improvement

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 S^2k-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)

introductory sentence

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)

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)

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)

Stereographic projection

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)

Polytwisters?

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)

So much abstraction....

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) (I thought I signed that) D3x0r (talk) 21:07, 2 April 2021 (UTC)

---

Really I'd love to know more about this whole thing; I obviously don't have perfect understandings; but this is certainly the 'natural metric' that someone above was seeking so very long ago; but then the answer is something about a map and a space S3, which is never really 'shown' just the idea of the space... since all these nested toroidal structures sort of look like a 'metric-able' thing?

I did play with converting the normal to a 2 angle definition, and projecting that, but it's hard to recover the original 'rotation space' covered so it all collapses to a small rectangular patch (at least for the latitude-longitude sphere I was mapping) which then makes the initial state just a flat XY grid; only that only works in the original case, and that grid warps, and gets a sin curve in it when adjusted with additional 'spin' around the normal... the normals were built such that for some point on the sphere 'up' or perpendicular to the surface of the sphere is a 'normal' and 'forward' is aligned with latitude lines and 'right' is aligned along longitude lines; from that state, the same normal, has a rotation basis that is the 'spin' around that normal... but the normal itself can be represented with 2 coordinates... however, mapping back to those angles to adjust doesn't seem practical yet - the Rodrigeus Composite rotation formula is quite complex, and substituting the sin/cos of the angles to define the axis in place of the normal X/Y/Z rotation axis didn't help the situation.

D3x0r (talk) 21:44, 2 April 2021 (UTC)

Complex projective line section

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)