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)

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)

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)