Visualizing the Hopf Fibration

Mathematica-generated images provide a novel visualization of an elusive
mathematical object. 
by Rick Kreminski

Mathematica-generated images provide a novel visualization of an elusive mathematical object. by Rick Kreminski 
The Hopf fibration is a versatile mathematical veteran, arising in numerous and seemingly unrelated situations. For instance, the Hopf fibration surfaces in particle physics: it underlies the mathematics of the Dirac monopole (see [[Ryder 1996]]). It also appears in general relativity, for instance in the Robinson congruence (see [[PenRin 1984]]). And of course it is found in several pure math contexts, such as in algebraic topology: roughly, there are infinitely many (topologically inequivalent) ways to map the three-dimensional sphere onto the two-dimensional sphere. More precisely, in the language of homotopy groups, (Subscript[π, 3])((("S")^2))="Z" (see almost any text in algebraic topology, for instance [Whitehead 1978]). 
2.5pt2.5pt0pt Even with all its applications, the Hopf fibration is a mathematical Kilroy: it's been all over the place, but it's difficult to literally "see". The Hopf fibration usually arises algebraically, rather than visually. The typical formulation involves (("S")^3), the unit sphere in (("C")^2) (or (("R")^4)), and this in turn is usually visualized by taking ordinary space, (("R")^3), and artificially adding a point out at infinity. This is the approach in [[Berger 1987 chapters 4 and 18]]. While this method of visualization is valuable and interesting and does lead to an actual picture of the Hopf fibration, it is not the approach of this paper. Instead, we present what we think is a new way, and more importantly an elementary way, to honestly visualize this elusive character, emphasizing that we do this without having to venture into (("C")^2), (("R")^4) or adding infinity to (("R")^3). In fact, we only use some linear algebra, and a "basketball" rendered with Mathematica 3.0. Code for all the illustrations is contained in HopfNoteBook.nb. 

 
FIGURE 1a:Trajectories in solid ball model of (("RP")^3), with dots showing where trajectories strike the plane y = 0. The trajectories are in fact "left cosets." 

 

FIGURE 1b:Top view of trajectories (or "left cosets"), with dots showing where trajectories strike the plane y = 0. 
We postpone discussion of the Hopf fibration until after we examine a close relative. Indeed, the object in Figure 1 is our main interest in this article; the Hopf fibration will arise almost as an afterthought. We therefore begin by explaining what the cut-away ball in Figure 1 really is, and then explain what those strings or trajectories inside it represent.

Solid ball model of the set of all rotations in (("R")^("3"))

(("R")^("3"))

π

π

p

-p

π

(("R")^n)

(("RP")^n)

(("RP")^3)

(("RP")^3)

(("RP")^3)

(("R")^3)

l

θ

π

l

θ

(("RP")^3)

θ

l

θ

(0, 0, π /2)

(("RP")^3)

(("R")^3)

z

π /2

x

y

y

x

(0, 0, -π /2)

x

y

y

x

0

r

(("RP")^3)

R

(("R")^3)

π

(("R")^3)

(("RP")^3)

Here is where the basketball mentioned in the introduction truly proves useful. Figure 2 depicts our virtual basketball. This ball has its center at the origin in (("R")^3); its radius is irrelevant. Imagine holding the ball with some pre-determined point on it pointing upwards, for instance the pinhole where the ball is inflated. For the rest of the article "pinhole" refers to this point that is originally on top of the ball. 


FIGURE 2: The plane perpendicular to the line segment whose endpoints are the topmost point (or "pinhole" of the "basketball") and a fixed point q. 
Let q be any other point on the surface of the basketball. Now imagine a rotation of (("R")^3) that rotates the basketball in such a way that the pinhole has moved to q's former location. Remember that rotations must keep the center of the ball fixed. The reader's duty is to try and find all such rotations that send the pinhole to q's former location. In demonstrations, people first come up with the rotation that seems the "quickest"; one simply rotates the pinhole down along the line of longitude connecting it to q. In other words, rotate the ball so that the pinhole stays in the plane that contains the origin as well as the original positions of the pinhole and q. This rotation will be referred to below as the "obvious" rotation. Trouble sometimes arises when it is pointed out that there are infinitely many other rotations that take the pinhole to q's former location. (The next rotation typically found is the one where one rotates by π about a point halfway down the line of longitude from the pinhole to q.) The reader equipped with a basketball is urged to find all rotations that take the pinhole to q, and characterize them, before reading on. 
A glance back at Figure 2 reveals the answer: all rotations of (("R")^3) that move the pinhole to q's former location have the following form. They have axis of rotation that lies in the plane τ that is the perpendicular bisector of the chord that connects the pinhole to q. [In fact, it is easy to physically see how much of a rotation angle is
needed for any axis of rotation l that lies in τ, if one is actually holding a basketball: simply grip the basketball tightly at the two points where l pierces the basketball, and swivel about l until indeed the pinhole reaches q's former location.] 
We can now state how the trajectories in Figure 1 come about: Fix a point q on the basketball, and consider the collection of all rotations of (("R")^3) which rotate the pinhole to q's former location. Plot these rotations in the solid ball model of (("RP")^3). The curve obtained is one of the trajectories. Moreover, all trajectories in Figure 1 are obtained this way. 
Since for a given q the relevant rotations all have axes that lie in a plane (called τ above), one sees that any one trajectory in Figure 1 should lie in a plane, namely τ. Using q as depicted in Figure 2, the plot of the trajectory obtained from it is indicated in Figure 3. 

 


FIGURE 3: Trajectory lying in the plane from Figure 2. The trajectory lies in the solid ball model of (("RP")^3). The point on the trajectory farthest from the origin in (("RP")^3) has coordinates {1.20223546, 0.0, 2.902453152}. This point represents a rotation (by π about its axis), with matrix given by

q

1. Each trajectory in the solid ball model of (("RP")^3) (depicted in Figure 1) lies in a plane. (This was discussed in the previous section.)

2. The set of all rotations that leave the top point, or "pinhole", fixed are precisely the rotations about the z-axis; and these rotations correspond to the vertical trajectory along the z-axis in Figure 1.

3. Each trajectory is topologically a circle. (This is true because when a trajectory reaches the boundary of the solid ball, it has reached a point that gets identified with its antipodal point.)

π

q

q

π

q

(("S")^2)

7. The collection of all trajectories, if it could be plotted, would be symmetric about the z-axis. That is, if Figure 1 depicted all trajectories, it would be rotationally symmetric about the z-axis. (Note that there is no x-axis or y-axis symmetry.)

8. Each trajectory is symmetric about its midpoint, in the plane that it lies in. Its midpoint in fact is where the trajectory strikes the plane z = 0 (which again represents the "obvious" rotation).

9. Denote the angle between the "pinhole" and the endpoint of a given trajectory (i.e. the point where the trajectory strikes the boundary of the solid ball in Figure 1) by ϕ. Then in polar coordinates, the equation of the trajectory in its plane is given by \(TraditionalForm
      \`r \((θ)\) = -\(cos\^\(-1\)\) 
        \((\(\(sin\^2\) θ - \(tan\^2\) ϕ\)\/\(\(sin\^2\) 
              θ + \(tan\^2\) ϕ\))\)\). We leave the direct (but somewhat tedious) computation as an exercise in linear algebra.

The reader is challenged to understand why the above statements hold true; detailed proofs can be obtained from the author, including proofs that use only linear algebra. 


[FIGURE 4a:]Two randomly chosen rotations, R and S, depicted as two randomly chosen points in the solid ball model of (("RP")^3). (Rotation R is a rotation with axis parallel to {-1,-2,5} and with amount of rotation π  / 3; rotation S is a rotation with axis parallel to {2,3,5}, with amount of rotation π  / 2. For ease of visualization, R and S are plotted along with vector from the origin in (("RP")^3), and color coded in same way as trajectories in (("RP")^3). Viewpoint is same as in Figures 2 and 3.) 
 
FIGURE 4b: The point q resulting from the rotation SR is coincidentally close to the point q of Figures 2 and 3; for ease in direct comparison, we use the same viewpoint here as in those figures. The trajectory, technically located in the solid ball model of (("RP")^3) and denoted α in the text, must contain the point representing the rotation SR.

Figure 1 as a multiplication table; or how to compose two rotations

R

S

R

S

R

S

S  R

α

α

α

(("RP")^3)

(("RP")^3)

R

S

(("RP")^3)

R

S

(("R")^3)

R

S

(q^′)

(q^′) = S(R("pinhole"))

S  R

S  R

α

(q^′)

α

S  R

α

S  R

n

R

m

S  R

n

m = S(R(n))

R(n) = n

S

n

n

m

α

α

(("RP")^3)

S  R

(("RP")^3)

(("RP")^3)

r

R

(R^(-1))

-r

r

Mathematical comments (including How to see a left coset, and The Hopf fibration)

This section assumes some familiarity with the notions of group, coset and topological homeomorphism. Let (("S")^n) denote the n-dimensional sphere, i.e. the set of all points in (("R")^(n + 1)) that are exactly one unit from the origin. Let S  O (n) denote the set of n by n orthogonal matrices of determinant 1, which is equivalently the matrices in (("R")^n) that correspond to rotations. S  O (3) is then the set of all rotations of (("R")^3) we have been examining (It is a fact, not immediately obvious (but surprisingly easy to show), that every element in S  O (3) does indeed represent a counterclockwise rotation about some axis L by some amount θ. This is often taught in a linear algebra course, and we leave it as an exercise for the reader.). We have discussed above how S  O(3)≃(("RP")^3), where  ≃  denotes topological homeomorphism. Rotations about the z-axis are given by block diagonal 3×3 matrices, with a `1' in the lower right entry and a 2×2 block consisting of an element of S  O (2) in the upper left. Abusing notation, we denote the rotations about the z-axis by S  O (2). These are precisely the rotations that preserve the point (0, 0, 1) (the pinhole). S  O (2) is known as the isotropy group of (0, 0, 1). Note that a (left) coset of S  O (2) in S  O (3) is exactly the set of all rotation matrices that send the point (0, 0, 1) to a common point. That is, all elements of a coset [g  S  O (2)] map (0, 0, 1) to the same point, call it q; and no other elements of S  O (3) send (0, 0, 1) to q. (For a more pedestrian way to consider this, due to the block-diagonal form of the group we call S  O (2), all elements of a left coset of S  O (2) must have the same third column; but this is precisely the point that (0, 0, 1) gets sent to by the action of a 3×3 matrix.) This implies that each left coset of S  O (2) in S  O (3) directly corresponds to one of the trajectories, and vice versa. 
Since we saw that the collection of all trajectories could be identified with (("S")^2), we realize that part of our discussion above yields an elementary way of understanding that S  O(3)/S  O(2)≃(("S")^2). 
We now briefly discuss some key properties of the concepts fibration and fibre bundle, skipping the formal (and unfortunately technical) definitions. Fibrations are useful in algebraic topology, for instance in considering homotopic maps between spaces (and hence arising in computations of higher homotopy groups, like (Subscript[π, 3])((("S")^2)).) Fibre bundles give rise to fibrations, and our discussion of S  O (3) and (("RP")^3) leads to two nontrivial fiber bundles. Let X, Y and Z be connected topological spaces. One consequence of a fibre bundle (and hence fibration) X -> Y -> Z is that space Y is composed of lots of pieces, each space looking like (or homeomorphic to) space X. Space Z is essentially a bookkeeping device for how many copies of space X to use: a different version of space X is used for each point in space Z. This process, of chopping Y into a Z's worth of disjoint copies of X, must be done "continuously". Earlier discussions imply that we have the fibre bundle (and hence fibration)

and this fibration is exactly what Figure 1 illustrates. In words, (("RP")^3) is comprised of disjoint pieces, each of which is an (("S")^1) or circle, and there are an (("S")^2)'s worth of these circles. This fibration is also exactly double covered by the fibration

(("S")^1)->(("S")^3)->(("S")^2),

which is the Hopf fibration. It is this nontrivial cutting up of (("S")^3) into a disjoint collection of (("S")^1)'s that arises in the situations cited in the introduction. 
To "see" the Hopf fibration, take two copies of Figure 1 and place them side by side; but slightly change the identification rules so that when trajectories reach the boundary of one solid ball they actually appear at the corresponding point in the other ball (instead of appearing at the antipodal point of the same ball). This is our (possibly never-before-actually-seen) visualization of the Hopf fibration. As claimed in the introduction, this rendition does not require visualization of (("R")^4) or adding a point at infinity to (("R")^3). Rather, (("S")^3) here is visualized as the result obtained when two solid balls are glued together on their boundary. (How gluing two solid balls together on their boundary truly yields (("S")^3) is most readily understood by analogy. If one takes two solid 2-dimensional "balls", which would really be two ordinary disks, and glues them together on their boundary, one gets an ordinary 2-dimensional sphere, at least topologically speaking. For instance, think of one of the solid 2-dimensional "balls" or disks as slightly stretched so as to be the lower hemisphere, and the other ball or disk as the upper hemisphere. The gluing occurs along their boundaries, and forms the equator of the sphere.) In this way, for instance, we can directly "see" the generator of the group (Subscript[π, 3])((("S")^2)).

Right cosets, double cosets, and nested tori

Let H denote the subgroup of rotations fixing the z-axis, i.e. the subgroup we have been denoting S  O (2). 
 
FIGURE 5a: A single right coset (thick black line), along with several of the left cosets it intersects. 

 

 FIGURE 5b:  A single right coset (thick black line), along with several of the left cosets it intersects. 
Figure 5 illustrates one typical right coset of H in S  O (3), along with a fairly random sample of the left cosets it intersects, in the (("RP")^3) model of S  O (3). One can show that the obvious observed symmetry is real, namely that the trajectory corresponding to a right coset [H  g] will: (a) lie in a plane; (b) only intersect left cosets' trajectories which are members of a certain "family", i.e. only intersect left cosets whose corresponding trajectories are all rotated versions of one another; and (c) be the mirror image of the left coset trajectory that lies in its plane. 
From a more purely algebraic viewpoint, (c) states: the elements of S  O (3) comprising the right coset are point-by-point the group-theoretic inverses of the elements of the left coset whose trajectory shares the plane of the right coset's trajectory. [Hint: Recall that group theoretic inversion corresponds to geometric inversion in the origin, and note that (((H  g))^(-1)) = (g^(-1)) H.] This in turn illustrates a situation arising in introductory abstract algebra: the natural bijection between the collection of left cosets of a subgroup H of a group G and the right cosets is given by \(TraditionalForm
      \`\([g  H]\) ↦ \([\((H  g)\)\^\(-1\)]\)\). Here, the geometric interpretation is simply: map a left coset to its inversion in the origin. 
Note in passing that the only right cosets that are also left cosets are (a) the coset of the identity element of the group and (b) the coset of the element representing rotation about any axis in the x - y plane. Geometrically, in Figure 1 the trajectory along the z-axis is both a left coset and a right coset, as is the equator. 
Lastly, consider the double cosets \(TraditionalForm
      \`\(S  O\) \((2)\)\\(S  O\) \((3)\)/\(S  O\) \((2)\)\), i.e. \(TraditionalForm\`H\\(S  O\) \((3)\)/H\). The reader might think that this would be difficult to "see"! However, one double coset is essentially depicted in Figure 5. A moment's thought indicates that H  g  H must be precisely the result of taking one trajectory and rotating about the z-axis. [Hint: h  g  H rotates the pinhole to wherever g did, followed by a rotation about the z-axis; the end result is a point on the same latitude line as g's end result. The trajectory associated with this final resting place for the pinhole is the rotation by h of the trajectory associated to g.] Recalling that antipodal points on the boundary of the ball in Figure 5 are identified, we see that such double cosets are topologically torii - with two exceptions: (a) the vertical trajectory is a double coset and (b) the equator is a double coset. These form "degenerate" torii, namely circles. Note that the torii are nested. Compare with the figure of the Hopf fibration given in chapter 18.9 of [Berger 1987] (who cites work of Penrose from the 1960's).
References 
Rerger M. Geometry, Springer-Verlag, New York, 1987. 
Penrose R. and Rindler  W. Spinors and Spacetime, Cambridge University Press, Cambridge, 1984. 
Ryder L. Quantum Field Theory, second edition, Cambridge University Press, Cambridge, 1996. 
Whitehead G. Elements of Homotopy Theory, Springer-Verlag, New York, NY, 1978. 
About the author 
Rick Kreminski is a mathematical physicist at Texas A & M University-Commerce. He is interested in interdisciplinary applied mathematics, in the general (and literal) sense of applying results from some branch of pure mathematics to physics, biology, chemistry, or economics, or even to other branches of mathematics. 
Rick Kreminski
Department of Mathematics,
Texas A & M University-Commerce,
Commerce, TX 75429