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VISUALIZING the HOPF FIBRATION Rick Kreminski Besides a Novel VISUALIZING THE HOPF FIBRATION Rick Kreminski Besides a novel visual rendering of an elusive mathematical object, fundamental concepts in abstract algebra are depicted in an important example The Hopf fibration is a versatile mathematical veteran, arising in numerous and seemingly unrelated situations. Even if you have never heard of it, you may still have encountered it. 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. 2 More precisely, in the language of homotopy groups, ¼3(S ) = Z (see almost any text in algebraic topology, for instance [Whitehead 1978]). 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 S3, the unit sphere in C2 (or R4), and this in turn is usually visualized by taking ordinary space, R3, 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 C2, R4 or adding infinity to R3. In fact, we only use some linear algebra, and a basketball. 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 R3: The “ball” in Figure 1 really depicts something called a projective space, which is obtained as follows. Begin with a solid ball in R3 that is centered at the origin and which has radius ¼, i.e. all points in ordinary space that are at most ¼ units from the origin. Then for each point p on the ball’s boundary, identify it with ¡p, its antipodal (diametrically opposite) point. Do this identification procedure only for the points on the boundary, i.e. only for those points in the ball that are exactly ¼ units from the origin. The resulting space is precisely how we are to interpret the ball-like object in the figure. [This situation is somewhat analogous to that which arises in various video games: if a curve or trajectory strikes the boundary of the solid ball at some point, it reappears at the antipodal point.] The space obtained by taking a solid ball in Rn and identifying antipodal boundary points is called a (real) projective space and denoted RP n. Thus, the curves in Figure 1 represent curves in the space denoted RP 3. To understand precisely how the curves in RP 3 that appear in Figure 1 are obtained, we first have to un- derstand the connection between RP 3 and the 3-dimensional rotation group. Every (orientation preserving) rotation of R3 can be completely characterized by two pieces of information, namely: (1) an axis of rotation given by some ray, l, and (2) a real number θ between 0 and ¼ describing the amount of (counterclockwise) rotation about that axis. (All rotations in this article are considered to keep the origin fixed.) Then to each rotation (with given l and θ) we associate a point in RP 3 as follows: simply go out θ units from the origin, along the ray l, making sure that when peering back at the origin the rotation would be counterclockwise by θ radians1. The origin itself depicts the identity transformation (or, the trivial rotation by 0 about any 1As an example, the point (0; 0; ¼=2) in RP 3 corresponds to the rotation of R3 about the z-axis by ¼=2 in the counterclockwise sense when viewed from above. This would rotate the x-axis to the former location of the y-axis, and the y-axis to the former Typeset by AMS-TEX 1 2 RICK KREMINSKI axis). Any point r in the solid ball model of RP 3 from Figure 1 therefore corresponds to a unique rotation R, and vice versa. Note that antipodal boundary points indeed correspond to identical rotations of R3 (since both points correspond to rotations about the same axis, with angle of rotation ¼). In this way, we see how the set of rotations of R3 corresponds to the solid ball model of RP 3 depicted in Figure 1. The Figure 1 trajectories: 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 R3; 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. Let q be any other point on the surface of the basketball. Now imagine a rotation of R3 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 postions 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 ¼ 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 R3 that move the pinhole to q’s former location have the following form. They have axis of rotation that lies in the plane ¿ 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 ¿, 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 R3 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 ¿ above), one sees that any one trajectory in Figure 1 should lie in a plane, namely ¿ 2. Using q as depicted in Figure 2, the plot of the trajectory obtained from it is indicated in Figure 3. Symmetry in Figure 1: We have stated that each trajectory in Figure 1 represents a certain set of rotations of R3, namely: each trajectory represents all rotations which rotate the top point on a given sphere centered at the origin to a fixed given point q on the same sphere. Many observations follow almost immediately from this: (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.) (4) One of the trajectories is the equator. (This corresponds to all rotations that rotate the top point, or “pinhole”, to the point q which is the bottommost point.) (5) The trajectories are disjoint. (6) The space of all trajectories can be identified with a 2-dimensional sphere. There are several ways to see this. (a) First, note that each trajectory, with the exception of the equator, strikes the upper hemisphere in exactly one point. (That point represents the “maximal” rotation, by ¼ about the location of the negative x-axis.
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