2074:
6633:
1683:
31:
67:
2069:{\displaystyle 2z_{0}z_{1}^{\ast }\cdot 2z_{0}^{\ast }z_{1}+\left(\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}\right)^{2}=4\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{0}\right|^{4}-2\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{1}\right|^{4}=\left(\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}\right)^{2}=1}
4780:
5507:
5771:
of fluid dynamics in which the fluid flows along the circles of the projection of the Hopf fibration in 3 dimensional space. The size of the velocities, the density and the pressure can be chosen at each point to satisfy the equations. All these quantities fall to zero going away from the centre. If
5739:. The Hopf mapping maps the rotation to the point on the 2-sphere given by θ and φ, and the associated circle is parametrized by ψ. Note that when θ = π the Euler angles φ and ψ are not well defined individually, so we do not have a one-to-one mapping (or a one-to-two mapping) between the
4400:
6706:: the former maps to a straight line, the latter to a unit circle perpendicular to, and centered on, this line, which may be viewed as a degenerate torus whose minor radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through
6007:
4471:
1610:
1191:
5302:
5220:
4244:
3702:
2836:
2750:
2664:
2578:
1322:
3894:
6140:
6260:
4924:
2482:
235:
2396:
5638:
5053:
5778:
7243:
650:
6824:
593:
536:
479:
826:
3126:
3068:
3945:
5690:
in three dimensions. The set of all possible quaternions produces the set of all possible rotations, which moves the tip of one unit vector of such a coordinate frame (say, the
4775:{\displaystyle {\begin{bmatrix}1-2(y^{2}+z^{2})&2(xy-wz)&2(xz+wy)\\2(xy+wz)&1-2(x^{2}+z^{2})&2(yz-wx)\\2(xz-wy)&2(yz+wx)&1-2(x^{2}+y^{2})\end{bmatrix}}.}
861:
422:
versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres:
3010:
1470:
1028:
6616:
as total space, base space, and fiber can occur only in these dimensions. Fiber bundles with similar properties, but different from the Hopf fibrations, were used by
3524:; its elements are angles of rotation leaving the given point unmoved, all sharing the axis connecting that point to the sphere's center. It follows easily that the
755:
712:
5502:{\displaystyle {\frac {1}{\sqrt {2(1+c)}}}{\Big (}(1+c)\cos(\theta ),a\sin(\theta )-b\cos(\theta ),a\cos(\theta )+b\sin(\theta ),(1+c)\sin(\theta ){\Big )}.\,\!}
5113:
4395:{\displaystyle {\begin{bmatrix}x_{1}+\mathbf {i} x_{2}&x_{3}+\mathbf {i} x_{4}\\-x_{3}+\mathbf {i} x_{4}&x_{1}-\mathbf {i} x_{2}\end{bmatrix}}.\,\!}
3615:
2756:
2670:
2584:
2498:
1202:
3820:
6013:
6146:
7465:
7132:
6979:
4801:
2402:
171:
2319:
7262:
5561:
5002:
5767:
If the Hopf fibration is treated as a vector field in 3 dimensional space then there is a solution to the (compressible, non-viscous)
6002:{\displaystyle \mathbf {v} (x,y,z)=A\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-2}\left(2(-ay+xz),2(ax+yz),a^{2}-x^{2}-y^{2}+z^{2}\right)}
7277:
6305:, admits several generalizations, which are also often known as Hopf fibrations. First, one can replace the projective line by an
6935:
6910:
363:
in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the stereographic projection of the
7476:
7535:
Banchoff, Thomas (1988). "Geometry of the Hopf
Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.).
726:, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the
377:
is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see
7647:
7602:
YouTube animation showing dynamic mapping of points on the 2-sphere to circles in the 3-sphere, by
Professor Niles Johnson.
7451:
6995:
599:
371:-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When
7561:
7504:
Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space".
6773:
6671:
which fill space. Here there is one exception: the Hopf circle containing the projection point maps to a straight line in
6451:
542:
485:
428:
5768:
7642:
7637:
7556:
7105:
Watterson, Michael; Kumar, Vijay (2020). Amato, Nancy M.; Hager, Greg; Thomas, Shawna; Torres-Torriti, Miguel (eds.).
5687:
7457:
7249:
3517:
3273:
760:
328:
3163:
3074:
3016:
6518:
5533:
acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle.
7657:
6649:
6412:
6408:
4785:
Here we find an explicit real formula for the bundle projection by noting that the fixed unit vector along the
402:
393:
342:
35:
6551:
Sometimes the term "Hopf fibration" is restricted to the fibrations between spheres obtained above, which are
7591:
4174:, will have the same effect. We put all these into one fibre, and the fibres can be mapped one-to-one to the
7313:
3529:
3507:
3366:
3228:
3137:
6667:). Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in
7652:
7220:
3909:
103:
7268:
6849:
6377:
3552:
3488:
831:
79:
3475:
Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the
7398:
Mosseri, R.; Dandoloff, R. (2001), "Geometry of entangled states, Bloch spheres and Hopf fibrations",
7485:
7419:
7400:
7352:
7020:
6845:
6702:
on these tori, with the exception of the circle through the projection point and the one through its
6350:
6282:
3484:
3174:
1672:, as may be shown by adding the squares of the absolute values of the complex and real components of
723:
7551:
2974:
6767:. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration
1605:{\displaystyle p(z_{0},z_{1})=(2z_{0}z_{1}^{\ast },\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}).}
719:
123:
1186:{\displaystyle (x_{1},x_{2},x_{3},x_{4})\leftrightarrow (z_{0},z_{1})=(x_{1}+ix_{2},x_{3}+ix_{4})}
7513:
7435:
7409:
7387:
7361:
7297:
7187:
7138:
7107:
6699:
6637:
1615:
The first component is a complex number, whereas the second component is real. Any point on the
356:
5772:
a is the distance to the inner ring, the velocities, pressure and density fields are given by:
7461:
7379:
7332:
7289:
7179:
7128:
7087:
7069:
6975:
6834:
6752:
6648:
The Hopf fibration has many implications, some purely attractive, others deeper. For example,
4463:
3292:
above defines an explicit diffeomorphism between the complex projective line and the ordinary
3283:
1455:
7623:
Interactive visualization of the mapping of points on the 2-sphere to circles in the 3-sphere
7344:
7523:
7493:
7427:
7371:
7322:
7281:
7229:
7171:
7120:
7077:
7061:
7028:
6897:
6760:
6314:
6310:
3544:
3361:. This means that it has a "local product structure", in the sense that every point of the
332:
2079:
Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if
733:
685:
327:
This has many implications: for example the existence of this bundle shows that the higher
290:-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is
7447:
6703:
3548:
3411:
3147:
715:
291:
7612:
7159:"Accurate High-Maneuvering Trajectory Tracking for Quadrotors: A Drag Utilization Method"
7489:
7423:
7024:
5215:{\displaystyle q_{(a,b,c)}={\frac {1}{\sqrt {2(1+c)}}}(1+c-\mathbf {i} b+\mathbf {j} a)}
4136:
Another way to look at this fibration is that every versor ω moves the plane spanned by
158:-sphere is composed of fibers, where each fiber is a circle — one for each point of the
7212:
7082:
7049:
6737:
6733:
6729:
6609:
3256:
2303:
1019:
668:
656:
7497:
7431:
5678:
Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the
7631:
7439:
7301:
7239:
7208:
7191:
7142:
6944:
6917:
6759:, and the Hopf fibration describes the topological structure of a quantum mechanical
6621:
5706:
vector does not specify the rotation fully; a further rotation is possible about the
3499:
3383:
3379:
2272:
364:
295:
351:, in which all of 3-dimensional space, except for the z-axis, is filled with nested
6967:
6833:). Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the
6756:
6690:(topologically, a torus is the product of two circles) and these project to nested
6525:
5725:
5078:
3521:
3352:
382:
336:
143:
119:
6632:
3697:{\displaystyle q=x_{1}+\mathbf {i} x_{2}+\mathbf {j} x_{3}+\mathbf {k} x_{4}.\,\!}
7124:
7588:
Chapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
6617:
6341:
in the usual way and by identifying antipodal points. This gives a fiber bundle
4191:
This approach is related to the direct construction by identifying a quaternion
2831:{\displaystyle x_{4}=\sin \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }
2745:{\displaystyle x_{3}=\cos \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }
2659:{\displaystyle x_{2}=\sin \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }
2573:{\displaystyle x_{1}=\cos \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }
411:
99:
7606:
7601:
7158:
4015:
determine the same rotation. As noted above, the rotations act transitively on
7311:(1935), "Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension",
7308:
7257:
7233:
6857:
6853:
6440:
5683:
3492:
2916:
415:
115:
7571:
7383:
7336:
7293:
7183:
7175:
7073:
7065:
1317:{\displaystyle (x_{1},x_{2},x_{3})\leftrightarrow (z,x)=(x_{1}+ix_{2},x_{3})}
392:
There are numerous generalizations of the Hopf fibration. The unit sphere in
7585:
7576:
7327:
7048:
Yershova, Anna; Jain, Swati; LaValle, Steven M.; Mitchell, Julie C. (2010).
6719:
4996:. Multiplication of unit quaternions produces composition of rotations, and
3889:{\displaystyle p=\mathbf {i} y_{1}+\mathbf {j} y_{2}+\mathbf {k} y_{3}.\,\!}
247:
30:
7527:
7091:
17:
7622:
6636:
The fibers of the Hopf fibration stereographically project to a family of
6135:{\displaystyle p(x,y,z)=-A^{2}B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-3},}
5643:
which completes the bundle. But note that this one-to-one mapping between
2259:
form the unit circle in the complex plane, it follows that for each point
66:
7414:
7050:"Generating Uniform Incremental Grids on SO (3) Using the Hopf Fibration"
6879:-sphere into disjoint great circles is possible because, unlike with the
6841:
6486:
6333:
A real version of the Hopf fibration is obtained by regarding the circle
6322:
4083:, and then the Hopf fibration can be defined as the map sending a versor
3957:
679:
419:
378:
95:
6255:{\displaystyle \rho (x,y,z)=3B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-1}}
4971:
consists of all those unit quaternions that send the unit vector there.
27:
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
7616:
7391:
7285:
7263:"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche"
6278:
5740:
4919:{\displaystyle {\Big (}2(xz+wy),2(yz-wx),1-2(x^{2}+y^{2}){\Big )},\,\!}
107:
71:
7033:
7008:
3136:
A geometric interpretation of the fibration may be obtained using the
2477:{\displaystyle z_{1}=e^{i\,{\frac {\xi _{2}-\xi _{1}}{2}}}\cos \eta .}
230:{\displaystyle S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}}S^{2},}
7518:
6613:
3993:
3712:
2391:{\displaystyle z_{0}=e^{i\,{\frac {\xi _{1}+\xi _{2}}{2}}}\sin \eta }
1335:
386:
360:
111:
7474:
Urbantke, H.K. (2003), "The Hopf fibration-seven times in physics",
7375:
203:
7366:
7156:
Jia, Jindou; Guo, Kexin; Yu, Xiang; Zhao, Weihua; Guo, Lei (2022).
4117:, get mapped to the same thing (which happens to be one of the two
6764:
6691:
6683:
6631:
3555:
2959:-sphere is as follows, with points on the circles parametrized by
352:
331:
are not trivial in general. It also provides a basic example of a
65:
29:
5633:{\displaystyle {\Big (}0,\cos(\theta ),-\sin(\theta ),0{\Big )},}
5048:{\displaystyle q_{\theta }=\cos \theta +\mathbf {k} \sin \theta }
4974:
We can also write an explicit formula for the fiber over a point
3539:
To make this more explicit, there are two approaches: the group
3146:, which is defined to be the set of all complex one-dimensional
7609:
By Gian Marco
Todesco shows the Hopf fibration of the 120-cell.
6852:
algorithm in motion planning. It also found application in the
2952:
causes you to make one full circle of both limbs of the torus.
2190:-sphere. These conclusions follow, because the complex factor
7595:
3417:
For the Hopf fibration, it is enough to remove a single point
3227:
there is a circle of unit norm, and so the restriction of the
3989:, and it is not hard to check that it preserves orientation.
722:. For concreteness, the central point can be taken to be the
6698:
which also fill space. The individual fibers map to linking
6313:. Second, one can replace the complex numbers by any (real)
5686:, which in turn is equivalent to a particular rotation of a
4073:. This is a circle subgroup. For concreteness, one can take
5659:
is not continuous on this circle, reflecting the fact that
3132:
Geometric interpretation using the complex projective line
7108:"Control of Quadrotors Using the Hopf Fibration on SO(3)"
6996:
sci.math.research 1993 thread "Spheres fibred by spheres"
6898:
https://ncatlab.org/nlab/show/quaternionic+Hopf+fibration
6663:, which in turn illuminates the topology of the bundle (
4409:, and the imaginary quaternions with the skew-hermitian
345:
of the Hopf fibration induces a remarkable structure on
4480:
4253:
7607:
YouTube animation of the construction of the 120-cell
6776:
6149:
6016:
5781:
5564:
5305:
5116:
5087:, our prototypical fiber. So long as the base point,
5005:
4804:
4474:
4247:
3912:
3823:
3618:
3077:
3019:
2977:
2759:
2673:
2587:
2501:
2405:
2322:
1686:
1473:
1205:
1031:
834:
763:
736:
688:
645:{\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}.}
602:
545:
488:
431:
174:
7539:. New York and Basel: Marcel Dekker. pp. 57–62.
7119:. Cham: Springer International Publishing: 199–215.
6819:{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4}.}
2176:. The converse is also true; any two points on the
659:
such fibrations can occur only in these dimensions.
588:{\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4},}
531:{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2},}
474:{\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1},}
6844:, where it was used to generate uniform samples on
62:
and their corresponding fibers with the same color.
7401:Journal of Physics A: Mathematical and Theoretical
7345:"An Elementary Introduction to the Hopf Fibration"
7261:
7245:The collected mathematical papers of Arthur Cayley
7157:
7106:
6818:
6254:
6134:
6001:
5632:
5501:
5214:
5047:
4918:
4774:
4394:
3939:
3888:
3696:
3120:
3062:
3004:
2830:
2744:
2658:
2572:
2476:
2390:
2068:
1604:
1316:
1185:
855:
820:
749:
706:
644:
587:
530:
473:
324:although locally it is indistinguishable from it.
229:
5622:
5567:
5498:
5489:
5335:
4959:-sphere where it sends the unit vector along the
4915:
4906:
4807:
4391:
3936:
3885:
3693:
118:in 1931, it is an influential early example of a
7592:An Elementary Introduction to the Hopf Fibration
6830:
6293:The Hopf construction, viewed as a fiber bundle
7506:Journal of Computational Design and Engineering
2180:-sphere that differ by a common complex factor
7054:The International Journal of Robotics Research
3255:-sphere: indeed it can be identified with the
2955:A mapping of the above parametrization to the
2313:-sphere employing the Hopf map is as follows.
7115:. Springer Proceedings in Advanced Robotics.
6446:-space) and factor out by unit quaternion (=
3300:-dimensional space. Alternatively, the point
3231:to the points of unit norm is a fibration of
821:{\displaystyle (x_{1},x_{2},\ldots ,x_{n+1})}
34:The Hopf fibration can be visualized using a
8:
7213:"On certain results relating to quaternions"
6937:Topological solitons in magnetohydrodynamics
2915:which specify circles, specifies a separate
682:, can be defined as the set of points in an
385:to circles, although they are not geometric
6972:Manifolds all of whose Geodesics are Closed
6395:The Hopf construction gives circle bundles
3121:{\displaystyle y=\sin(2\eta )\sin \xi _{1}}
3063:{\displaystyle x=\sin(2\eta )\cos \xi _{1}}
410:with circles as fibers, and there are also
6911:"Benjamin H. Smith's Hopf fibration notes"
6896:quaternionic Hopf Fibration, ncatlab.org.
6840:Hopf fibration also found applications in
6411:. This is actually the restriction of the
6277:. Similar patterns of fields are found as
5724:The rotation can be represented using the
4431:The rotation induced by a unit quaternion
4405:This identifies the group of versors with
718:which are a fixed distance from a central
7613:Video of one 30-cell ring of the 600-cell
7517:
7413:
7365:
7326:
7081:
7032:
6807:
6794:
6781:
6775:
6243:
6232:
6219:
6206:
6193:
6148:
6120:
6109:
6096:
6083:
6070:
6051:
6015:
5988:
5975:
5962:
5949:
5874:
5863:
5850:
5837:
5824:
5782:
5780:
5700:-sphere. However, fixing the tip of the
5696:vector) to all possible points on a unit
5621:
5620:
5566:
5565:
5563:
5497:
5488:
5487:
5334:
5333:
5306:
5304:
5201:
5190:
5148:
5121:
5115:
5031:
5010:
5004:
4914:
4905:
4904:
4895:
4882:
4806:
4805:
4803:
4752:
4739:
4632:
4619:
4512:
4499:
4475:
4473:
4390:
4373:
4364:
4355:
4343:
4334:
4325:
4308:
4299:
4290:
4278:
4269:
4260:
4248:
4246:
4168:is one of the circle of versors that fix
4111:is one of the circle of versors that fix
3935:
3929:
3911:
3884:
3875:
3866:
3857:
3848:
3839:
3830:
3822:
3692:
3683:
3674:
3665:
3656:
3647:
3638:
3629:
3617:
3536:-sphere, and this is the Hopf fibration.
3112:
3076:
3054:
3018:
2976:
2803:
2790:
2783:
2764:
2758:
2717:
2704:
2697:
2678:
2672:
2631:
2618:
2611:
2592:
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2506:
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2448:
2435:
2428:
2427:
2423:
2410:
2404:
2365:
2352:
2345:
2344:
2340:
2327:
2321:
2054:
2043:
2033:
2015:
2005:
1981:
1971:
1953:
1943:
1928:
1918:
1897:
1887:
1869:
1859:
1844:
1834:
1813:
1802:
1792:
1774:
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1740:
1730:
1725:
1709:
1704:
1694:
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1529:
1519:
1497:
1484:
1472:
1305:
1292:
1276:
1239:
1226:
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1204:
1174:
1158:
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1129:
1110:
1097:
1078:
1065:
1052:
1039:
1030:
841:
837:
836:
833:
803:
784:
771:
762:
741:
735:
687:
633:
620:
607:
601:
576:
563:
550:
544:
519:
506:
493:
487:
462:
449:
436:
430:
218:
210:
198:
192:
179:
173:
6678:The fibers over a circle of latitude on
3814:can be interpreted as a pure quaternion
3543:can either be identified with the group
3471:Geometric interpretation using rotations
976:-sphere can be defined in several ways.
122:. Technically, Hopf found a many-to-one
6883:-sphere, distinct great circles of the
6868:
165:This fiber bundle structure is denoted
56:to a ball. This image shows points on
7321:, Warsaw: Polish Acad. Sci.: 427–440,
3992:In fact, this identifies the group of
3900:
3715:, the quaternions of unit norm, those
6755:, the Riemann sphere is known as the
6675:— a "circle through infinity".
6664:
4965:axis. The fiber for a given point on
1619:-sphere must have the property that
1386:is identified with the subset of all
7:
7617:http://page.math.tu-berlin.de/~gunn/
7164:IEEE Robotics and Automation Letters
5261:is given by quaternions of the form
3940:{\displaystyle p\mapsto qpq^{*}\,\!}
3711:-sphere is then identified with the
2224:component and in the real component
335:, by identifying the fiber with the
151:
5665:is not topologically equivalent to
4002:, modulo the fact that the versors
2196:cancels with its complex conjugate
7009:"Historical note on fiber bundles"
6728:Hopf proved that the Hopf map has
6718:. Two such linking circles form a
6659:induces a remarkable structure in
6524:. Although one can also define an
4929:which is a continuous function of
4795:, rotates to another unit vector,
856:{\displaystyle \mathbb {R} ^{n+1}}
142:-sphere is mapped from a distinct
25:
5721:-sphere, plus a single rotation.
3467:has a neighborhood of this form.
3410:. Such a fibration is said to be
2245:Since the set of complex numbers
887: = 1. For example, the
74:mimic part of the Hopf fibration.
6485:A similar construction with the
5783:
5202:
5191:
5032:
4365:
4335:
4300:
4270:
4182:rotations which is the range of
3867:
3849:
3831:
3675:
3657:
3639:
3561:In the first approach, a vector
3520:of a point is isomorphic to the
2309:A direct parametrization of the
891:-sphere consists of the points (
359:. Here each fiber projects to a
134:-sphere such that each distinct
7477:Journal of Geometry and Physics
7007:Friedman, John L. (June 2015).
4027:which fix a given right versor
3996:with the group of rotations of
3599:is interpreted as a quaternion
3219:for any nonzero complex number
367:of a circle of latitude of the
7343:Lyons, David W. (April 2003),
6800:
6787:
6365:is diffeomorphic to a circle.
6361:is diffeomorphic to a sphere,
6171:
6153:
6038:
6020:
5939:
5921:
5912:
5891:
5805:
5787:
5611:
5605:
5590:
5584:
5484:
5478:
5469:
5457:
5451:
5445:
5430:
5424:
5409:
5403:
5388:
5382:
5367:
5361:
5352:
5340:
5327:
5315:
5209:
5175:
5169:
5157:
5140:
5122:
4901:
4875:
4860:
4842:
4833:
4815:
4758:
4732:
4718:
4700:
4692:
4674:
4664:
4646:
4638:
4612:
4598:
4580:
4570:
4552:
4544:
4526:
4518:
4492:
3916:
3099:
3090:
3041:
3032:
3005:{\displaystyle z=\cos(2\eta )}
2999:
2990:
1596:
1509:
1503:
1477:
1420:. (Here, for a complex number
1311:
1269:
1263:
1251:
1248:
1245:
1206:
1180:
1122:
1116:
1090:
1087:
1084:
1032:
815:
764:
701:
689:
626:
613:
569:
556:
512:
499:
455:
442:
185:
1:
7498:10.1016/S0393-0440(02)00121-3
7453:The Topology of Fibre Bundles
7237:; reprinted as article 20 in
6452:quaternionic projective space
3899:Then, as is well-known since
3506:-sphere. The spin group acts
3429:and the corresponding circle
3351:The Hopf fibration defines a
2923:-sphere, and one round trip (
2186:map to the same point on the
1454:, where the star denotes the
240:meaning that the fiber space
7125:10.1007/978-3-030-28619-4_20
6831:Mosseri & Dandoloff 2001
6450:) multiplication to get the
6427:Quaternionic Hopf fibrations
3783:On the other hand, a vector
7557:Encyclopedia of Mathematics
7432:10.1088/0305-4470/34/47/324
6887:-sphere need not intersect.
6736:. In fact it generates the
6526:octonionic projective plane
5717:-sphere is mapped onto the
5682:-sphere is equivalent to a
5540:, can be given by defining
4462:is given explicitly by the
3374:whose inverse image in the
3318:can be mapped to the ratio
1458:.) Then the Hopf fibration
663:Definition and construction
7674:
7458:Princeton University Press
7250:Cambridge University Press
6934:Kamchatnov, A. M. (1982),
6748:) and has infinite order.
6547:Fibrations between spheres
6519:octonionic projective line
6481:Octonionic Hopf fibrations
6431:Similarly, one can regard
5688:Cartesian coordinate frame
5077:varies, this sweeps out a
4144:to a new plane spanned by
3956:: indeed it is clearly an
3274:one point compactification
2306:of these circular fibers.
401:fibers naturally over the
329:homotopy groups of spheres
7248:, vol. (1841–1853),
7234:10.1080/14786444508562684
6628:Geometry and applications
6357:= {1, −1}. Just as
4021:, and the set of versors
3355:, with bundle projection
3286:). The formula given for
3223:. On any complex line in
2867:runs over the range from
2847:runs over the range from
2302:-sphere is realized as a
757:, consists of the points
7176:10.1109/LRA.2022.3176449
7066:10.1177/0278364909352700
6732:1, and therefore is not
6650:stereographic projection
6457:. In particular, since
6413:tautological line bundle
6409:complex projective space
6265:for arbitrary constants
5512:Since multiplication by
4949:. That is, the image of
4413:matrices (isomorphic to
3483:-dimensional space. The
2887:can take any value from
2162:for some complex number
678:-dimensional sphere, or
403:complex projective space
394:complex coordinate space
343:Stereographic projection
36:stereographic projection
7328:10.4064/fm-25-1-427-440
7314:Fundamenta Mathematicae
6391:Complex Hopf fibrations
5769:Navier–Stokes equations
5103:, is not the antipode,
3530:principal circle bundle
3138:complex projective line
1334:is identified with the
7221:Philosophical Magazine
6923:on September 14, 2016.
6875:This partition of the
6820:
6645:
6419:to the unit sphere in
6256:
6136:
6003:
5634:
5503:
5216:
5049:
4920:
4776:
4396:
4099:. All the quaternions
4062:are real numbers with
3941:
3890:
3698:
3347:Fiber bundle structure
3334:in the Riemann sphere
3282:(obtained by adding a
3251:is diffeomorphic to a
3122:
3064:
3006:
2832:
2746:
2660:
2574:
2478:
2392:
2070:
1637:. If that is so, then
1606:
1318:
1187:
857:
822:
751:
708:
646:
589:
532:
475:
298:. However it is not a
274:(Hopf's map) projects
231:
104:four-dimensional space
75:
63:
7648:Differential geometry
7269:Mathematische Annalen
6850:probabilistic roadmap
6821:
6635:
6612:, fiber bundles with
6378:real projective space
6257:
6137:
6004:
5635:
5536:The final fiber, for
5504:
5217:
5050:
4921:
4777:
4397:
4127:to the same place as
3942:
3891:
3699:
3553:special unitary group
3123:
3065:
3007:
2833:
2747:
2661:
2575:
2479:
2393:
2071:
1607:
1319:
1188:
877: + ⋯+
858:
823:
752:
750:{\displaystyle S^{n}}
709:
707:{\displaystyle (n+1)}
647:
590:
533:
476:
379:Topology and geometry
232:
80:differential topology
69:
50:and then compressing
33:
7537:Computers in Algebra
7528:10.1093/jcde/qwab018
7353:Mathematics Magazine
6774:
6608:As a consequence of
6535:does not fiber over
6505:does not fiber over
6465:, there is a bundle
6368:More generally, the
6351:real projective line
6329:Real Hopf fibrations
6283:magnetohydrodynamics
6147:
6014:
5779:
5562:
5303:
5245:. Thus the fiber of
5114:
5003:
4955:is the point on the
4802:
4472:
4245:
3910:
3821:
3744:, which is equal to
3616:
3485:rotation group SO(3)
3479:-sphere in ordinary
3446:; thus one can take
3175:equivalence relation
3075:
3017:
2975:
2757:
2671:
2585:
2499:
2403:
2320:
1684:
1471:
1203:
1029:
832:
761:
734:
686:
600:
543:
486:
429:
302:fiber bundle, i.e.,
280:onto the base space
172:
126:(or "map") from the
7594:by David W. Lyons (
7490:2003JGP....46..125U
7424:2001JPhA...3410243M
7408:(47): 10243–10252,
7025:2015PhT....68f..11F
6974:. Springer-Verlag.
4121:rotations rotating
3461:, and any point in
2285:is a circle, i.e.,
1735:
1714:
1539:
980:Direct construction
954:The Hopf fibration
250:in the total space
211:
124:continuous function
86:(also known as the
7643:Geometric topology
7638:Algebraic topology
7460:(published 1999),
7286:10.1007/BF01457962
7252:, pp. 123–126
6816:
6700:Villarceau circles
6646:
6638:Villarceau circles
6252:
6132:
5999:
5630:
5499:
5212:
5045:
4916:
4772:
4763:
4392:
4381:
3937:
3886:
3694:
3516:by rotations. The
3118:
3060:
3002:
2898:. Every value of
2828:
2742:
2656:
2570:
2474:
2388:
2066:
1721:
1700:
1602:
1525:
1314:
1183:
853:
818:
747:
704:
642:
585:
528:
471:
381:). The loops are
357:Villarceau circles
227:
76:
64:
7467:978-0-691-00548-5
7134:978-3-030-28619-4
7113:Robotics Research
7034:10.1063/PT.3.2799
6986:(§0.26 on page 6)
6981:978-3-540-08158-6
6909:Smith, Benjamin.
6854:automatic control
6753:quantum mechanics
6513:. One can regard
6501:. But the sphere
6317:, including (for
5713:axis. Thus, the
5331:
5330:
5173:
5172:
5107:, the quaternion
5058:is a rotation by
4464:orthogonal matrix
4427:Explicit formulae
4156:. Any quaternion
3950:is a rotation in
3365:-sphere has some
3284:point at infinity
3177:which identifies
2813:
2727:
2641:
2555:
2458:
2375:
2208:: in the complex
2202:in both parts of
1658:lies on the unit
1456:complex conjugate
972:-sphere over the
212:
206:
130:-sphere onto the
16:(Redirected from
7665:
7582:
7581:
7572:"Hopf fibration"
7565:
7552:"Hopf fibration"
7540:
7531:
7521:
7500:
7470:
7448:Steenrod, Norman
7442:
7417:
7415:quant-ph/0108137
7394:
7369:
7349:
7339:
7330:
7304:
7265:
7253:
7236:
7228:(171): 141–145,
7217:
7196:
7195:
7170:(3): 6966–6973.
7161:
7153:
7147:
7146:
7110:
7102:
7096:
7095:
7085:
7045:
7039:
7038:
7036:
7004:
6998:
6993:
6987:
6985:
6964:
6958:
6957:
6956:
6955:
6949:
6943:, archived from
6942:
6931:
6925:
6924:
6922:
6916:. Archived from
6915:
6906:
6900:
6894:
6888:
6886:
6882:
6878:
6873:
6825:
6823:
6822:
6817:
6812:
6811:
6799:
6798:
6786:
6785:
6761:two-level system
6710:circle, both in
6489:yields a bundle
6315:division algebra
6311:projective space
6276:
6270:
6261:
6259:
6258:
6253:
6251:
6250:
6242:
6238:
6237:
6236:
6224:
6223:
6211:
6210:
6198:
6197:
6141:
6139:
6138:
6133:
6128:
6127:
6119:
6115:
6114:
6113:
6101:
6100:
6088:
6087:
6075:
6074:
6056:
6055:
6008:
6006:
6005:
6000:
5998:
5994:
5993:
5992:
5980:
5979:
5967:
5966:
5954:
5953:
5882:
5881:
5873:
5869:
5868:
5867:
5855:
5854:
5842:
5841:
5829:
5828:
5786:
5720:
5716:
5712:
5705:
5699:
5695:
5681:
5674:
5664:
5658:
5648:
5639:
5637:
5636:
5631:
5626:
5625:
5571:
5570:
5554:
5548:
5539:
5532:
5508:
5506:
5505:
5500:
5493:
5492:
5339:
5338:
5332:
5311:
5307:
5295:
5290:, which are the
5289:
5260:
5244:
5228:
5221:
5219:
5218:
5213:
5205:
5194:
5174:
5153:
5149:
5144:
5143:
5106:
5102:
5086:
5076:
5070:
5064:
5054:
5052:
5051:
5046:
5035:
5015:
5014:
4995:
4989:
4970:
4964:
4958:
4954:
4948:
4925:
4923:
4922:
4917:
4910:
4909:
4900:
4899:
4887:
4886:
4811:
4810:
4794:
4790:
4781:
4779:
4778:
4773:
4768:
4767:
4757:
4756:
4744:
4743:
4637:
4636:
4624:
4623:
4517:
4516:
4504:
4503:
4461:
4422:
4412:
4408:
4401:
4399:
4398:
4393:
4386:
4385:
4378:
4377:
4368:
4360:
4359:
4348:
4347:
4338:
4330:
4329:
4313:
4312:
4303:
4295:
4294:
4283:
4282:
4273:
4265:
4264:
4237:
4233:
4187:
4181:
4177:
4173:
4167:
4161:
4155:
4143:
4132:
4126:
4120:
4116:
4110:
4104:
4098:
4082:
4072:
4061:
4055:
4049:
4032:
4026:
4020:
4014:
4007:
4001:
3988:
3955:
3946:
3944:
3943:
3938:
3934:
3933:
3895:
3893:
3892:
3887:
3880:
3879:
3870:
3862:
3861:
3852:
3844:
3843:
3834:
3813:
3807:
3779:
3773:
3743:
3732:
3724:
3710:
3703:
3701:
3700:
3695:
3688:
3687:
3678:
3670:
3669:
3660:
3652:
3651:
3642:
3634:
3633:
3608:
3598:
3592:
3549:unit quaternions
3542:
3535:
3527:
3515:
3505:
3497:
3482:
3478:
3466:
3460:
3445:
3439:
3428:
3422:
3409:
3391:
3377:
3373:
3364:
3360:
3342:
3333:
3317:
3299:
3295:
3291:
3281:
3271:
3254:
3250:
3242:
3236:
3218:
3194:
3172:
3161:
3156:. Equivalently,
3155:
3145:
3127:
3125:
3124:
3119:
3117:
3116:
3069:
3067:
3066:
3061:
3059:
3058:
3011:
3009:
3008:
3003:
2967:
2958:
2951:
2942:
2933:
2926:
2922:
2914:
2907:
2903:
2897:
2890:
2886:
2877:
2870:
2866:
2857:
2850:
2846:
2837:
2835:
2834:
2829:
2818:
2814:
2809:
2808:
2807:
2795:
2794:
2784:
2769:
2768:
2751:
2749:
2748:
2743:
2732:
2728:
2723:
2722:
2721:
2709:
2708:
2698:
2683:
2682:
2665:
2663:
2662:
2657:
2646:
2642:
2637:
2636:
2635:
2623:
2622:
2612:
2597:
2596:
2579:
2577:
2576:
2571:
2560:
2556:
2551:
2550:
2549:
2537:
2536:
2526:
2511:
2510:
2492:
2487:or in Euclidean
2483:
2481:
2480:
2475:
2461:
2460:
2459:
2454:
2453:
2452:
2440:
2439:
2429:
2415:
2414:
2397:
2395:
2394:
2389:
2378:
2377:
2376:
2371:
2370:
2369:
2357:
2356:
2346:
2332:
2331:
2312:
2301:
2297:
2284:
2270:
2264:
2258:
2250:
2241:
2223:
2207:
2201:
2195:
2189:
2185:
2179:
2175:
2167:
2161:
2135:
2117:
2075:
2073:
2072:
2067:
2059:
2058:
2053:
2049:
2048:
2047:
2042:
2038:
2037:
2020:
2019:
2014:
2010:
2009:
1986:
1985:
1980:
1976:
1975:
1958:
1957:
1952:
1948:
1947:
1933:
1932:
1927:
1923:
1922:
1902:
1901:
1896:
1892:
1891:
1874:
1873:
1868:
1864:
1863:
1849:
1848:
1843:
1839:
1838:
1818:
1817:
1812:
1808:
1807:
1806:
1801:
1797:
1796:
1779:
1778:
1773:
1769:
1768:
1745:
1744:
1734:
1729:
1713:
1708:
1699:
1698:
1677:
1671:
1661:
1657:
1636:
1618:
1611:
1609:
1608:
1603:
1595:
1594:
1589:
1585:
1584:
1567:
1566:
1561:
1557:
1556:
1538:
1533:
1524:
1523:
1502:
1501:
1489:
1488:
1463:
1453:
1419:
1407:
1397:
1385:
1379:
1361:
1355:
1333:
1323:
1321:
1320:
1315:
1310:
1309:
1297:
1296:
1281:
1280:
1244:
1243:
1231:
1230:
1218:
1217:
1192:
1190:
1189:
1184:
1179:
1178:
1163:
1162:
1150:
1149:
1134:
1133:
1115:
1114:
1102:
1101:
1083:
1082:
1070:
1069:
1057:
1056:
1044:
1043:
1017:
1011:
1001:
995:
989:
975:
971:
967:
951: = 1.
890:
862:
860:
859:
854:
852:
851:
840:
827:
825:
824:
819:
814:
813:
789:
788:
776:
775:
756:
754:
753:
748:
746:
745:
713:
711:
710:
705:
651:
649:
648:
643:
638:
637:
625:
624:
612:
611:
594:
592:
591:
586:
581:
580:
568:
567:
555:
554:
537:
535:
534:
529:
524:
523:
511:
510:
498:
497:
480:
478:
477:
472:
467:
466:
454:
453:
441:
440:
409:
400:
376:
370:
355:made of linking
350:
333:principal bundle
323:
317:
307:
289:
285:
279:
273:
259:
255:
245:
236:
234:
233:
228:
223:
222:
213:
204:
199:
197:
196:
184:
183:
161:
157:
149:
141:
133:
129:
114:. Discovered by
110:and an ordinary
70:Pairwise linked
61:
55:
49:
43:
21:
7673:
7672:
7668:
7667:
7666:
7664:
7663:
7662:
7658:Homotopy theory
7628:
7627:
7586:Dimensions Math
7570:Rowland, Todd.
7569:
7568:
7550:
7547:
7534:
7503:
7473:
7468:
7446:
7397:
7376:10.2307/3219300
7347:
7342:
7307:
7256:
7238:
7215:
7207:
7204:
7199:
7155:
7154:
7150:
7135:
7104:
7103:
7099:
7047:
7046:
7042:
7006:
7005:
7001:
6994:
6990:
6982:
6966:
6965:
6961:
6953:
6951:
6947:
6940:
6933:
6932:
6928:
6920:
6913:
6908:
6907:
6903:
6895:
6891:
6884:
6880:
6876:
6874:
6870:
6866:
6803:
6790:
6777:
6772:
6771:
6743:
6630:
6610:Adams's theorem
6549:
6483:
6429:
6393:
6337:as a subset of
6331:
6291:
6289:Generalizations
6272:
6266:
6228:
6215:
6202:
6189:
6188:
6184:
6183:
6145:
6144:
6105:
6092:
6079:
6066:
6065:
6061:
6060:
6047:
6012:
6011:
5984:
5971:
5958:
5945:
5887:
5883:
5859:
5846:
5833:
5820:
5819:
5815:
5814:
5777:
5776:
5765:
5763:Fluid mechanics
5718:
5714:
5707:
5701:
5697:
5691:
5679:
5666:
5660:
5650:
5644:
5560:
5559:
5550:
5547:
5541:
5537:
5531:
5513:
5301:
5300:
5291:
5288:
5280:
5262:
5246:
5230:
5226:
5117:
5112:
5111:
5104:
5088:
5082:
5072:
5066:
5059:
5006:
5001:
5000:
4991:
4975:
4966:
4960:
4956:
4950:
4930:
4891:
4878:
4800:
4799:
4792:
4786:
4762:
4761:
4748:
4735:
4721:
4695:
4668:
4667:
4641:
4628:
4615:
4601:
4574:
4573:
4547:
4521:
4508:
4495:
4476:
4470:
4469:
4432:
4429:
4414:
4410:
4406:
4380:
4379:
4369:
4351:
4349:
4339:
4321:
4315:
4314:
4304:
4286:
4284:
4274:
4256:
4249:
4243:
4242:
4235:
4232:
4222:
4212:
4202:
4192:
4183:
4179:
4175:
4169:
4163:
4157:
4145:
4137:
4128:
4122:
4118:
4112:
4106:
4100:
4084:
4074:
4063:
4057:
4051:
4034:
4028:
4022:
4016:
4009:
4003:
3997:
3961:
3951:
3925:
3908:
3907:
3871:
3853:
3835:
3819:
3818:
3809:
3805:
3798:
3791:
3784:
3775:
3772:
3765:
3758:
3751:
3745:
3734:
3726:
3716:
3708:
3679:
3661:
3643:
3625:
3614:
3613:
3600:
3594:
3590:
3583:
3576:
3569:
3562:
3540:
3533:
3525:
3511:
3503:
3495:
3480:
3476:
3473:
3462:
3447:
3441:
3430:
3424:
3418:
3412:locally trivial
3393:
3387:
3378:-sphere can be
3375:
3369:
3362:
3356:
3349:
3341:
3335:
3332:
3325:
3319:
3315:
3308:
3301:
3297:
3293:
3287:
3277:
3272:, which is the
3265:
3259:
3252:
3246:
3238:
3232:
3216:
3206:
3196:
3192:
3185:
3178:
3167:
3157:
3151:
3141:
3134:
3108:
3073:
3072:
3050:
3015:
3014:
2973:
2972:
2966:
2960:
2956:
2950:
2944:
2941:
2935:
2928:
2924:
2920:
2909:
2905:
2899:
2892:
2888:
2885:
2879:
2872:
2868:
2865:
2859:
2852:
2848:
2842:
2799:
2786:
2785:
2779:
2760:
2755:
2754:
2713:
2700:
2699:
2693:
2674:
2669:
2668:
2627:
2614:
2613:
2607:
2588:
2583:
2582:
2541:
2528:
2527:
2521:
2502:
2497:
2496:
2488:
2444:
2431:
2430:
2419:
2406:
2401:
2400:
2361:
2348:
2347:
2336:
2323:
2318:
2317:
2310:
2299:
2286:
2275:
2266:
2260:
2257:| = 1
2252:
2246:
2239:
2233:| − |
2232:
2225:
2222:
2216:
2209:
2203:
2197:
2191:
2187:
2181:
2177:
2174:| = 1
2169:
2163:
2159:
2148:
2137:
2133:
2126:
2119:
2115:
2108:
2097:
2090:
2080:
2029:
2025:
2024:
2001:
1997:
1996:
1995:
1991:
1990:
1967:
1963:
1962:
1939:
1935:
1934:
1914:
1910:
1909:
1883:
1879:
1878:
1855:
1851:
1850:
1830:
1826:
1825:
1788:
1784:
1783:
1760:
1756:
1755:
1754:
1750:
1749:
1736:
1690:
1682:
1681:
1673:
1663:
1659:
1655:
1648:
1638:
1634:
1628:| + |
1627:
1620:
1616:
1576:
1572:
1571:
1548:
1544:
1543:
1515:
1493:
1480:
1469:
1468:
1459:
1421:
1409:
1399:
1387:
1381:
1377:
1370:
1363:
1357:
1353:
1346:
1339:
1329:
1301:
1288:
1272:
1235:
1222:
1209:
1201:
1200:
1170:
1154:
1141:
1125:
1106:
1093:
1074:
1061:
1048:
1035:
1027:
1026:
1020:complex numbers
1013:
1003:
997:
991:
985:
982:
973:
969:
955:
950:
943:
936:
929:
918:
911:
904:
897:
888:
886:
876:
869:
835:
830:
829:
799:
780:
767:
759:
758:
737:
732:
731:
684:
683:
665:
657:Adams's theorem
629:
616:
603:
598:
597:
572:
559:
546:
541:
540:
515:
502:
489:
484:
483:
458:
445:
432:
427:
426:
405:
396:
372:
368:
346:
319:
313:
303:
287:
281:
275:
261:
257:
251:
241:
214:
188:
175:
170:
169:
159:
155:
147:
139:
131:
127:
57:
51:
45:
39:
28:
23:
22:
15:
12:
11:
5:
7671:
7669:
7661:
7660:
7655:
7650:
7645:
7640:
7630:
7629:
7626:
7625:
7620:
7610:
7604:
7599:
7589:
7583:
7566:
7546:
7545:External links
7543:
7542:
7541:
7532:
7512:(3): 836–854.
7501:
7484:(2): 125–150,
7471:
7466:
7444:
7395:
7340:
7305:
7254:
7240:Cayley, Arthur
7209:Cayley, Arthur
7203:
7200:
7198:
7197:
7148:
7133:
7097:
7060:(7): 801–812.
7040:
6999:
6988:
6980:
6959:
6926:
6901:
6889:
6867:
6865:
6862:
6835:Dirac monopole
6827:
6826:
6815:
6810:
6806:
6802:
6797:
6793:
6789:
6784:
6780:
6741:
6738:homotopy group
6734:null-homotopic
6730:Hopf invariant
6704:opposite point
6629:
6626:
6622:exotic spheres
6606:
6605:
6592:
6579:
6566:
6548:
6545:
6482:
6479:
6428:
6425:
6392:
6389:
6330:
6327:
6290:
6287:
6263:
6262:
6249:
6246:
6241:
6235:
6231:
6227:
6222:
6218:
6214:
6209:
6205:
6201:
6196:
6192:
6187:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6142:
6131:
6126:
6123:
6118:
6112:
6108:
6104:
6099:
6095:
6091:
6086:
6082:
6078:
6073:
6069:
6064:
6059:
6054:
6050:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6025:
6022:
6019:
6009:
5997:
5991:
5987:
5983:
5978:
5974:
5970:
5965:
5961:
5957:
5952:
5948:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5902:
5899:
5896:
5893:
5890:
5886:
5880:
5877:
5872:
5866:
5862:
5858:
5853:
5849:
5845:
5840:
5836:
5832:
5827:
5823:
5818:
5813:
5810:
5807:
5804:
5801:
5798:
5795:
5792:
5789:
5785:
5764:
5761:
5641:
5640:
5629:
5624:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5580:
5577:
5574:
5569:
5545:
5517:
5510:
5509:
5496:
5491:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5354:
5351:
5348:
5345:
5342:
5337:
5329:
5326:
5323:
5320:
5317:
5314:
5310:
5284:
5266:
5223:
5222:
5211:
5208:
5204:
5200:
5197:
5193:
5189:
5186:
5183:
5180:
5177:
5171:
5168:
5165:
5162:
5159:
5156:
5152:
5147:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5120:
5056:
5055:
5044:
5041:
5038:
5034:
5030:
5027:
5024:
5021:
5018:
5013:
5009:
4927:
4926:
4913:
4908:
4903:
4898:
4894:
4890:
4885:
4881:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4809:
4783:
4782:
4771:
4766:
4760:
4755:
4751:
4747:
4742:
4738:
4734:
4731:
4728:
4725:
4722:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4640:
4635:
4631:
4627:
4622:
4618:
4614:
4611:
4608:
4605:
4602:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4520:
4515:
4511:
4507:
4502:
4498:
4494:
4491:
4488:
4485:
4482:
4481:
4479:
4428:
4425:
4403:
4402:
4389:
4384:
4376:
4372:
4367:
4363:
4358:
4354:
4350:
4346:
4342:
4337:
4333:
4328:
4324:
4320:
4317:
4316:
4311:
4307:
4302:
4298:
4293:
4289:
4285:
4281:
4277:
4272:
4268:
4263:
4259:
4255:
4254:
4252:
4230:
4220:
4210:
4200:
4033:have the form
3948:
3947:
3932:
3928:
3924:
3921:
3918:
3915:
3903:, the mapping
3897:
3896:
3883:
3878:
3874:
3869:
3865:
3860:
3856:
3851:
3847:
3842:
3838:
3833:
3829:
3826:
3803:
3796:
3789:
3770:
3763:
3756:
3749:
3705:
3704:
3691:
3686:
3682:
3677:
3673:
3668:
3664:
3659:
3655:
3650:
3646:
3641:
3637:
3632:
3628:
3624:
3621:
3588:
3581:
3574:
3567:
3551:, or with the
3472:
3469:
3401:) ≅
3392:and a circle:
3348:
3345:
3339:
3330:
3323:
3313:
3306:
3263:
3257:Riemann sphere
3214:
3204:
3190:
3183:
3133:
3130:
3129:
3128:
3115:
3111:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3070:
3057:
3053:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3012:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2964:
2948:
2939:
2883:
2863:
2839:
2838:
2827:
2824:
2821:
2817:
2812:
2806:
2802:
2798:
2793:
2789:
2782:
2778:
2775:
2772:
2767:
2763:
2752:
2741:
2738:
2735:
2731:
2726:
2720:
2716:
2712:
2707:
2703:
2696:
2692:
2689:
2686:
2681:
2677:
2666:
2655:
2652:
2649:
2645:
2640:
2634:
2630:
2626:
2621:
2617:
2610:
2606:
2603:
2600:
2595:
2591:
2580:
2569:
2566:
2563:
2559:
2554:
2548:
2544:
2540:
2535:
2531:
2524:
2520:
2517:
2514:
2509:
2505:
2485:
2484:
2473:
2470:
2467:
2464:
2457:
2451:
2447:
2443:
2438:
2434:
2426:
2422:
2418:
2413:
2409:
2398:
2387:
2384:
2381:
2374:
2368:
2364:
2360:
2355:
2351:
2343:
2339:
2335:
2330:
2326:
2304:disjoint union
2237:
2230:
2220:
2214:
2157:
2146:
2131:
2124:
2113:
2106:
2095:
2088:
2077:
2076:
2065:
2062:
2057:
2052:
2046:
2041:
2036:
2032:
2028:
2023:
2018:
2013:
2008:
2004:
2000:
1994:
1989:
1984:
1979:
1974:
1970:
1966:
1961:
1956:
1951:
1946:
1942:
1938:
1931:
1926:
1921:
1917:
1913:
1908:
1905:
1900:
1895:
1890:
1886:
1882:
1877:
1872:
1867:
1862:
1858:
1854:
1847:
1842:
1837:
1833:
1829:
1824:
1821:
1816:
1811:
1805:
1800:
1795:
1791:
1787:
1782:
1777:
1772:
1767:
1763:
1759:
1753:
1748:
1743:
1739:
1733:
1728:
1724:
1720:
1717:
1712:
1707:
1703:
1697:
1693:
1689:
1653:
1646:
1632:
1625:
1613:
1612:
1601:
1598:
1593:
1588:
1583:
1579:
1575:
1570:
1565:
1560:
1555:
1551:
1547:
1542:
1537:
1532:
1528:
1522:
1518:
1514:
1511:
1508:
1505:
1500:
1496:
1492:
1487:
1483:
1479:
1476:
1464:is defined by
1429: + i
1414:| +
1375:
1368:
1351:
1344:
1326:
1325:
1313:
1308:
1304:
1300:
1295:
1291:
1287:
1284:
1279:
1275:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1242:
1238:
1234:
1229:
1225:
1221:
1216:
1212:
1208:
1194:
1193:
1182:
1177:
1173:
1169:
1166:
1161:
1157:
1153:
1148:
1144:
1140:
1137:
1132:
1128:
1124:
1121:
1118:
1113:
1109:
1105:
1100:
1096:
1092:
1089:
1086:
1081:
1077:
1073:
1068:
1064:
1060:
1055:
1051:
1047:
1042:
1038:
1034:
1022:) by writing:
981:
978:
948:
941:
934:
927:
916:
909:
902:
895:
881:
874:
867:
850:
847:
844:
839:
817:
812:
809:
806:
802:
798:
795:
792:
787:
783:
779:
774:
770:
766:
744:
740:
703:
700:
697:
694:
691:
669:natural number
664:
661:
653:
652:
641:
636:
632:
628:
623:
619:
615:
610:
606:
595:
584:
579:
575:
571:
566:
562:
558:
553:
549:
538:
527:
522:
518:
514:
509:
505:
501:
496:
492:
481:
470:
465:
461:
457:
452:
448:
444:
439:
435:
286:(the ordinary
260:-sphere), and
246:(a circle) is
238:
237:
226:
221:
217:
209:
202:
195:
191:
187:
182:
178:
106:) in terms of
94:) describes a
84:Hopf fibration
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7670:
7659:
7656:
7654:
7653:Fiber bundles
7651:
7649:
7646:
7644:
7641:
7639:
7636:
7635:
7633:
7624:
7621:
7618:
7614:
7611:
7608:
7605:
7603:
7600:
7597:
7593:
7590:
7587:
7584:
7579:
7578:
7573:
7567:
7563:
7559:
7558:
7553:
7549:
7548:
7544:
7538:
7533:
7529:
7525:
7520:
7515:
7511:
7507:
7502:
7499:
7495:
7491:
7487:
7483:
7479:
7478:
7472:
7469:
7463:
7459:
7455:
7454:
7449:
7445:
7441:
7437:
7433:
7429:
7425:
7421:
7416:
7411:
7407:
7403:
7402:
7396:
7393:
7389:
7385:
7381:
7377:
7373:
7368:
7363:
7359:
7355:
7354:
7346:
7341:
7338:
7334:
7329:
7324:
7320:
7316:
7315:
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7276:(1), Berlin:
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7018:
7014:
7013:Physics Today
7010:
7003:
7000:
6997:
6992:
6989:
6983:
6977:
6973:
6969:
6968:Besse, Arthur
6963:
6960:
6950:on 2016-01-28
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6625:
6623:
6620:to construct
6619:
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6316:
6312:
6309:-dimensional
6308:
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6300:
6296:
6288:
6286:
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6281:solutions of
6280:
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3901:Cayley (1845)
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3528:-sphere is a
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2296:
2293: ≅
2292:
2289:
2282:
2278:
2274:
2273:inverse image
2269:
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2256:
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2016:
2011:
2006:
2002:
1998:
1992:
1987:
1982:
1977:
1972:
1968:
1964:
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1954:
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866:
848:
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768:
742:
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729:
725:
721:
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714:-dimensional
698:
695:
692:
681:
677:
673:
670:
662:
660:
658:
639:
634:
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621:
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399:
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384:
380:
375:
366:
365:inverse image
362:
358:
354:
349:
344:
340:
338:
334:
330:
325:
322:
316:
312:a product of
311:
306:
301:
297:
296:product space
293:
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268:
265: :
264:
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193:
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137:
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113:
109:
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101:
97:
93:
89:
85:
81:
73:
68:
60:
54:
48:
42:
37:
32:
19:
7575:
7555:
7536:
7519:2003.09236v2
7509:
7505:
7481:
7475:
7452:
7405:
7399:
7360:(2): 87–98,
7357:
7351:
7318:
7312:
7273:
7267:
7244:
7225:
7219:
7167:
7163:
7151:
7116:
7112:
7100:
7057:
7053:
7043:
7016:
7012:
7002:
6991:
6971:
6962:
6952:, retrieved
6945:the original
6936:
6929:
6918:the original
6904:
6892:
6871:
6839:
6828:
6757:Bloch sphere
6750:
6745:
6727:
6723:
6715:
6711:
6707:
6695:
6687:
6679:
6677:
6672:
6668:
6660:
6656:
6652:
6647:
6641:
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6602:
6598:
6594:
6589:
6585:
6581:
6576:
6572:
6568:
6563:
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6555:
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6540:
6536:
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6528:
6521:
6514:
6510:
6506:
6502:
6498:
6494:
6490:
6484:
6474:
6470:
6466:
6462:
6458:
6454:
6447:
6443:
6441:quaternionic
6436:
6435:as lying in
6432:
6430:
6420:
6416:
6404:
6400:
6396:
6394:
6384:
6380:
6376:fibers over
6373:
6369:
6367:
6362:
6358:
6354:
6346:
6342:
6338:
6334:
6332:
6318:
6306:
6302:
6298:
6294:
6292:
6273:
6267:
6264:
5766:
5756:
5752:
5748:
5744:
5736:
5732:
5728:
5726:Euler angles
5723:
5708:
5702:
5692:
5677:
5671:
5667:
5661:
5655:
5651:
5645:
5642:
5555:, producing
5551:
5542:
5535:
5527:
5523:
5519:
5514:
5511:
5292:
5285:
5281:
5276:
5272:
5268:
5263:
5256:
5252:
5248:
5240:
5236:
5232:
5224:
5098:
5094:
5090:
5083:
5079:great circle
5073:
5067:
5061:
5057:
4992:
4985:
4981:
4977:
4973:
4967:
4961:
4951:
4944:
4940:
4936:
4932:
4928:
4787:
4784:
4458:
4455:
4451:
4448:
4444:
4441:
4437:
4433:
4430:
4419:
4415:
4404:
4227:
4224:
4217:
4214:
4207:
4204:
4197:
4193:
4190:
4184:
4170:
4164:
4158:
4151:
4147:
4139:
4135:
4129:
4123:
4113:
4107:
4101:
4095:
4092:
4089:
4085:
4079:
4075:
4068:
4064:
4058:
4052:
4046:
4043:
4039:
4035:
4029:
4023:
4017:
4011:
4004:
3998:
3991:
3984:
3980:
3977:
3973:
3970:
3967:
3963:
3952:
3949:
3898:
3810:
3800:
3793:
3786:
3782:
3776:
3767:
3760:
3753:
3746:
3740:
3736:
3728:
3721:
3717:
3706:
3605:
3601:
3595:
3585:
3578:
3571:
3564:
3560:
3538:
3522:circle group
3512:
3508:transitively
3489:double cover
3474:
3463:
3456:
3452:
3448:
3442:
3435:
3431:
3425:
3419:
3416:
3406:
3402:
3398:
3394:
3388:
3370:
3367:neighborhood
3357:
3353:fiber bundle
3350:
3336:
3327:
3320:
3310:
3303:
3288:
3278:
3267:
3260:
3247:
3245:
3239:
3233:
3229:quotient map
3224:
3220:
3211:
3208:
3201:
3198:
3187:
3180:
3168:
3158:
3152:
3142:
3135:
2961:
2954:
2945:
2936:
2934:) of either
2930:
2910:
2900:
2894:
2880:
2874:
2860:
2853:
2843:
2840:
2489:
2486:
2308:
2294:
2290:
2287:
2280:
2276:
2267:
2261:
2254:
2247:
2244:
2234:
2227:
2217:
2211:
2204:
2198:
2192:
2182:
2171:
2164:
2154:
2150:
2143:
2139:
2128:
2121:
2110:
2103:
2099:
2092:
2085:
2081:
2078:
1674:
1668:
1664:
1650:
1643:
1639:
1629:
1622:
1614:
1460:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1415:
1411:
1404:
1400:
1393:
1389:
1382:
1372:
1365:
1358:
1348:
1341:
1330:
1327:
1195:
1018:denotes the
1014:
1008:
1004:
998:
992:
986:
983:
964:
960:
956:
953:
945:
938:
931:
924:
920:
913:
906:
899:
892:
882:
878:
871:
864:
727:
675:
671:
666:
654:
416:quaternionic
406:
397:
391:
383:homeomorphic
373:
347:
341:
337:circle group
326:
320:
314:
309:
304:
299:
282:
276:
270:
266:
262:
252:
242:
239:
164:
154:). Thus the
144:great circle
135:
120:fiber bundle
91:
87:
83:
77:
58:
52:
46:
40:
7309:Hopf, Heinz
7280:: 637–665,
7258:Hopf, Heinz
6618:John Milnor
6601:with fiber
6588:with fiber
6575:with fiber
6562:with fiber
6539:with fiber
6509:with fiber
6497:with fiber
6473:with fiber
6383:with fiber
6353:with fiber
5065:around the
4178:-sphere of
3609:by writing
3296:-sphere in
2298:. Thus the
2136:must equal
1662:-sphere in
100:hypersphere
88:Hopf bundle
18:Hopf bundle
7632:Categories
7456:, PMS 14,
7367:2212.01642
7202:References
6954:2011-08-03
6858:quadrotors
6665:Lyons 2003
5684:quaternion
5538:(0, 0, −1)
5225:will send
5105:(0, 0, −1)
3780:as above.
3725:for which
3518:stabilizer
3493:spin group
3380:identified
2917:flat torus
1635:| = 1
1408:such that
1378:| = 1
1362:such that
420:octonionic
116:Heinz Hopf
7577:MathWorld
7562:EMS Press
7440:119462869
7384:0025-570X
7337:0016-2736
7302:123533891
7294:0025-5831
7192:249550496
7184:2377-3766
7143:195852176
7074:0278-3649
7019:(6): 11.
6801:→
6788:↪
6720:Hopf link
6487:octonions
6349:over the
6323:octonions
6321:= 1) the
6245:−
6151:ρ
6122:−
6045:−
5969:−
5956:−
5895:−
5876:−
5609:θ
5603:
5597:−
5588:θ
5582:
5549:to equal
5482:θ
5476:
5449:θ
5443:
5428:θ
5422:
5407:θ
5401:
5392:−
5386:θ
5380:
5365:θ
5359:
5227:(0, 0, 1)
5188:−
5071:axis. As
5043:θ
5040:
5026:θ
5023:
5012:θ
4870:−
4852:−
4727:−
4684:−
4656:−
4607:−
4536:−
4487:−
4362:−
4319:−
4234:with the
3931:∗
3917:↦
3532:over the
3382:with the
3148:subspaces
3110:ξ
3106:
3097:η
3088:
3052:ξ
3048:
3039:η
3030:
2997:η
2988:
2904:, except
2826:η
2823:
2801:ξ
2797:−
2788:ξ
2777:
2740:η
2737:
2715:ξ
2711:−
2702:ξ
2691:
2654:η
2651:
2629:ξ
2616:ξ
2605:
2568:η
2565:
2543:ξ
2530:ξ
2519:
2469:η
2466:
2446:ξ
2442:−
2433:ξ
2386:η
2383:
2363:ξ
2350:ξ
1904:−
1781:−
1732:∗
1716:⋅
1711:∗
1569:−
1536:∗
1437:| =
1249:↔
1088:↔
984:Identify
794:…
730:-sphere,
627:→
614:↪
570:→
557:↪
513:→
500:↪
456:→
443:↪
186:↪
162:-sphere.
152:Hopf 1931
150:-sphere (
7450:(1951),
7278:Springer
7260:(1931),
7242:(1889),
7211:(1845),
7092:20607113
6970:(1978).
6848:for the
6842:robotics
6399: :
6372:-sphere
5546:(0,0,−1)
4238:matrix:
4162:, where
4105:, where
4050:, where
3960:, since
3958:isometry
3733:, where
3164:quotient
1445: =
1425: =
680:n-sphere
667:For any
310:globally
248:embedded
201:→
96:3-sphere
92:Hopf map
72:keyrings
7564:, 2001
7486:Bibcode
7420:Bibcode
7392:3219300
7083:2896220
7021:Bibcode
6714:and in
6692:toruses
6682:form a
6614:spheres
6517:as the
6279:soliton
5741:3-torus
5296:points
4793:(0,0,1)
4133:does).
3994:versors
3713:versors
3541:Spin(3)
3502:to the
3496:Spin(3)
3384:product
3173:by the
3162:is the
2919:in the
2118:, then
1338:of all
1012:(where
968:of the
912:,
905:,
898:,
387:circles
308:is not
300:trivial
292:locally
146:of the
138:of the
108:circles
7464:
7438:
7390:
7382:
7335:
7300:
7292:
7190:
7182:
7141:
7131:
7090:
7080:
7072:
6978:
5755:) and
5735:, and
4791:axis,
3491:, the
3487:has a
2878:, and
2841:Where
2271:, the
2153:
2142:
1441:
1380:, and
1336:subset
724:origin
418:, and
361:circle
205:
112:sphere
82:, the
7615:from
7514:arXiv
7436:S2CID
7410:arXiv
7388:JSTOR
7362:arXiv
7348:(PDF)
7298:S2CID
7216:(PDF)
7188:S2CID
7139:S2CID
6948:(PDF)
6941:(PDF)
6921:(PDF)
6914:(PDF)
6864:Notes
6846:SO(3)
6765:qubit
6708:every
6684:torus
6415:over
6407:over
4407:SU(2)
3978:q p p
3968:q p q
3964:q p q
3731:| = 1
3556:SU(2)
3545:Sp(1)
3440:from
3423:from
3270:∪ {∞}
3237:over
3195:with
2251:with
2168:with
1371:| + |
1328:Thus
1002:with
990:with
923:with
919:) in
863:with
720:point
716:space
674:, an
256:(the
136:point
7462:ISBN
7380:ISSN
7333:ISSN
7290:ISSN
7180:ISSN
7129:ISBN
7088:PMID
7070:ISSN
6976:ISBN
6271:and
5743:of (
5649:and
4180:180°
4138:{1,
4119:180°
4056:and
4008:and
3966:| =
3774:for
3739:| =
3707:The
3171:\{0}
2908:and
2098:) =
1196:and
996:and
412:real
353:tori
318:and
7596:PDF
7524:doi
7494:doi
7428:doi
7372:doi
7323:doi
7282:doi
7274:104
7230:doi
7172:doi
7121:doi
7078:PMC
7062:doi
7029:doi
6856:of
6763:or
6751:In
6722:in
6694:in
6686:in
6640:in
5600:sin
5579:cos
5473:sin
5440:sin
5419:cos
5398:cos
5377:sin
5356:cos
5229:to
5081:of
5037:sin
5020:cos
4990:in
4423:).
4411:2×2
4236:2×2
4185:ωkω
4088:to
4071:= 1
3983:= |
3971:q p
3808:in
3741:q q
3593:in
3547:of
3510:on
3386:of
3276:of
3166:of
3150:of
3103:sin
3085:sin
3045:cos
3027:sin
2985:cos
2943:or
2927:to
2891:to
2871:to
2851:to
2820:cos
2774:sin
2734:cos
2688:cos
2648:sin
2602:sin
2562:sin
2516:cos
2463:cos
2380:sin
2265:in
1433:, |
1418:= 1
1398:in
1356:in
885:+ 1
828:in
655:By
102:in
98:(a
90:or
78:In
44:to
38:of
7634::
7574:.
7560:,
7554:,
7522:.
7508:.
7492:,
7482:46
7480:,
7434:,
7426:,
7418:,
7406:34
7404:,
7386:,
7378:,
7370:,
7358:76
7356:,
7350:,
7331:,
7319:25
7317:,
7296:,
7288:,
7272:,
7266:,
7226:26
7224:,
7218:,
7186:.
7178:.
7166:.
7162:.
7137:.
7127:.
7117:10
7111:.
7086:.
7076:.
7068:.
7058:29
7056:.
7052:.
7027:.
7017:68
7015:.
7011:.
6860:.
6837:.
6655:→
6624:.
6597:→
6584:→
6571:→
6558:→
6543:.
6537:OP
6529:OP
6522:OP
6493:→
6477:.
6469:→
6463:HP
6461:=
6455:HP
6423:.
6417:CP
6405:CP
6403:→
6387:.
6381:RP
6363:RP
6359:CP
6347:RP
6345:→
6325:.
6303:CP
6301:→
6297::
6285::
5759:.
5751:,
5747:,
5731:,
5675:.
5275:,
5271:,
5255:,
5251:,
5239:,
5235:,
5097:,
5093:,
4984:,
4980:,
4943:,
4939:,
4935:,
4454:+
4447:+
4440:+
4436:=
4418:×
4223:+
4213:+
4203:+
4196:=
4188:.
4159:ωq
4152:ωk
4150:,
4102:ωq
4078:=
4067:+
4042:+
4038:=
3976:=
3799:,
3792:,
3766:+
3759:+
3752:+
3720:∈
3604:∈
3584:,
3577:,
3570:,
3558:.
3498:,
3455:\{
3451:=
3414:.
3405:×
3343:.
3309:,
3266:=
3248:CP
3243:.
3240:CP
3207:,
3186:,
3159:CP
3143:CP
3140:,
2968:.
2913:/2
2858:,
2856:/2
2242:.
2149:,
2127:,
2109:,
2091:,
1667:×
1649:,
1392:,
1347:,
1007:×
963:→
959::
622:15
414:,
407:CP
389:.
339:.
294:a
269:→
7619:.
7598:)
7580:.
7530:.
7526::
7516::
7510:8
7496::
7488::
7443:.
7430::
7422::
7412::
7374::
7364::
7325::
7284::
7232::
7194:.
7174::
7168:7
7145:.
7123::
7094:.
7064::
7037:.
7031::
7023::
6984:.
6885:3
6881:2
6877:3
6829:(
6814:.
6809:4
6805:S
6796:7
6792:S
6783:3
6779:S
6746:S
6744:(
6742:3
6740:π
6724:R
6716:S
6712:R
6696:R
6688:S
6680:S
6673:R
6669:R
6661:R
6657:R
6653:S
6644:.
6642:R
6603:S
6599:S
6595:S
6590:S
6586:S
6582:S
6577:S
6573:S
6569:S
6564:S
6560:S
6556:S
6541:S
6533:S
6515:S
6511:S
6507:S
6503:S
6499:S
6495:S
6491:S
6475:S
6471:S
6467:S
6459:S
6448:S
6444:n
6439:(
6437:H
6433:S
6421:C
6401:S
6397:p
6385:S
6374:S
6370:n
6355:S
6343:S
6339:R
6335:S
6319:n
6307:n
6299:S
6295:p
6274:B
6268:A
6248:1
6240:)
6234:2
6230:z
6226:+
6221:2
6217:y
6213:+
6208:2
6204:x
6200:+
6195:2
6191:a
6186:(
6181:B
6178:3
6175:=
6172:)
6169:z
6166:,
6163:y
6160:,
6157:x
6154:(
6130:,
6125:3
6117:)
6111:2
6107:z
6103:+
6098:2
6094:y
6090:+
6085:2
6081:x
6077:+
6072:2
6068:a
6063:(
6058:B
6053:2
6049:A
6042:=
6039:)
6036:z
6033:,
6030:y
6027:,
6024:x
6021:(
6018:p
5996:)
5990:2
5986:z
5982:+
5977:2
5973:y
5964:2
5960:x
5951:2
5947:a
5943:,
5940:)
5937:z
5934:y
5931:+
5928:x
5925:a
5922:(
5919:2
5916:,
5913:)
5910:z
5907:x
5904:+
5901:y
5898:a
5892:(
5889:2
5885:(
5879:2
5871:)
5865:2
5861:z
5857:+
5852:2
5848:y
5844:+
5839:2
5835:x
5831:+
5826:2
5822:a
5817:(
5812:A
5809:=
5806:)
5803:z
5800:,
5797:y
5794:,
5791:x
5788:(
5784:v
5757:S
5753:ψ
5749:φ
5745:θ
5737:ψ
5733:φ
5729:θ
5719:2
5715:3
5711:-
5709:z
5703:z
5698:2
5693:z
5680:3
5672:S
5670:×
5668:S
5662:S
5656:S
5654:×
5652:S
5646:S
5628:,
5623:)
5618:0
5615:,
5612:)
5606:(
5594:,
5591:)
5585:(
5576:,
5573:0
5568:(
5552:i
5543:q
5530:)
5528:c
5526:,
5524:b
5522:,
5520:a
5518:(
5515:q
5495:.
5490:)
5485:)
5479:(
5470:)
5467:c
5464:+
5461:1
5458:(
5455:,
5452:)
5446:(
5437:b
5434:+
5431:)
5425:(
5416:a
5413:,
5410:)
5404:(
5395:b
5389:)
5383:(
5374:a
5371:,
5368:)
5362:(
5353:)
5350:c
5347:+
5344:1
5341:(
5336:(
5328:)
5325:c
5322:+
5319:1
5316:(
5313:2
5309:1
5293:S
5286:θ
5282:q
5279:)
5277:c
5273:b
5269:a
5267:(
5264:q
5259:)
5257:c
5253:b
5249:a
5247:(
5243:)
5241:c
5237:b
5233:a
5231:(
5210:)
5207:a
5203:j
5199:+
5196:b
5192:i
5185:c
5182:+
5179:1
5176:(
5170:)
5167:c
5164:+
5161:1
5158:(
5155:2
5151:1
5146:=
5141:)
5138:c
5135:,
5132:b
5129:,
5126:a
5123:(
5119:q
5101:)
5099:c
5095:b
5091:a
5089:(
5084:S
5074:θ
5068:z
5062:θ
5060:2
5033:k
5029:+
5017:=
5008:q
4993:S
4988:)
4986:c
4982:b
4978:a
4976:(
4968:S
4962:z
4957:2
4952:q
4947:)
4945:z
4941:y
4937:x
4933:w
4931:(
4912:,
4907:)
4902:)
4897:2
4893:y
4889:+
4884:2
4880:x
4876:(
4873:2
4867:1
4864:,
4861:)
4858:x
4855:w
4849:z
4846:y
4843:(
4840:2
4837:,
4834:)
4831:y
4828:w
4825:+
4822:z
4819:x
4816:(
4813:2
4808:(
4788:z
4770:.
4765:]
4759:)
4754:2
4750:y
4746:+
4741:2
4737:x
4733:(
4730:2
4724:1
4719:)
4716:x
4713:w
4710:+
4707:z
4704:y
4701:(
4698:2
4693:)
4690:y
4687:w
4681:z
4678:x
4675:(
4672:2
4665:)
4662:x
4659:w
4653:z
4650:y
4647:(
4644:2
4639:)
4634:2
4630:z
4626:+
4621:2
4617:x
4613:(
4610:2
4604:1
4599:)
4596:z
4593:w
4590:+
4587:y
4584:x
4581:(
4578:2
4571:)
4568:y
4565:w
4562:+
4559:z
4556:x
4553:(
4550:2
4545:)
4542:z
4539:w
4533:y
4530:x
4527:(
4524:2
4519:)
4514:2
4510:z
4506:+
4501:2
4497:y
4493:(
4490:2
4484:1
4478:[
4459:z
4456:k
4452:y
4449:j
4445:x
4442:i
4438:w
4434:q
4420:R
4416:C
4388:.
4383:]
4375:2
4371:x
4366:i
4357:1
4353:x
4345:4
4341:x
4336:i
4332:+
4327:3
4323:x
4310:4
4306:x
4301:i
4297:+
4292:3
4288:x
4280:2
4276:x
4271:i
4267:+
4262:1
4258:x
4251:[
4231:4
4228:x
4225:k
4221:3
4218:x
4215:j
4211:2
4208:x
4205:i
4201:1
4198:x
4194:q
4176:2
4171:k
4165:q
4154:}
4148:ω
4146:{
4142:}
4140:k
4130:ω
4124:k
4114:k
4108:q
4096:ω
4093:k
4090:ω
4086:ω
4080:k
4076:p
4069:v
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