36:
1986:. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.
3628:
3075:
120:
2368:
3512:
5724:
1989:
Note that the interiors of the 3-balls are not glued to each other. One way to think of the fourth dimension is as a continuous real-valued function of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2-sphere
1165:
is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the
1993:
This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their
5459:
4826:
4397:
4147:
2041:. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius
6252:
A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the two-sphere a
1990:
and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.
2175:
518:
725:
3292:
3309:
1926:
93:
2701:
907:
2919:
2002:
After removing a single point from the 2-sphere, what remains is homeomorphic to the
Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via
3881:
5556:
4969:
4647:
3973:
5249:
4666:
1169:
In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.
4243:
3988:
5824:
5070:
985:
3030:
2782:
2363:{\displaystyle {\begin{aligned}x_{0}&=r\cos \psi \\x_{1}&=r\sin \psi \cos \theta \\x_{2}&=r\sin \psi \sin \theta \cos \varphi \\x_{3}&=r\sin \psi \sin \theta \sin \varphi \end{aligned}}}
277:
539:
3993:
3314:
3176:
2180:
3171:
3507:{\displaystyle {\begin{aligned}x_{0}&=\cos \xi _{1}\sin \eta \\x_{1}&=\sin \xi _{1}\sin \eta \\x_{2}&=\cos \xi _{2}\cos \eta \\x_{3}&=\sin \xi _{2}\cos \eta .\end{aligned}}}
1156:
1994:
boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.
3622:
1826:
177:
It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. This means that a 3-sphere is an example of a
4506:
with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes
1094:
2030:
with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is
2591:
2606:
2432:
762:
2823:
3764:
1194:. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The
5719:{\displaystyle {\begin{pmatrix}e^{i\,\xi _{1}}\sin \eta &e^{i\,\xi _{2}}\cos \eta \\-e^{-i\,\xi _{2}}\cos \eta &e^{-i\,\xi _{1}}\sin \eta \end{pmatrix}}.}
1758:). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the
1166:
hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.
1966:
There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.
4875:
4553:
3892:
5454:{\displaystyle x_{1}+x_{2}i+x_{3}j+x_{4}k\mapsto {\begin{pmatrix}\;\;\,x_{1}+ix_{2}&x_{3}+ix_{4}\\-x_{3}+ix_{4}&x_{1}-ix_{2}\end{pmatrix}}.}
5093:
inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication,
4821:{\displaystyle p=\left({\frac {-1+\|v\|^{2}}{1+\|v\|^{2}}},{\frac {2\mathbf {v} }{1+\|v\|^{2}}}\right)={\frac {-1+\mathbf {v} }{1+\mathbf {v} }}}
4392:{\displaystyle p=\left({\frac {1-\|u\|^{2}}{1+\|u\|^{2}}},{\frac {2\mathbf {u} }{1+\|u\|^{2}}}\right)={\frac {1+\mathbf {u} }{1-\mathbf {u} }}}
4142:{\displaystyle {\begin{aligned}z_{1}&=e^{i\,(\xi _{1}+\xi _{2})}\sin \eta \\z_{2}&=e^{i\,(\xi _{2}-\xi _{1})}\cos \eta .\end{aligned}}}
6194:
6160:
6082:
6042:
2038:
5746:
2045:
are sent to the north pole. Since the open unit disk is homeomorphic to the
Euclidean plane, this is again a one-point compactification.
57:
6338:
5500:
The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the
5020:
918:
79:
103:, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect
3052:
2938:
2724:
6227:
513:{\displaystyle \sum _{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.}
5908:, Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics.
6002:
720:{\displaystyle S^{3}=\left\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb {R} ^{4}:x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\}.}
2048:
The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the
6259:
Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space".
6241:
6030:
3287:{\displaystyle {\begin{aligned}z_{1}&=e^{i\,\xi _{1}}\sin \eta \\z_{2}&=e^{i\,\xi _{2}}\cos \eta .\end{aligned}}}
6107:
2117:
it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use
6348:
6343:
5883:
5960:
1307:
50:
44:
1105:
6353:
5176:
which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that
1782:
1389:
1210:
1099:
while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) is
1979:
3570:
99:
of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is
61:
4205:
3079:
2031:
2004:
1921:{\displaystyle (z_{1},z_{2})\cdot \lambda =(z_{1}\lambda ,z_{2}\lambda )\quad \forall \lambda \in \mathbb {T} }
1775:
96:
1018:
for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of
1801:
1763:
1751:
1195:
262:
2696:{\displaystyle dV=r^{3}\left(\sin ^{2}\psi \,\sin \theta \right)\,d\psi \wedge d\theta \wedge d\varphi .}
1049:
5501:
2144:
2455:
1014:, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See
5887:, Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in
6358:
6119:
5733:
as a exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element
5468:
5224:
1731:
Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural
1673:
1385:
5990:
5851:
5100:
1701:
1697:
1023:
1015:
3729:
above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when
166:, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a
6268:
6223:
2408:
1983:
6059:
2803:
902:{\displaystyle S^{3}=\left\{(z_{1},z_{2})\in \mathbb {C} ^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\right\}}
2914:{\displaystyle \tau =(\cos \theta )i+(\sin \theta \cos \varphi )j+(\sin \theta \sin \varphi )k}
2103:
by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as
1338:
in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.
6299:
6200:
6190:
6156:
6088:
6078:
6038:
6034:
5889:
5198:
1297:
1220:
1019:
1011:
6150:
5729:
Another way to state this result is if we express the matrix representation of an element of
3876:{\displaystyle ds^{2}=d\eta ^{2}+\sin ^{2}\eta \,d\xi _{1}^{2}+\cos ^{2}\eta \,d\xi _{2}^{2}}
6363:
6278:
6127:
5133:
5118:
5005:
4995:
2122:
1683:
1199:
1191:
6020:
5894:
5104:
5084:
5014:. Note that the transition function between these two charts on their overlap is given by
1669:
1342:
1293:
1267:
1183:
737:
163:
6123:
5970:
5965:
5955:
5946:
5478:
5240:
5161:
3979:
1767:
1755:
1739:). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see
1231:
127:
into 3D space and covered with surface grid, showing structure as stack of 3D spheres (
6332:
5898:
5867:
5122:
2446:
1693:
1203:
1179:
1003:
107:
have infinite radius (= straight line). In this picture, the whole 3D space maps the
100:
6233:
6135:
1786:
1747:
1740:
1311:
6245:
1770:. For a general discussion of the number of linear independent vector fields on a
6317:: This article uses the alternate naming scheme for spheres in which a sphere in
6302:
6184:
6024:
5951:
5917:
5490:
5486:
5187:
5154:
5149:
4964:{\displaystyle \mathbf {v} ={\frac {1}{1-x_{0}}}\left(x_{1},x_{2},x_{3}\right).}
4642:{\displaystyle \mathbf {u} ={\frac {1}{1+x_{0}}}\left(x_{1},x_{2},x_{3}\right).}
3968:{\displaystyle dV=\sin \eta \cos \eta \,d\eta \wedge d\xi _{1}\wedge d\xi _{2}.}
3562:
2597:
1933:
1805:
1759:
1335:
1007:
190:
136:
92:
2037:
A somewhat different way to think of the one-point compactification is via the
6016:
5937:
5862:
5514:
4220:
3659:
2792:
2707:
1162:
995:
991:
747:
178:
6247:
The Shape of Space: How to
Visualize Surfaces and Three-dimensional Manifolds
6204:
6092:
5477:
to the set of 2 × 2 complex matrices. It has the property that the
6307:
5933:
5465:
5114:
3627:
3074:
2108:
2049:
1732:
17:
6282:
1306:, but then he himself constructed a non-homeomorphic one, now known as the
1202:, provides that the 3-sphere is the only three-dimensional manifold (up to
111:
of the hypersphere, whereas in the next picture the 3D space contained the
5186:, would form a Lie group, but this fails since octonion multiplication is
1310:. Infinitely many homology spheres are now known to exist. For example, a
5925:
5856:
5183:
3561:. These coordinates are useful in the description of the 3-sphere as the
2134:
2104:
1975:
1187:
156:
119:
5941:
1735:
structure given by quaternion multiplication (see the section below on
6273:
6229:
Experiencing
Geometry: In Euclidean, Spherical, and Hyperbolic Spaces
5921:
4196:
equates to a round trip of the torus in the 2 respective directions.
3752:
2806:. Now the unit imaginary quaternions all lie on the unit 2-sphere in
2715:
1388:
but otherwise follow no discernible pattern. For more discussion see
999:
152:
6131:
5819:{\displaystyle U=\exp \left(\sum _{i=1}^{3}\alpha _{i}J_{i}\right).}
1161:
Every non-empty intersection of a 3-sphere with a three-dimensional
3758:
The round metric on the 3-sphere in these coordinates is given by
3708:
3626:
3073:
118:
91:
5202:. It turns out that the only spheres that are parallelizable are
2097:. As a 3-dimensional manifold one should be able to parameterize
1292:. Any topological space with these homology groups is known as a
4174:
specifies the position along each circle. One round trip (0 to 2
523:
The 3-sphere centered at the origin with radius 1 is called the
5065:{\displaystyle \mathbf {v} ={\frac {1}{\|u\|^{2}}}\mathbf {u} }
3649:) direction, represented by the blue arrow, although the terms
980:{\displaystyle S^{3}=\left\{q\in \mathbb {H} :\|q\|=1\right\}.}
3110:
For unit radius another choice of hyperspherical coordinates,
29:
3640:) direction, represented by the red arrow, and the toroidal (
1300:
conjectured that all homology 3-spheres are homeomorphic to
3025:{\displaystyle q=e^{\tau \psi }=x_{0}+x_{1}i+x_{2}j+x_{3}k}
2777:{\displaystyle q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }
2066:
are redundant since they are subject to the condition that
4204:
Another convenient set of coordinates can be obtained via
2706:
These coordinates have an elegant description in terms of
1700:. As with all spheres, the 3-sphere has constant positive
6226:(2001). "Chapter 20: 3-spheres and hyperbolic 3-spaces".
5125:
Lie group of dimension 3. When thought of as a Lie group
2125:. Some different choices of coordinates are given below.
2034:, round spheres are sent to round spheres or to planes.)
1039:
The 3-dimensional surface volume of a 3-sphere of radius
1766:. It follows that the tangent bundle of the 3-sphere is
5565:
5319:
1746:
Unlike the 2-sphere, the 3-sphere admits nonvanishing
5749:
5559:
5252:
5023:
4878:
4669:
4556:
4246:
3991:
3895:
3767:
3573:
3312:
3174:
2941:
2826:
2727:
2609:
2458:
2411:
2178:
1829:
1108:
1052:
921:
765:
542:
280:
4652:
We could just as well have projected from the point
3982:, make a simple substitution in the equations above
5170:, the set of unit quaternions (The degenerate case
4466:. In the second equality above, we have identified
3106:
and their corresponding fibers with the same color.
2153:. One such choice — by no means unique — is to use
2007:. We first describe the lower-dimensional version.
1762:of the 3-sphere. This implies that the 3-sphere is
5818:
5718:
5453:
5064:
4963:
4820:
4641:
4391:
4141:
3967:
3875:
3616:
3506:
3286:
3024:
2913:
2776:
2695:
2585:
2426:
2362:
1920:
1150:
1088:
979:
901:
719:
512:
5103:. Moreover, since quaternionic multiplication is
2449:on the 3-sphere in these coordinates is given by
1936:of this action is homeomorphic to the two-sphere
1223:that is homeomorphic to the 3-sphere is called a
6232:(second ed.). Prentice-Hall. Archived from
1377:is infinite cyclic. The higher-homotopy groups (
6261:Journal of Computational Design and Engineering
4214:from a pole onto the corresponding equatorial
2010:Rest the south pole of a unit 2-sphere on the
1151:{\displaystyle H={\frac {1}{2}}\pi ^{2}r^{4}.}
6186:Art meets mathematics in the fourth dimension
5906:Art Meets Mathematics in the Fourth Dimension
3078:The Hopf fibration can be visualized using a
1696:on the 3-sphere giving it the structure of a
8:
5993:(1948). "Quaternions et espace elliptique".
5045:
5038:
4765:
4758:
4724:
4717:
4700:
4693:
4339:
4332:
4298:
4291:
4274:
4267:
4223:. For example, if we project from the point
960:
954:
3051:is used to describe spatial rotations (cf.
5323:
5322:
5008:or "patches", which together cover all of
3617:{\displaystyle S^{1}\to S^{3}\to S^{2}.\,}
6272:
6155:. Cambridge: Cambridge University Press.
5802:
5792:
5782:
5771:
5748:
5688:
5683:
5676:
5653:
5648:
5641:
5613:
5608:
5604:
5581:
5576:
5572:
5560:
5558:
5434:
5418:
5406:
5390:
5373:
5357:
5345:
5329:
5324:
5314:
5302:
5286:
5270:
5257:
5251:
5233:, one obtains a matrix representation of
5057:
5048:
5032:
5024:
5022:
4947:
4934:
4921:
4903:
4887:
4879:
4877:
4810:
4797:
4785:
4768:
4745:
4739:
4727:
4703:
4681:
4668:
4625:
4612:
4599:
4581:
4565:
4557:
4555:
4381:
4368:
4359:
4342:
4319:
4313:
4301:
4277:
4258:
4245:
4112:
4099:
4091:
4087:
4070:
4042:
4029:
4021:
4017:
4000:
3992:
3990:
3956:
3940:
3923:
3894:
3867:
3862:
3854:
3842:
3829:
3824:
3816:
3804:
3791:
3775:
3766:
3613:
3604:
3591:
3578:
3572:
3482:
3459:
3436:
3413:
3390:
3367:
3344:
3321:
3313:
3311:
3260:
3255:
3251:
3234:
3209:
3204:
3200:
3183:
3175:
3173:
3013:
2997:
2981:
2968:
2952:
2940:
2825:
2738:
2726:
2665:
2650:
2638:
2623:
2608:
2567:
2559:
2547:
2534:
2510:
2497:
2479:
2466:
2457:
2410:
2313:
2262:
2220:
2187:
2179:
2177:
1914:
1913:
1891:
1875:
1850:
1837:
1828:
1139:
1129:
1115:
1107:
1085:
1079:
1069:
1051:
947:
946:
926:
920:
882:
877:
870:
861:
852:
847:
840:
831:
822:
818:
817:
804:
791:
770:
764:
697:
692:
679:
674:
661:
656:
643:
638:
625:
621:
620:
607:
594:
581:
568:
547:
541:
501:
488:
478:
465:
449:
439:
426:
410:
400:
387:
371:
361:
348:
332:
322:
309:
296:
285:
279:
80:Learn how and when to remove this message
5239:. One convenient choice is given by the
2434:, except for the degenerate cases, when
1394:
43:This article includes a list of general
6244:(1985). "Chapter 14: The Hypersphere".
5982:
4980:coordinates are defined everywhere but
3978:To get the interlocking circles of the
3100:to a ball. This image shows points on
2802:. This is the quaternionic analogue of
2795:; that is, a quaternion that satisfies
2442:, in which case they describe a point.
2014:-plane in three-space. We map a point
1800:giving the 3-sphere the structure of a
6033:. Vol. 2 (7th ed.). Moscow:
5842:are constrained to lie on a 3-sphere.
5829:The condition that the determinant of
2133:It is convenient to have some sort of
756:). The unit 3-sphere is then given by
5897:viewed the Universe in the same way;
5874:, and a 4-sphere is referred to as a
5190:. The octonionic structure does give
5152:that admit a Lie group structure are
2111:). Due to the nontrivial topology of
998:one identifies the 3-sphere with the
7:
6060:"The Flat Torus in the Three-Sphere"
5870:, the 3-sphere is referred to as an
5833:is +1 implies that the coefficients
2393:. Note that, for any fixed value of
2018:of the sphere (minus the north pole
1736:
1209:The 3-sphere is homeomorphic to the
6149:Rovelli, Carlo (9 September 2021).
5227:representation of the quaternions,
3557:can take any values between 0 and 2
2060:The four Euclidean coordinates for
1089:{\displaystyle SV=2\pi ^{2}r^{3}\,}
1016:polar decomposition of a quaternion
6152:General Relativity: The Essentials
3631:A diagram depicting the poloidal (
2928:in this form, the unit quaternion
2405:parameterize a 2-sphere of radius
2056:Coordinate systems on the 3-sphere
1904:
49:it lacks sufficient corresponding
25:
5546:we can then write any element of
3055:), it describes a rotation about
3053:quaternions and spatial rotations
2586:{\displaystyle ds^{2}=r^{2}\left}
1958:, the Hopf bundle is nontrivial.
730:It is often convenient to regard
151:is a 4-dimensional analogue of a
6321:-dimensional space is termed an
6189:(2 ed.). Berlin: Springer.
5160:, thought of as the set of unit
5058:
5025:
4880:
4865:. The inverse of this map takes
4811:
4798:
4746:
4558:
4382:
4369:
4320:
3297:This could also be expressed in
3132:, makes use of the embedding of
2026:to the intersection of the line
1234:of the 3-sphere are as follows:
34:
5866:, a 1965 sequel to Flatland by
3751:, these coordinates describe a
1903:
6073:Schwichtenberg, Jakob (2015).
6003:Pontifical Academy of Sciences
5311:
5083:When considered as the set of
4118:
4092:
4048:
4022:
3597:
3584:
2905:
2884:
2875:
2854:
2845:
2833:
1982:the boundaries of a pair of 3-
1974:A 3-sphere can be constructed
1900:
1868:
1856:
1830:
878:
862:
848:
832:
810:
784:
613:
561:
485:
458:
446:
419:
407:
380:
368:
341:
329:
302:
1:
6031:Course of Theoretical Physics
3707:parameterize a 2-dimensional
1190:without boundary. It is also
5860:, published in 1884, and in
5099:takes on the structure of a
1668:The 3-sphere is naturally a
6112:American Journal of Physics
5884:American Journal of Physics
5524:Using our Hopf coordinates
5148:It turns out that the only
4990:coordinates everywhere but
4470:with a unit quaternion and
2427:{\displaystyle r\sin \psi }
155:, and is the 3-dimensional
6380:
6183:Lipscomb, Stephen (2014).
6106:Peterson, Mark A. (1979).
6026:Classical Theory of Fields
5113:can be regarded as a real
4656:, in which case the point
4165:specify which circle, and
2135:hyperspherical coordinates
2129:Hyperspherical coordinates
2022:) to the plane by sending
1998:One-point compactification
1390:homotopy groups of spheres
1211:one-point compactification
6339:Four-dimensional geometry
4200:Stereographic coordinates
3521:runs over the range 0 to
3144:. In complex coordinates
2793:unit imaginary quaternion
1820:, the action is given by
1774:-sphere, see the article
1206:) with these properties.
229:is the set of all points
193:, a 3-sphere with center
6108:"Dante and the 3-sphere"
5901:supports the same idea.
5196:one important property:
4206:stereographic projection
3080:stereographic projection
2381:run over the range 0 to
2143:in analogy to the usual
2005:stereographic projection
1962:Topological construction
1781:There is an interesting
1776:vector fields on spheres
1308:Poincaré homology sphere
1010:is important for planar
990:This description as the
115:of the bulk hypersphere.
97:Stereographic projection
5493:of the matrix image of
3886:and the volume form by
3667:For any fixed value of
1948:is not homeomorphic to
1802:principal circle bundle
527:and is usually denoted
105:⟨0,0,0,1⟩
64:more precise citations.
5820:
5787:
5720:
5455:
5066:
4965:
4822:
4643:
4393:
4143:
3969:
3877:
3664:
3657:are arbitrary in this
3618:
3508:
3288:
3107:
3026:
2915:
2778:
2710:. Any unit quaternion
2697:
2587:
2428:
2364:
1922:
1288:for all other indices
1174:Topological properties
1152:
1090:
981:
903:
721:
514:
301:
132:
116:
6075:Physics from symmetry
5821:
5767:
5721:
5502:special unitary group
5456:
5067:
4966:
4859:is another vector in
4823:
4644:
4394:
4227:we can write a point
4144:
3970:
3878:
3630:
3619:
3509:
3289:
3094:and then compressing
3077:
3027:
2916:
2779:
2698:
2588:
2429:
2365:
2145:spherical coordinates
2052:of unit quaternions.
1923:
1153:
1091:
1035:Elementary properties
982:
904:
722:
515:
281:
123:Direct projection of
122:
95:
6283:10.1093/jcde/qwab018
6138:on 23 February 2013.
5747:
5557:
5469:algebra homomorphism
5250:
5021:
4876:
4667:
4554:
4244:
3989:
3893:
3765:
3711:. Rings of constant
3571:
3310:
3172:
3059:through an angle of
2939:
2824:
2725:
2714:can be written as a
2607:
2456:
2409:
2176:
1827:
1674:embedded submanifold
1672:, in fact, a closed
1664:Geometric properties
1225:topological 3-sphere
1198:, proved in 2003 by
1106:
1050:
919:
763:
736:as the space with 2
540:
278:
6224:Henderson, David W.
6124:1979AmJPh..47.1031P
6021:Lifshitz, Evgeny M.
5852:Edwin Abbott Abbott
3872:
3834:
1808:. If one thinks of
1702:sectional curvature
1698:Riemannian manifold
1402:
1396:Homotopy groups of
1196:Poincaré conjecture
702:
684:
666:
648:
263:4-dimensional space
162:. In 4-dimensional
27:Mathematical object
6349:Geometric topology
6344:Algebraic topology
6300:Weisstein, Eric W.
6077:. Cham: Springer.
6058:Banchoff, Thomas.
5816:
5740:can be written as
5716:
5707:
5513:as a Lie group is
5464:This map gives an
5451:
5442:
5182:, the set of unit
5062:
5004:consisting of two
4994:. This defines an
4961:
4818:
4639:
4389:
4139:
4137:
3965:
3873:
3858:
3820:
3689:, the coordinates
3665:
3614:
3504:
3502:
3284:
3282:
3108:
3022:
2911:
2774:
2693:
2583:
2424:
2360:
2358:
1918:
1395:
1219:. In general, any
1148:
1086:
1002:in the quaternion
977:
899:
738:complex dimensions
717:
688:
670:
652:
634:
510:
133:
117:
6354:Analytic geometry
6242:Weeks, Jeffrey R.
6196:978-3-319-06254-9
6162:978-1-00-901369-7
6118:(12): 1031–1035.
6084:978-3-319-19201-7
6044:978-5-02-014420-0
5991:Lemaître, Georges
5890:The Divine Comedy
5199:parallelizability
5131:is often denoted
5055:
5006:coordinate charts
4910:
4816:
4775:
4734:
4588:
4387:
4349:
4308:
2123:coordinate charts
1980:"gluing" together
1661:
1660:
1294:homology 3-sphere
1221:topological space
1123:
1012:polar coordinates
90:
89:
82:
16:(Redirected from
6371:
6324:
6320:
6313:
6312:
6286:
6276:
6255:
6237:
6209:
6208:
6180:
6174:
6173:
6171:
6169:
6146:
6140:
6139:
6134:. Archived from
6103:
6097:
6096:
6070:
6064:
6063:
6055:
6049:
6048:
6013:
6007:
6006:
5987:
5841:
5832:
5825:
5823:
5822:
5817:
5812:
5808:
5807:
5806:
5797:
5796:
5786:
5781:
5739:
5732:
5725:
5723:
5722:
5717:
5712:
5711:
5695:
5694:
5693:
5692:
5660:
5659:
5658:
5657:
5620:
5619:
5618:
5617:
5588:
5587:
5586:
5585:
5549:
5545:
5520:
5512:
5506:
5496:
5485:is equal to the
5484:
5481:of a quaternion
5476:
5460:
5458:
5457:
5452:
5447:
5446:
5439:
5438:
5423:
5422:
5411:
5410:
5395:
5394:
5378:
5377:
5362:
5361:
5350:
5349:
5334:
5333:
5307:
5306:
5291:
5290:
5275:
5274:
5262:
5261:
5238:
5232:
5219:
5213:
5207:
5195:
5181:
5175:
5169:
5159:
5144:
5136:
5130:
5112:
5098:
5092:
5085:unit quaternions
5075:and vice versa.
5071:
5069:
5068:
5063:
5061:
5056:
5054:
5053:
5052:
5033:
5028:
5013:
5003:
4993:
4989:
4983:
4979:
4970:
4968:
4967:
4962:
4957:
4953:
4952:
4951:
4939:
4938:
4926:
4925:
4911:
4909:
4908:
4907:
4888:
4883:
4868:
4864:
4858:
4827:
4825:
4824:
4819:
4817:
4815:
4814:
4802:
4801:
4786:
4781:
4777:
4776:
4774:
4773:
4772:
4750:
4749:
4740:
4735:
4733:
4732:
4731:
4709:
4708:
4707:
4682:
4659:
4655:
4648:
4646:
4645:
4640:
4635:
4631:
4630:
4629:
4617:
4616:
4604:
4603:
4589:
4587:
4586:
4585:
4566:
4561:
4546:
4540:
4505:
4469:
4465:
4443:
4435:
4429:
4398:
4396:
4395:
4390:
4388:
4386:
4385:
4373:
4372:
4360:
4355:
4351:
4350:
4348:
4347:
4346:
4324:
4323:
4314:
4309:
4307:
4306:
4305:
4283:
4282:
4281:
4259:
4236:
4230:
4226:
4219:
4213:
4195:
4186:
4177:
4173:
4164:
4155:
4148:
4146:
4145:
4140:
4138:
4122:
4121:
4117:
4116:
4104:
4103:
4075:
4074:
4052:
4051:
4047:
4046:
4034:
4033:
4005:
4004:
3974:
3972:
3971:
3966:
3961:
3960:
3945:
3944:
3882:
3880:
3879:
3874:
3871:
3866:
3847:
3846:
3833:
3828:
3809:
3808:
3796:
3795:
3780:
3779:
3750:
3748:
3747:
3744:
3741:
3740:
3732:
3728:
3719:
3706:
3688:
3686:
3685:
3682:
3679:
3678:
3670:
3648:
3639:
3623:
3621:
3620:
3615:
3609:
3608:
3596:
3595:
3583:
3582:
3560:
3556:
3547:
3538:
3536:
3535:
3532:
3529:
3528:
3520:
3513:
3511:
3510:
3505:
3503:
3487:
3486:
3464:
3463:
3441:
3440:
3418:
3417:
3395:
3394:
3372:
3371:
3349:
3348:
3326:
3325:
3302:
3293:
3291:
3290:
3285:
3283:
3267:
3266:
3265:
3264:
3239:
3238:
3216:
3215:
3214:
3213:
3188:
3187:
3164:
3143:
3137:
3131:
3105:
3099:
3093:
3087:
3070:Hopf coordinates
3065:
3058:
3050:
3043:
3031:
3029:
3028:
3023:
3018:
3017:
3002:
3001:
2986:
2985:
2973:
2972:
2960:
2959:
2931:
2927:
2920:
2918:
2917:
2912:
2817:can be written:
2816:
2812:
2801:
2790:
2783:
2781:
2780:
2775:
2746:
2745:
2713:
2702:
2700:
2699:
2694:
2664:
2660:
2643:
2642:
2628:
2627:
2592:
2590:
2589:
2584:
2582:
2578:
2577:
2573:
2572:
2571:
2552:
2551:
2539:
2538:
2515:
2514:
2502:
2501:
2484:
2483:
2471:
2470:
2441:
2437:
2433:
2431:
2430:
2425:
2404:
2400:
2396:
2392:
2389:runs over 0 to 2
2388:
2384:
2380:
2376:
2369:
2367:
2366:
2361:
2359:
2318:
2317:
2267:
2266:
2225:
2224:
2192:
2191:
2168:
2152:
2142:
2116:
2102:
2096:
2065:
2044:
2029:
2025:
2021:
2017:
2013:
1957:
1947:
1941:
1927:
1925:
1924:
1919:
1917:
1896:
1895:
1880:
1879:
1855:
1854:
1842:
1841:
1819:
1813:
1799:
1793:
1773:
1727:
1723:
1722:
1720:
1719:
1714:
1711:
1691:
1684:Euclidean metric
1681:
1657:
1647:
1630:
1606:
1589:
1572:
1562:
1552:
1542:
1532:
1522:
1512:
1502:
1492:
1476:
1408:
1403:
1401:
1383:
1376:
1364:
1333:
1332:
1330:
1329:
1324:
1321:
1305:
1291:
1287:
1265:
1249:
1218:
1200:Grigori Perelman
1192:simply connected
1186:, 3-dimensional
1178:A 3-sphere is a
1157:
1155:
1154:
1149:
1144:
1143:
1134:
1133:
1124:
1116:
1095:
1093:
1092:
1087:
1084:
1083:
1074:
1073:
1042:
1024:Georges Lemaître
1022:as developed by
986:
984:
983:
978:
973:
969:
950:
931:
930:
908:
906:
905:
900:
898:
894:
887:
886:
881:
875:
874:
865:
857:
856:
851:
845:
844:
835:
827:
826:
821:
809:
808:
796:
795:
775:
774:
755:
745:
735:
726:
724:
723:
718:
713:
709:
701:
696:
683:
678:
665:
660:
647:
642:
630:
629:
624:
612:
611:
599:
598:
586:
585:
573:
572:
552:
551:
532:
519:
517:
516:
511:
506:
505:
493:
492:
483:
482:
470:
469:
454:
453:
444:
443:
431:
430:
415:
414:
405:
404:
392:
391:
376:
375:
366:
365:
353:
352:
337:
336:
327:
326:
314:
313:
300:
295:
270:
260:
228:
224:
106:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
6379:
6378:
6374:
6373:
6372:
6370:
6369:
6368:
6329:
6328:
6322:
6318:
6298:
6297:
6294:
6289:
6258:
6240:
6222:
6218:
6216:Further reading
6213:
6212:
6197:
6182:
6181:
6177:
6167:
6165:
6163:
6148:
6147:
6143:
6132:10.1119/1.11968
6105:
6104:
6100:
6085:
6072:
6071:
6067:
6057:
6056:
6052:
6045:
6037:. p. 385.
6015:
6014:
6010:
5989:
5988:
5984:
5979:
5961:Poincaré sphere
5914:
5881:Writing in the
5848:
5840:
5834:
5830:
5798:
5788:
5766:
5762:
5745:
5744:
5734:
5730:
5706:
5705:
5684:
5672:
5670:
5649:
5637:
5631:
5630:
5609:
5600:
5598:
5577:
5568:
5561:
5555:
5554:
5547:
5543:
5536:
5525:
5518:
5508:
5504:
5494:
5482:
5472:
5441:
5440:
5430:
5414:
5412:
5402:
5386:
5380:
5379:
5369:
5353:
5351:
5341:
5325:
5315:
5298:
5282:
5266:
5253:
5248:
5247:
5234:
5228:
5215:
5209:
5203:
5191:
5177:
5171:
5165:
5162:complex numbers
5153:
5138:
5132:
5126:
5108:
5094:
5088:
5081:
5079:Group structure
5044:
5037:
5019:
5018:
5009:
4999:
4991:
4985:
4981:
4975:
4943:
4930:
4917:
4916:
4912:
4899:
4892:
4874:
4873:
4866:
4860:
4856:
4849:
4842:
4832:
4803:
4787:
4764:
4751:
4741:
4723:
4710:
4699:
4683:
4680:
4676:
4665:
4664:
4657:
4653:
4621:
4608:
4595:
4594:
4590:
4577:
4570:
4552:
4551:
4542:
4538:
4531:
4524:
4517:
4507:
4501:
4491:
4481:
4471:
4467:
4464:
4457:
4450:
4439:
4437:
4431:
4430:is a vector in
4427:
4420:
4413:
4403:
4374:
4361:
4338:
4325:
4315:
4297:
4284:
4273:
4260:
4257:
4253:
4242:
4241:
4232:
4228:
4224:
4215:
4209:
4202:
4194:
4188:
4185:
4179:
4175:
4172:
4166:
4163:
4157:
4153:
4136:
4135:
4108:
4095:
4083:
4076:
4066:
4063:
4062:
4038:
4025:
4013:
4006:
3996:
3987:
3986:
3952:
3936:
3891:
3890:
3838:
3800:
3787:
3771:
3763:
3762:
3745:
3742:
3738:
3737:
3736:
3734:
3730:
3727:
3721:
3718:
3712:
3704:
3697:
3690:
3683:
3680:
3676:
3675:
3674:
3672:
3668:
3647:
3641:
3638:
3632:
3600:
3587:
3574:
3569:
3568:
3558:
3555:
3549:
3546:
3540:
3533:
3530:
3526:
3525:
3524:
3522:
3518:
3501:
3500:
3478:
3465:
3455:
3452:
3451:
3432:
3419:
3409:
3406:
3405:
3386:
3373:
3363:
3360:
3359:
3340:
3327:
3317:
3308:
3307:
3298:
3281:
3280:
3256:
3247:
3240:
3230:
3227:
3226:
3205:
3196:
3189:
3179:
3170:
3169:
3159:
3152:
3145:
3139:
3133:
3129:
3122:
3111:
3101:
3095:
3089:
3083:
3072:
3060:
3056:
3048:
3042:
3036:
3009:
2993:
2977:
2964:
2948:
2937:
2936:
2929:
2925:
2822:
2821:
2814:
2807:
2804:Euler's formula
2796:
2788:
2734:
2723:
2722:
2711:
2634:
2633:
2629:
2619:
2605:
2604:
2563:
2543:
2530:
2526:
2522:
2506:
2493:
2489:
2485:
2475:
2462:
2454:
2453:
2439:
2435:
2407:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2357:
2356:
2319:
2309:
2306:
2305:
2268:
2258:
2255:
2254:
2226:
2216:
2213:
2212:
2193:
2183:
2174:
2173:
2154:
2148:
2138:
2131:
2112:
2098:
2094:
2087:
2080:
2073:
2067:
2061:
2058:
2042:
2039:exponential map
2027:
2023:
2019:
2015:
2011:
2000:
1972:
1964:
1949:
1943:
1937:
1887:
1871:
1846:
1833:
1825:
1824:
1815:
1814:as a subset of
1809:
1795:
1789:
1771:
1737:group structure
1728:is the radius.
1725:
1715:
1712:
1709:
1708:
1706:
1705:
1687:
1677:
1670:smooth manifold
1666:
1656:
1650:
1646:
1639:
1633:
1629:
1622:
1615:
1609:
1605:
1598:
1592:
1588:
1581:
1575:
1571:
1565:
1561:
1555:
1551:
1545:
1541:
1535:
1531:
1525:
1521:
1515:
1511:
1505:
1501:
1495:
1488:
1470:
1464:
1406:
1397:
1378:
1370:
1366:
1358:
1350:
1346:
1343:homotopy groups
1325:
1322:
1319:
1318:
1316:
1315:
1301:
1289:
1277:
1271:
1268:infinite cyclic
1255:
1251:
1239:
1235:
1232:homology groups
1214:
1176:
1135:
1125:
1104:
1103:
1075:
1065:
1048:
1047:
1040:
1037:
1032:
939:
935:
922:
917:
916:
876:
866:
846:
836:
816:
800:
787:
783:
779:
766:
761:
760:
751:
741:
731:
619:
603:
590:
577:
564:
560:
556:
543:
538:
537:
528:
497:
484:
474:
461:
445:
435:
422:
406:
396:
383:
367:
357:
344:
328:
318:
305:
276:
275:
266:
258:
251:
244:
237:
230:
226:
222:
215:
208:
201:
194:
187:
164:Euclidean space
104:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
6377:
6375:
6367:
6366:
6361:
6356:
6351:
6346:
6341:
6331:
6330:
6327:
6326:
6293:
6292:External links
6290:
6288:
6287:
6267:(3): 836–854.
6256:
6238:
6236:on 2018-06-19.
6219:
6217:
6214:
6211:
6210:
6195:
6175:
6161:
6141:
6098:
6083:
6065:
6050:
6043:
6017:Landau, Lev D.
6008:
5981:
5980:
5978:
5975:
5974:
5973:
5971:Clifford torus
5968:
5966:Reeb foliation
5963:
5958:
5956:Riemann sphere
5949:
5947:Pauli matrices
5944:
5931:
5913:
5910:
5893:that suggests
5847:
5844:
5838:
5827:
5826:
5815:
5811:
5805:
5801:
5795:
5791:
5785:
5780:
5777:
5774:
5770:
5765:
5761:
5758:
5755:
5752:
5727:
5726:
5715:
5710:
5704:
5701:
5698:
5691:
5687:
5682:
5679:
5675:
5671:
5669:
5666:
5663:
5656:
5652:
5647:
5644:
5640:
5636:
5633:
5632:
5629:
5626:
5623:
5616:
5612:
5607:
5603:
5599:
5597:
5594:
5591:
5584:
5580:
5575:
5571:
5567:
5566:
5564:
5541:
5534:
5479:absolute value
5462:
5461:
5450:
5445:
5437:
5433:
5429:
5426:
5421:
5417:
5413:
5409:
5405:
5401:
5398:
5393:
5389:
5385:
5382:
5381:
5376:
5372:
5368:
5365:
5360:
5356:
5352:
5348:
5344:
5340:
5337:
5332:
5328:
5321:
5320:
5318:
5313:
5310:
5305:
5301:
5297:
5294:
5289:
5285:
5281:
5278:
5273:
5269:
5265:
5260:
5256:
5241:Pauli matrices
5188:nonassociative
5080:
5077:
5073:
5072:
5060:
5051:
5047:
5043:
5040:
5036:
5031:
5027:
4974:Note that the
4972:
4971:
4960:
4956:
4950:
4946:
4942:
4937:
4933:
4929:
4924:
4920:
4915:
4906:
4902:
4898:
4895:
4891:
4886:
4882:
4854:
4847:
4840:
4829:
4828:
4813:
4809:
4806:
4800:
4796:
4793:
4790:
4784:
4780:
4771:
4767:
4763:
4760:
4757:
4754:
4748:
4744:
4738:
4730:
4726:
4722:
4719:
4716:
4713:
4706:
4702:
4698:
4695:
4692:
4689:
4686:
4679:
4675:
4672:
4650:
4649:
4638:
4634:
4628:
4624:
4620:
4615:
4611:
4607:
4602:
4598:
4593:
4584:
4580:
4576:
4573:
4569:
4564:
4560:
4536:
4529:
4522:
4515:
4499:
4489:
4479:
4462:
4455:
4448:
4425:
4418:
4411:
4400:
4399:
4384:
4380:
4377:
4371:
4367:
4364:
4358:
4354:
4345:
4341:
4337:
4334:
4331:
4328:
4322:
4318:
4312:
4304:
4300:
4296:
4293:
4290:
4287:
4280:
4276:
4272:
4269:
4266:
4263:
4256:
4252:
4249:
4201:
4198:
4192:
4183:
4170:
4161:
4150:
4149:
4134:
4131:
4128:
4125:
4120:
4115:
4111:
4107:
4102:
4098:
4094:
4090:
4086:
4082:
4079:
4077:
4073:
4069:
4065:
4064:
4061:
4058:
4055:
4050:
4045:
4041:
4037:
4032:
4028:
4024:
4020:
4016:
4012:
4009:
4007:
4003:
3999:
3995:
3994:
3980:Hopf fibration
3976:
3975:
3964:
3959:
3955:
3951:
3948:
3943:
3939:
3935:
3932:
3929:
3926:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3884:
3883:
3870:
3865:
3861:
3857:
3853:
3850:
3845:
3841:
3837:
3832:
3827:
3823:
3819:
3815:
3812:
3807:
3803:
3799:
3794:
3790:
3786:
3783:
3778:
3774:
3770:
3725:
3716:
3702:
3695:
3671:between 0 and
3645:
3636:
3625:
3624:
3612:
3607:
3603:
3599:
3594:
3590:
3586:
3581:
3577:
3553:
3544:
3515:
3514:
3499:
3496:
3493:
3490:
3485:
3481:
3477:
3474:
3471:
3468:
3466:
3462:
3458:
3454:
3453:
3450:
3447:
3444:
3439:
3435:
3431:
3428:
3425:
3422:
3420:
3416:
3412:
3408:
3407:
3404:
3401:
3398:
3393:
3389:
3385:
3382:
3379:
3376:
3374:
3370:
3366:
3362:
3361:
3358:
3355:
3352:
3347:
3343:
3339:
3336:
3333:
3330:
3328:
3324:
3320:
3316:
3315:
3295:
3294:
3279:
3276:
3273:
3270:
3263:
3259:
3254:
3250:
3246:
3243:
3241:
3237:
3233:
3229:
3228:
3225:
3222:
3219:
3212:
3208:
3203:
3199:
3195:
3192:
3190:
3186:
3182:
3178:
3177:
3157:
3150:
3127:
3120:
3071:
3068:
3044:are as above.
3040:
3033:
3032:
3021:
3016:
3012:
3008:
3005:
3000:
2996:
2992:
2989:
2984:
2980:
2976:
2971:
2967:
2963:
2958:
2955:
2951:
2947:
2944:
2922:
2921:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2785:
2784:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2744:
2741:
2737:
2733:
2730:
2704:
2703:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2663:
2659:
2656:
2653:
2649:
2646:
2641:
2637:
2632:
2626:
2622:
2618:
2615:
2612:
2594:
2593:
2581:
2576:
2570:
2566:
2562:
2558:
2555:
2550:
2546:
2542:
2537:
2533:
2529:
2525:
2521:
2518:
2513:
2509:
2505:
2500:
2496:
2492:
2488:
2482:
2478:
2474:
2469:
2465:
2461:
2423:
2420:
2417:
2414:
2371:
2370:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2320:
2316:
2312:
2308:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2269:
2265:
2261:
2257:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2227:
2223:
2219:
2215:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2194:
2190:
2186:
2182:
2181:
2130:
2127:
2092:
2085:
2078:
2071:
2057:
2054:
1999:
1996:
1971:
1968:
1963:
1960:
1930:
1929:
1916:
1912:
1909:
1906:
1902:
1899:
1894:
1890:
1886:
1883:
1878:
1874:
1870:
1867:
1864:
1861:
1858:
1853:
1849:
1845:
1840:
1836:
1832:
1764:parallelizable
1756:tangent bundle
1665:
1662:
1659:
1658:
1654:
1648:
1644:
1637:
1631:
1627:
1620:
1613:
1607:
1603:
1596:
1590:
1586:
1579:
1573:
1569:
1563:
1559:
1553:
1549:
1543:
1539:
1533:
1529:
1523:
1519:
1513:
1509:
1503:
1499:
1493:
1486:
1483:
1480:
1477:
1466:
1461:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1386:finite abelian
1368:
1356:
1348:
1273:
1253:
1237:
1175:
1172:
1159:
1158:
1147:
1142:
1138:
1132:
1128:
1122:
1119:
1114:
1111:
1097:
1096:
1082:
1078:
1072:
1068:
1064:
1061:
1058:
1055:
1036:
1033:
1031:
1028:
1020:elliptic space
1006:. Just as the
988:
987:
976:
972:
968:
965:
962:
959:
956:
953:
949:
945:
942:
938:
934:
929:
925:
910:
909:
897:
893:
890:
885:
880:
873:
869:
864:
860:
855:
850:
843:
839:
834:
830:
825:
820:
815:
812:
807:
803:
799:
794:
790:
786:
782:
778:
773:
769:
728:
727:
716:
712:
708:
705:
700:
695:
691:
687:
682:
677:
673:
669:
664:
659:
655:
651:
646:
641:
637:
633:
628:
623:
618:
615:
610:
606:
602:
597:
593:
589:
584:
580:
576:
571:
567:
563:
559:
555:
550:
546:
521:
520:
509:
504:
500:
496:
491:
487:
481:
477:
473:
468:
464:
460:
457:
452:
448:
442:
438:
434:
429:
425:
421:
418:
413:
409:
403:
399:
395:
390:
386:
382:
379:
374:
370:
364:
360:
356:
351:
347:
343:
340:
335:
331:
325:
321:
317:
312:
308:
304:
299:
294:
291:
288:
284:
256:
249:
242:
235:
220:
213:
206:
199:
186:
183:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6376:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6336:
6334:
6316:
6310:
6309:
6304:
6303:"Hypersphere"
6301:
6296:
6295:
6291:
6284:
6280:
6275:
6270:
6266:
6262:
6257:
6254:
6253:three-sphere.
6249:
6248:
6243:
6239:
6235:
6231:
6230:
6225:
6221:
6220:
6215:
6206:
6202:
6198:
6192:
6188:
6187:
6179:
6176:
6164:
6158:
6154:
6153:
6145:
6142:
6137:
6133:
6129:
6125:
6121:
6117:
6113:
6109:
6102:
6099:
6094:
6090:
6086:
6080:
6076:
6069:
6066:
6061:
6054:
6051:
6046:
6040:
6036:
6032:
6028:
6027:
6022:
6018:
6012:
6009:
6004:
6000:
5996:
5992:
5986:
5983:
5976:
5972:
5969:
5967:
5964:
5962:
5959:
5957:
5953:
5950:
5948:
5945:
5943:
5939:
5935:
5932:
5930:
5928:
5923:
5919:
5916:
5915:
5911:
5909:
5907:
5902:
5900:
5899:Carlo Rovelli
5896:
5892:
5891:
5886:
5885:
5879:
5877:
5873:
5869:
5868:Dionys Burger
5865:
5864:
5859:
5858:
5853:
5846:In literature
5845:
5843:
5837:
5813:
5809:
5803:
5799:
5793:
5789:
5783:
5778:
5775:
5772:
5768:
5763:
5759:
5756:
5753:
5750:
5743:
5742:
5741:
5737:
5713:
5708:
5702:
5699:
5696:
5689:
5685:
5680:
5677:
5673:
5667:
5664:
5661:
5654:
5650:
5645:
5642:
5638:
5634:
5627:
5624:
5621:
5614:
5610:
5605:
5601:
5595:
5592:
5589:
5582:
5578:
5573:
5569:
5562:
5553:
5552:
5551:
5540:
5533:
5529:
5522:
5516:
5511:
5503:
5498:
5492:
5488:
5480:
5475:
5470:
5467:
5448:
5443:
5435:
5431:
5427:
5424:
5419:
5415:
5407:
5403:
5399:
5396:
5391:
5387:
5383:
5374:
5370:
5366:
5363:
5358:
5354:
5346:
5342:
5338:
5335:
5330:
5326:
5316:
5308:
5303:
5299:
5295:
5292:
5287:
5283:
5279:
5276:
5271:
5267:
5263:
5258:
5254:
5246:
5245:
5244:
5242:
5237:
5231:
5226:
5221:
5218:
5212:
5206:
5201:
5200:
5194:
5189:
5185:
5180:
5174:
5168:
5163:
5158:
5157:
5151:
5146:
5142:
5135:
5129:
5124:
5120:
5116:
5111:
5106:
5102:
5097:
5091:
5086:
5078:
5076:
5049:
5041:
5034:
5029:
5017:
5016:
5015:
5012:
5007:
5002:
4997:
4988:
4982:(−1, 0, 0, 0)
4978:
4958:
4954:
4948:
4944:
4940:
4935:
4931:
4927:
4922:
4918:
4913:
4904:
4900:
4896:
4893:
4889:
4884:
4872:
4871:
4870:
4863:
4853:
4846:
4839:
4835:
4807:
4804:
4794:
4791:
4788:
4782:
4778:
4769:
4761:
4755:
4752:
4742:
4736:
4728:
4720:
4714:
4711:
4704:
4696:
4690:
4687:
4684:
4677:
4673:
4670:
4663:
4662:
4661:
4636:
4632:
4626:
4622:
4618:
4613:
4609:
4605:
4600:
4596:
4591:
4582:
4578:
4574:
4571:
4567:
4562:
4550:
4549:
4548:
4545:
4535:
4528:
4521:
4514:
4510:
4504:
4498:
4494:
4488:
4484:
4478:
4474:
4461:
4454:
4447:
4442:
4434:
4424:
4417:
4410:
4406:
4378:
4375:
4365:
4362:
4356:
4352:
4343:
4335:
4329:
4326:
4316:
4310:
4302:
4294:
4288:
4285:
4278:
4270:
4264:
4261:
4254:
4250:
4247:
4240:
4239:
4238:
4235:
4225:(−1, 0, 0, 0)
4222:
4218:
4212:
4207:
4199:
4197:
4191:
4182:
4169:
4160:
4152:In this case
4132:
4129:
4126:
4123:
4113:
4109:
4105:
4100:
4096:
4088:
4084:
4080:
4078:
4071:
4067:
4059:
4056:
4053:
4043:
4039:
4035:
4030:
4026:
4018:
4014:
4010:
4008:
4001:
3997:
3985:
3984:
3983:
3981:
3962:
3957:
3953:
3949:
3946:
3941:
3937:
3933:
3930:
3927:
3924:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3889:
3888:
3887:
3868:
3863:
3859:
3855:
3851:
3848:
3843:
3839:
3835:
3830:
3825:
3821:
3817:
3813:
3810:
3805:
3801:
3797:
3792:
3788:
3784:
3781:
3776:
3772:
3768:
3761:
3760:
3759:
3756:
3754:
3724:
3715:
3710:
3701:
3694:
3662:
3661:
3656:
3652:
3644:
3635:
3629:
3610:
3605:
3601:
3592:
3588:
3579:
3575:
3567:
3566:
3565:
3564:
3552:
3543:
3497:
3494:
3491:
3488:
3483:
3479:
3475:
3472:
3469:
3467:
3460:
3456:
3448:
3445:
3442:
3437:
3433:
3429:
3426:
3423:
3421:
3414:
3410:
3402:
3399:
3396:
3391:
3387:
3383:
3380:
3377:
3375:
3368:
3364:
3356:
3353:
3350:
3345:
3341:
3337:
3334:
3331:
3329:
3322:
3318:
3306:
3305:
3304:
3301:
3277:
3274:
3271:
3268:
3261:
3257:
3252:
3248:
3244:
3242:
3235:
3231:
3223:
3220:
3217:
3210:
3206:
3201:
3197:
3193:
3191:
3184:
3180:
3168:
3167:
3166:
3163:
3156:
3149:
3142:
3136:
3126:
3119:
3115:
3104:
3098:
3092:
3086:
3081:
3076:
3069:
3067:
3064:
3054:
3045:
3039:
3019:
3014:
3010:
3006:
3003:
2998:
2994:
2990:
2987:
2982:
2978:
2974:
2969:
2965:
2961:
2956:
2953:
2949:
2945:
2942:
2935:
2934:
2933:
2908:
2902:
2899:
2896:
2893:
2890:
2887:
2881:
2878:
2872:
2869:
2866:
2863:
2860:
2857:
2851:
2848:
2842:
2839:
2836:
2830:
2827:
2820:
2819:
2818:
2811:
2805:
2799:
2794:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2742:
2739:
2735:
2731:
2728:
2721:
2720:
2719:
2717:
2709:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2661:
2657:
2654:
2651:
2647:
2644:
2639:
2635:
2630:
2624:
2620:
2616:
2613:
2610:
2603:
2602:
2601:
2599:
2579:
2574:
2568:
2564:
2560:
2556:
2553:
2548:
2544:
2540:
2535:
2531:
2527:
2523:
2519:
2516:
2511:
2507:
2503:
2498:
2494:
2490:
2486:
2480:
2476:
2472:
2467:
2463:
2459:
2452:
2451:
2450:
2448:
2443:
2421:
2418:
2415:
2412:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2321:
2314:
2310:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2270:
2263:
2259:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2228:
2221:
2217:
2209:
2206:
2203:
2200:
2197:
2195:
2188:
2184:
2172:
2171:
2170:
2166:
2162:
2158:
2151:
2146:
2141:
2136:
2128:
2126:
2124:
2120:
2115:
2110:
2106:
2101:
2091:
2084:
2077:
2070:
2064:
2055:
2053:
2051:
2046:
2040:
2035:
2033:
2008:
2006:
1997:
1995:
1991:
1987:
1985:
1981:
1977:
1976:topologically
1969:
1967:
1961:
1959:
1956:
1952:
1946:
1940:
1935:
1910:
1907:
1897:
1892:
1888:
1884:
1881:
1876:
1872:
1865:
1862:
1859:
1851:
1847:
1843:
1838:
1834:
1823:
1822:
1821:
1818:
1812:
1807:
1804:known as the
1803:
1798:
1792:
1788:
1784:
1779:
1777:
1769:
1765:
1761:
1757:
1753:
1749:
1748:vector fields
1744:
1742:
1738:
1734:
1729:
1718:
1703:
1699:
1695:
1690:
1685:
1680:
1675:
1671:
1663:
1653:
1649:
1643:
1636:
1632:
1626:
1619:
1612:
1608:
1602:
1595:
1591:
1585:
1578:
1574:
1568:
1564:
1558:
1554:
1548:
1544:
1538:
1534:
1528:
1524:
1518:
1514:
1508:
1504:
1498:
1494:
1491:
1487:
1484:
1481:
1478:
1474:
1469:
1463:
1462:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1405:
1404:
1400:
1393:
1391:
1387:
1381:
1374:
1362:
1354:
1344:
1339:
1337:
1328:
1313:
1309:
1304:
1299:
1295:
1285:
1281:
1276:
1269:
1263:
1259:
1247:
1243:
1233:
1228:
1226:
1222:
1217:
1212:
1207:
1205:
1204:homeomorphism
1201:
1197:
1193:
1189:
1185:
1181:
1173:
1171:
1167:
1164:
1145:
1140:
1136:
1130:
1126:
1120:
1117:
1112:
1109:
1102:
1101:
1100:
1080:
1076:
1070:
1066:
1062:
1059:
1056:
1053:
1046:
1045:
1044:
1034:
1029:
1027:
1025:
1021:
1017:
1013:
1009:
1005:
1004:division ring
1001:
997:
993:
974:
970:
966:
963:
957:
951:
943:
940:
936:
932:
927:
923:
915:
914:
913:
895:
891:
888:
883:
871:
867:
858:
853:
841:
837:
828:
823:
813:
805:
801:
797:
792:
788:
780:
776:
771:
767:
759:
758:
757:
754:
749:
744:
739:
734:
714:
710:
706:
703:
698:
693:
689:
685:
680:
675:
671:
667:
662:
657:
653:
649:
644:
639:
635:
631:
626:
616:
608:
604:
600:
595:
591:
587:
582:
578:
574:
569:
565:
557:
553:
548:
544:
536:
535:
534:
531:
526:
525:unit 3-sphere
507:
502:
498:
494:
489:
479:
475:
471:
466:
462:
455:
450:
440:
436:
432:
427:
423:
416:
411:
401:
397:
393:
388:
384:
377:
372:
362:
358:
354:
349:
345:
338:
333:
323:
319:
315:
310:
306:
297:
292:
289:
286:
282:
274:
273:
272:
269:
264:
255:
248:
241:
234:
219:
212:
205:
198:
192:
184:
182:
180:
175:
173:
169:
165:
161:
159:
154:
150:
146:
142:
138:
130:
126:
121:
114:
110:
102:
98:
94:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
6314:
6306:
6274:2003.09236v2
6264:
6260:
6251:
6246:
6234:the original
6228:
6185:
6178:
6168:13 September
6166:. Retrieved
6151:
6144:
6136:the original
6115:
6111:
6101:
6074:
6068:
6053:
6025:
6011:
5998:
5994:
5985:
5926:
5905:
5903:
5888:
5882:
5880:
5875:
5871:
5861:
5855:
5849:
5835:
5828:
5735:
5728:
5550:in the form
5538:
5531:
5527:
5523:
5509:
5499:
5473:
5463:
5235:
5229:
5222:
5216:
5210:
5204:
5197:
5192:
5178:
5172:
5166:
5155:
5147:
5140:
5127:
5109:
5095:
5089:
5082:
5074:
5010:
5000:
4992:(1, 0, 0, 0)
4986:
4976:
4973:
4861:
4851:
4844:
4837:
4833:
4830:
4660:is given by
4654:(1, 0, 0, 0)
4651:
4543:
4533:
4526:
4519:
4512:
4508:
4502:
4496:
4492:
4486:
4482:
4476:
4472:
4459:
4452:
4445:
4440:
4432:
4422:
4415:
4408:
4404:
4401:
4233:
4216:
4210:
4203:
4189:
4180:
4167:
4158:
4151:
3977:
3885:
3757:
3733:equals 0 or
3722:
3713:
3699:
3692:
3666:
3658:
3654:
3650:
3642:
3633:
3550:
3541:
3516:
3299:
3296:
3161:
3154:
3147:
3140:
3134:
3124:
3117:
3113:
3109:
3102:
3096:
3090:
3084:
3062:
3046:
3037:
3034:
2932:is given by
2923:
2813:so any such
2809:
2797:
2786:
2705:
2595:
2447:round metric
2444:
2438:equals 0 or
2372:
2164:
2160:
2156:
2149:
2139:
2132:
2118:
2113:
2099:
2089:
2082:
2075:
2068:
2062:
2059:
2047:
2036:
2009:
2001:
1992:
1988:
1973:
1965:
1954:
1950:
1944:
1938:
1931:
1816:
1810:
1796:
1790:
1787:circle group
1780:
1745:
1741:circle group
1730:
1716:
1688:
1678:
1667:
1651:
1641:
1634:
1624:
1617:
1610:
1600:
1593:
1583:
1576:
1566:
1556:
1546:
1536:
1526:
1516:
1506:
1496:
1489:
1472:
1467:
1398:
1379:
1372:
1360:
1352:
1340:
1326:
1312:Dehn filling
1302:
1296:. Initially
1283:
1279:
1274:
1261:
1257:
1245:
1241:
1229:
1224:
1215:
1208:
1177:
1168:
1160:
1098:
1038:
989:
911:
752:
742:
732:
729:
529:
524:
522:
271:) such that
267:
253:
246:
239:
232:
217:
210:
203:
196:
188:
176:
171:
167:
157:
148:
144:
140:
134:
128:
124:
112:
108:
76:
67:
48:
18:Three-sphere
6359:Quaternions
5952:Hopf bundle
5876:hypersphere
5491:determinant
5487:square root
5223:By using a
4444:‖ =
3563:Hopf bundle
2708:quaternions
2598:volume form
1934:orbit space
1806:Hopf bundle
1760:Lie algebra
1314:with slope
1008:unit circle
992:quaternions
748:quaternions
225:and radius
191:coordinates
141:hypersphere
137:mathematics
62:introducing
6333:Categories
5977:References
5938:polychoron
5872:oversphere
5863:Sphereland
5515:isomorphic
5119:nonabelian
5117:. It is a
4221:hyperplane
3660:flat torus
1692:induces a
1384:) are all
1345:, we have
1341:As to the
1163:hyperplane
1030:Properties
185:Definition
179:3-manifold
45:references
6308:MathWorld
6205:893872366
6093:910917227
5934:tesseract
5790:α
5769:∑
5760:
5703:η
5700:
5686:ξ
5678:−
5668:η
5665:
5651:ξ
5643:−
5635:−
5628:η
5625:
5611:ξ
5596:η
5593:
5579:ξ
5466:injective
5425:−
5384:−
5312:↦
5184:octonions
5115:Lie group
5046:‖
5039:‖
4897:−
4789:−
4766:‖
4759:‖
4725:‖
4718:‖
4701:‖
4694:‖
4685:−
4379:−
4340:‖
4333:‖
4299:‖
4292:‖
4275:‖
4268:‖
4265:−
4130:η
4127:
4110:ξ
4106:−
4097:ξ
4060:η
4057:
4040:ξ
4027:ξ
3954:ξ
3947:∧
3938:ξ
3931:∧
3928:η
3921:η
3918:
3912:η
3909:
3860:ξ
3852:η
3849:
3822:ξ
3814:η
3811:
3789:η
3598:→
3585:→
3495:η
3492:
3480:ξ
3476:
3449:η
3446:
3434:ξ
3430:
3403:η
3400:
3388:ξ
3384:
3357:η
3354:
3342:ξ
3338:
3275:η
3272:
3258:ξ
3224:η
3221:
3207:ξ
3165:we write
2957:ψ
2954:τ
2903:φ
2900:
2894:θ
2891:
2873:φ
2870:
2864:θ
2861:
2843:θ
2840:
2828:τ
2772:ψ
2769:
2763:τ
2757:ψ
2754:
2743:ψ
2740:τ
2688:φ
2682:∧
2679:θ
2673:∧
2670:ψ
2658:θ
2655:
2648:ψ
2645:
2565:φ
2557:θ
2554:
2532:θ
2520:ψ
2517:
2495:ψ
2422:ψ
2419:
2354:φ
2351:
2345:θ
2342:
2336:ψ
2333:
2303:φ
2300:
2294:θ
2291:
2285:ψ
2282:
2252:θ
2249:
2243:ψ
2240:
2210:ψ
2207:
2109:longitude
2050:Lie group
2032:conformal
1911:∈
1908:λ
1905:∀
1898:λ
1882:λ
1863:λ
1860:⋅
1733:Lie group
1704:equal to
1266:are both
1184:connected
1127:π
1067:π
961:‖
955:‖
944:∈
814:∈
746:) or the
617:∈
472:−
433:−
394:−
355:−
316:−
283:∑
261:in real,
129:2-spheres
101:conformal
70:June 2016
6325:-sphere.
6023:(1988).
6005:: 57–78.
5922:2-sphere
5918:1-sphere
5912:See also
5857:Flatland
5507:. Thus,
4984:and the
4438:‖
3655:toroidal
3651:poloidal
2596:and the
2169:, where
2119:at least
2105:latitude
1942:. Since
1752:sections
1298:Poincaré
1270:, while
1188:manifold
145:3-sphere
125:3-sphere
6364:Spheres
6120:Bibcode
5942:simplex
5929:-sphere
5738:∈ SU(2)
5489:of the
5150:spheres
5123:compact
3749:
3735:
3687:
3673:
3537:
3523:
3041:0,1,2,3
1785:of the
1768:trivial
1754:of its
1721:
1707:
1334:on any
1331:
1317:
1180:compact
1000:versors
170:, or a
160:-sphere
109:surface
58:improve
6203:
6193:
6159:
6091:
6081:
6041:
5225:matrix
5214:, and
5164:, and
5105:smooth
4831:where
4402:where
4156:, and
3753:circle
3539:, and
3035:where
2787:where
2716:versor
2385:, and
2373:where
1970:Gluing
1783:action
1724:where
1694:metric
1682:. The
1363:) = {}
1286:) = {}
172:gongyl
168:4-ball
153:sphere
113:shadow
47:, but
6269:arXiv
6035:Nauka
5895:Dante
5731:SU(2)
5548:SU(2)
5519:SU(2)
5505:SU(2)
5471:from
5139:U(1,
5134:Sp(1)
5101:group
4996:atlas
4178:) of
3709:torus
3663:case.
3517:Here
3047:When
2924:With
2791:is a
1984:balls
1355:) = π
149:glome
147:, or
6315:Note
6201:OCLC
6191:ISBN
6170:2021
6157:ISBN
6089:OCLC
6079:ISBN
6039:ISBN
5995:Acta
4436:and
3720:and
3653:and
3548:and
3160:) ∈
2800:= −1
2445:The
2401:and
2377:and
2121:two
2107:and
1932:The
1365:and
1336:knot
1250:and
1230:The
996:norm
139:, a
6279:doi
6128:doi
5904:In
5854:'s
5850:In
5757:exp
5697:sin
5662:cos
5622:cos
5590:sin
5517:to
5137:or
4998:on
4869:to
4836:= (
4547:to
4541:in
4511:= (
4407:= (
4237:as
4231:in
4208:of
4187:or
4124:cos
4054:sin
3915:cos
3906:sin
3840:cos
3802:sin
3489:cos
3473:sin
3443:cos
3427:cos
3397:sin
3381:sin
3351:sin
3335:cos
3303:as
3269:cos
3218:sin
3138:in
3088:to
3082:of
2897:sin
2888:sin
2867:cos
2858:sin
2837:cos
2808:Im
2766:sin
2751:cos
2652:sin
2636:sin
2600:by
2545:sin
2508:sin
2416:sin
2348:sin
2339:sin
2330:sin
2297:cos
2288:sin
2279:sin
2246:cos
2237:sin
2204:cos
2147:on
2137:on
2095:= 1
1978:by
1794:on
1743:).
1686:on
1676:of
1459:16
1382:≥ 4
1213:of
1043:is
994:of
912:or
189:In
135:In
6335::
6305:.
6277:.
6263:.
6250:.
6199:.
6126:.
6116:47
6114:.
6110:.
6087:.
6029:.
6019:;
6001:.
5999:12
5997:.
5954:,
5940:,
5936:,
5924:,
5920:,
5878:.
5537:,
5530:,
5521:.
5497:.
5243::
5220:.
5208:,
5145:.
5121:,
5107:,
5087:,
4850:,
4843:,
4532:,
4525:,
4518:,
4495:+
4485:+
4475:=
4458:+
4451:+
4421:,
4414:,
3755:.
3698:,
3153:,
3123:,
3116:,
3066:.
2718::
2397:,
2163:,
2159:,
2088:+
2081:+
2074:+
2028:NP
2012:xy
1953:×
1778:.
1614:84
1597:12
1560:15
1520:12
1456:15
1453:14
1450:13
1447:12
1444:11
1441:10
1392:.
1282:,
1260:,
1244:,
1227:.
1182:,
1026:.
533::
252:,
245:,
238:,
216:,
209:,
202:,
181:.
174:.
143:,
6323:n
6319:n
6311:.
6285:.
6281::
6271::
6265:8
6207:.
6172:.
6130::
6122::
6095:.
6062:.
6047:.
5927:n
5839:1
5836:α
5831:U
5814:.
5810:)
5804:i
5800:J
5794:i
5784:3
5779:1
5776:=
5773:i
5764:(
5754:=
5751:U
5736:U
5714:.
5709:)
5690:1
5681:i
5674:e
5655:2
5646:i
5639:e
5615:2
5606:i
5602:e
5583:1
5574:i
5570:e
5563:(
5544:)
5542:2
5539:ξ
5535:1
5532:ξ
5528:η
5526:(
5510:S
5495:q
5483:q
5474:H
5449:.
5444:)
5436:2
5432:x
5428:i
5420:1
5416:x
5408:4
5404:x
5400:i
5397:+
5392:3
5388:x
5375:4
5371:x
5367:i
5364:+
5359:3
5355:x
5347:2
5343:x
5339:i
5336:+
5331:1
5327:x
5317:(
5309:k
5304:4
5300:x
5296:+
5293:j
5288:3
5284:x
5280:+
5277:i
5272:2
5268:x
5264:+
5259:1
5255:x
5236:S
5230:H
5217:S
5211:S
5205:S
5193:S
5179:S
5173:S
5167:S
5156:S
5143:)
5141:H
5128:S
5110:S
5096:S
5090:S
5059:u
5050:2
5042:u
5035:1
5030:=
5026:v
5011:S
5001:S
4987:v
4977:u
4959:.
4955:)
4949:3
4945:x
4941:,
4936:2
4932:x
4928:,
4923:1
4919:x
4914:(
4905:0
4901:x
4894:1
4890:1
4885:=
4881:v
4867:p
4862:R
4857:)
4855:3
4852:v
4848:2
4845:v
4841:1
4838:v
4834:v
4812:v
4808:+
4805:1
4799:v
4795:+
4792:1
4783:=
4779:)
4770:2
4762:v
4756:+
4753:1
4747:v
4743:2
4737:,
4729:2
4721:v
4715:+
4712:1
4705:2
4697:v
4691:+
4688:1
4678:(
4674:=
4671:p
4658:p
4637:.
4633:)
4627:3
4623:x
4619:,
4614:2
4610:x
4606:,
4601:1
4597:x
4592:(
4583:0
4579:x
4575:+
4572:1
4568:1
4563:=
4559:u
4544:S
4539:)
4537:3
4534:x
4530:2
4527:x
4523:1
4520:x
4516:0
4513:x
4509:p
4503:k
4500:3
4497:u
4493:j
4490:2
4487:u
4483:i
4480:1
4477:u
4473:u
4468:p
4463:3
4460:u
4456:2
4453:u
4449:1
4446:u
4441:u
4433:R
4428:)
4426:3
4423:u
4419:2
4416:u
4412:1
4409:u
4405:u
4383:u
4376:1
4370:u
4366:+
4363:1
4357:=
4353:)
4344:2
4336:u
4330:+
4327:1
4321:u
4317:2
4311:,
4303:2
4295:u
4289:+
4286:1
4279:2
4271:u
4262:1
4255:(
4251:=
4248:p
4234:S
4229:p
4217:R
4211:S
4193:2
4190:ξ
4184:1
4181:ξ
4176:π
4171:2
4168:ξ
4162:1
4159:ξ
4154:η
4133:.
4119:)
4114:1
4101:2
4093:(
4089:i
4085:e
4081:=
4072:2
4068:z
4049:)
4044:2
4036:+
4031:1
4023:(
4019:i
4015:e
4011:=
4002:1
3998:z
3963:.
3958:2
3950:d
3942:1
3934:d
3925:d
3903:=
3900:V
3897:d
3869:2
3864:2
3856:d
3844:2
3836:+
3831:2
3826:1
3818:d
3806:2
3798:+
3793:2
3785:d
3782:=
3777:2
3773:s
3769:d
3746:2
3743:/
3739:π
3731:η
3726:2
3723:ξ
3717:1
3714:ξ
3705:)
3703:2
3700:ξ
3696:1
3693:ξ
3691:(
3684:2
3681:/
3677:π
3669:η
3646:2
3643:ξ
3637:1
3634:ξ
3611:.
3606:2
3602:S
3593:3
3589:S
3580:1
3576:S
3559:π
3554:2
3551:ξ
3545:1
3542:ξ
3534:2
3531:/
3527:π
3519:η
3498:.
3484:2
3470:=
3461:3
3457:x
3438:2
3424:=
3415:2
3411:x
3392:1
3378:=
3369:1
3365:x
3346:1
3332:=
3323:0
3319:x
3300:R
3278:.
3262:2
3253:i
3249:e
3245:=
3236:2
3232:z
3211:1
3202:i
3198:e
3194:=
3185:1
3181:z
3162:C
3158:2
3155:z
3151:1
3148:z
3146:(
3141:C
3135:S
3130:)
3128:2
3125:ξ
3121:1
3118:ξ
3114:η
3112:(
3103:S
3097:R
3091:R
3085:S
3063:ψ
3061:2
3057:τ
3049:q
3038:x
3020:k
3015:3
3011:x
3007:+
3004:j
2999:2
2995:x
2991:+
2988:i
2983:1
2979:x
2975:+
2970:0
2966:x
2962:=
2950:e
2946:=
2943:q
2930:q
2926:τ
2909:k
2906:)
2885:(
2882:+
2879:j
2876:)
2855:(
2852:+
2849:i
2846:)
2834:(
2831:=
2815:τ
2810:H
2798:τ
2789:τ
2760:+
2748:=
2736:e
2732:=
2729:q
2712:q
2691:.
2685:d
2676:d
2667:d
2662:)
2640:2
2631:(
2625:3
2621:r
2617:=
2614:V
2611:d
2580:]
2575:)
2569:2
2561:d
2549:2
2541:+
2536:2
2528:d
2524:(
2512:2
2504:+
2499:2
2491:d
2487:[
2481:2
2477:r
2473:=
2468:2
2464:s
2460:d
2440:π
2436:ψ
2413:r
2403:φ
2399:θ
2395:ψ
2391:π
2387:φ
2383:π
2379:θ
2375:ψ
2327:r
2324:=
2315:3
2311:x
2276:r
2273:=
2264:2
2260:x
2234:r
2231:=
2222:1
2218:x
2201:r
2198:=
2189:0
2185:x
2167:)
2165:φ
2161:θ
2157:ψ
2155:(
2150:S
2140:S
2114:S
2100:S
2093:3
2090:x
2086:2
2083:x
2079:1
2076:x
2072:0
2069:x
2063:S
2043:π
2024:P
2020:N
2016:P
1955:S
1951:S
1945:S
1939:S
1928:.
1915:T
1901:)
1893:2
1889:z
1885:,
1877:1
1873:z
1869:(
1866:=
1857:)
1852:2
1848:z
1844:,
1839:1
1835:z
1831:(
1817:C
1811:S
1797:S
1791:T
1772:n
1750:(
1726:r
1717:r
1713:/
1710:1
1689:R
1679:R
1655:6
1652:Z
1645:2
1642:Z
1640:⊕
1638:2
1635:Z
1628:2
1625:Z
1623:⊕
1621:2
1618:Z
1616:⊕
1611:Z
1604:2
1601:Z
1599:⊕
1594:Z
1587:2
1584:Z
1582:⊕
1580:2
1577:Z
1570:2
1567:Z
1557:Z
1550:3
1547:Z
1540:2
1537:Z
1530:2
1527:Z
1517:Z
1510:2
1507:Z
1500:2
1497:Z
1490:Z
1485:0
1482:0
1479:0
1475:)
1473:S
1471:(
1468:k
1465:π
1438:9
1435:8
1432:7
1429:6
1426:5
1423:4
1420:3
1417:2
1414:1
1411:0
1407:k
1399:S
1380:k
1375:)
1373:S
1371:(
1369:3
1367:π
1361:S
1359:(
1357:2
1353:S
1351:(
1349:1
1347:π
1327:n
1323:/
1320:1
1303:S
1290:i
1284:Z
1280:S
1278:(
1275:i
1272:H
1264:)
1262:Z
1258:S
1256:(
1254:3
1252:H
1248:)
1246:Z
1242:S
1240:(
1238:0
1236:H
1216:R
1146:.
1141:4
1137:r
1131:2
1121:2
1118:1
1113:=
1110:H
1081:3
1077:r
1071:2
1063:2
1060:=
1057:V
1054:S
1041:r
975:.
971:}
967:1
964:=
958:q
952::
948:H
941:q
937:{
933:=
928:3
924:S
896:}
892:1
889:=
884:2
879:|
872:2
868:z
863:|
859:+
854:2
849:|
842:1
838:z
833:|
829::
824:2
819:C
811:)
806:2
802:z
798:,
793:1
789:z
785:(
781:{
777:=
772:3
768:S
753:H
750:(
743:C
740:(
733:R
715:.
711:}
707:1
704:=
699:2
694:3
690:x
686:+
681:2
676:2
672:x
668:+
663:2
658:1
654:x
650:+
645:2
640:0
636:x
632::
627:4
622:R
614:)
609:3
605:x
601:,
596:2
592:x
588:,
583:1
579:x
575:,
570:0
566:x
562:(
558:{
554:=
549:3
545:S
530:S
508:.
503:2
499:r
495:=
490:2
486:)
480:3
476:C
467:3
463:x
459:(
456:+
451:2
447:)
441:2
437:C
428:2
424:x
420:(
417:+
412:2
408:)
402:1
398:C
389:1
385:x
381:(
378:+
373:2
369:)
363:0
359:C
350:0
346:x
342:(
339:=
334:2
330:)
324:i
320:C
311:i
307:x
303:(
298:3
293:0
290:=
287:i
268:R
265:(
259:)
257:3
254:x
250:2
247:x
243:1
240:x
236:0
233:x
231:(
227:r
223:)
221:3
218:C
214:2
211:C
207:1
204:C
200:0
197:C
195:(
158:n
131:)
83:)
77:(
72:)
68:(
54:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
