• Thumbnail for Hopf fibration
    In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)...
    36 KB (4,827 words) - 10:48, 2 July 2025
  • Thumbnail for Heinz Hopf
    Hopfian object Hopf algebra Quantum group Hopf fibration Alexandroff P., Hopf H. Topologie Bd.1 — B: 1935 Hopf, Heinz (1964), Selecta Heinz Hopf, Herausgegeben...
    11 KB (974 words) - 16:00, 9 July 2025
  • : 198  The Hopf fibration S 1 → S 3 → S 2 {\displaystyle S^{1}\to S^{3}\to S^{2}} is a non-trivial fiber bundle and, specifically, a Serre fibration. A mapping...
    18 KB (3,463 words) - 01:34, 29 May 2025
  • Thumbnail for Hopf link
    a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial fibration. This...
    9 KB (892 words) - 16:20, 15 November 2022
  • Thumbnail for Homotopy groups of spheres
    The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial...
    83 KB (8,124 words) - 04:10, 28 March 2025
  • Thumbnail for Villarceau circles
    presented at a congress in 1891. Hopf fibration Toric section Vesica piscis Coxeter 1969. Hirsch 2002. Dorst 2019, §6. Hopf Fibration and Stereographic Projection...
    11 KB (1,496 words) - 10:46, 2 July 2025
  • Thumbnail for 3-sphere
    d\xi _{1}\wedge d\xi _{2}.} To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above z 1 = e i ( ξ 1 +...
    28 KB (4,048 words) - 04:16, 9 May 2025
  • Thumbnail for 600-cell
    All the fiber bundles are aspects of the same discrete Hopf fibration, because the fibration is the various expressions of the same distinct left-right...
    141 KB (18,460 words) - 16:02, 15 July 2025
  • Example: the Hopf fibration. Let B equal S 2 {\displaystyle S^{2}} and E equal S 3 . {\displaystyle S^{3}.} Let p be the Hopf fibration, which has fiber...
    20 KB (3,432 words) - 14:48, 25 May 2025
  • 4
    four-group is the smallest noncyclic group,... Shokurov, A.V. (2002). "Hopf fibration". In Michiel Hazewinkel (ed.). Encyclopedia of Mathematics. Helsinki:...
    18 KB (1,744 words) - 12:56, 12 July 2025
  • {\displaystyle d_{k}:E_{p,q}^{k}\to E_{p-k,q-1+k}^{k}} . Recall that the Hopf fibration is given by S 1 ↪ S 3 → S 2 {\displaystyle S^{1}\hookrightarrow S^{3}\to...
    12 KB (2,641 words) - 13:35, 29 February 2024
  • Thumbnail for Bloch sphere
    space C 2 {\displaystyle \mathbb {C} ^{2}} to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere....
    23 KB (3,795 words) - 18:29, 25 June 2025
  • Thumbnail for 120-cell
    skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell). The Petrie...
    131 KB (14,831 words) - 16:19, 14 July 2025
  • Thumbnail for 24-cell
    discrete Hopf fibration of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of...
    237 KB (30,662 words) - 18:54, 14 July 2025
  • Thumbnail for Complex projective space
    be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study...
    26 KB (3,929 words) - 11:29, 22 April 2025
  • singularities comes from considering the flow along the lines of a Hopf fibration. Let r {\textstyle r} be a constant radius of the inner coil. One set...
    97 KB (15,479 words) - 14:00, 4 July 2025
  • commonly referred to as the Hopf fibration of S3, a topological construction pointed out by Heinz Hopf (1931). But Hopf’s procedure was explicitly based...
    8 KB (1,203 words) - 01:20, 3 May 2025
  • Thumbnail for Principal U(1)-bundle
    two-dimensional sphere S 2 {\displaystyle S^{2}} , which include the complex Hopf fibration, can be used to describe hypothetical magnetic monopoles in three dimensions...
    10 KB (1,567 words) - 03:32, 6 July 2025
  • Thumbnail for Fiber bundle
    Fiber bundle (redirect from Fibre bundle)
    then the quotient map is a fiber bundle. One example of this is the Hopf fibration, S 3 → S 2 {\displaystyle S^{3}\to S^{2}} , which is a fiber bundle...
    29 KB (4,152 words) - 05:48, 27 June 2025
  • Thumbnail for Euler angles
    Grids on SO(3) Using the Hopf Fibration". The International Journal of Robotics Research. 29 (7). Section 8 – Derivation of Hopf parametrisation. doi:10...
    48 KB (5,168 words) - 06:53, 28 May 2025
  • over the 2-sphere, known as Hopf fibration in the case r = i  ; other vectors give isomorphic, but not identical fibrations. Lyons (2003) gives an elementary...
    22 KB (2,945 words) - 03:51, 4 June 2025
  • other naturally due to the Hopf fibration.[failed verification] The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices...
    11 KB (990 words) - 19:00, 5 June 2025
  • trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration K ( π n + 1 ( X ) , n + 1 ) ≃ F n + 1 → S n + 1 n → S n n ≃...
    20 KB (3,834 words) - 20:43, 19 June 2025
  • CG]. Zamboj, Michal (8 January 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational...
    37 KB (5,718 words) - 03:50, 1 March 2025
  • four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles in five dimensions...
    15 KB (2,518 words) - 12:59, 7 July 2025
  • known as the Bloch sphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case. The Cartesian...
    6 KB (826 words) - 08:04, 6 July 2025
  • horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study...
    55 KB (8,944 words) - 02:31, 9 June 2025
  • diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration. Explicit expressions for coordinates...
    7 KB (1,245 words) - 17:30, 5 June 2023
  • "Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". International Journal of Theoretical Physics. 16 (9): 561–565. Bibcode:1977IJTP...
    35 KB (5,111 words) - 09:32, 1 July 2025
  • conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited...
    4 KB (442 words) - 15:10, 16 January 2025