Hopf fibration (ConceptTopic, 3)

From Hi.gher. Space

The Hopf fibration is a 1-to-1 mapping between points on the 2-sphere (i.e., 3D sphere) and circles on the 3-sphere (4D sphere, or glome). It has many interesting characteristics, both mathematically and geometrically.

These circles are all mutually disjoint, so none of them intersect each other; in other words, you can cut up the 3-sphere into these circles, so to speak, or conversely, the union of these circles equals the entire surface of the 3-sphere. This in itself isn't that remarkable, but what makes it remarkable is the way these circles are laid out on the 3-sphere. Unlike the 3D case, where you can cut the 2-sphere into, say, latitude circles, all the circles in the Hopf fibration are great circles, or geodesics (their radius is equal to the radius of the entire 3-sphere). Furthermore, they are laid out in a spiraling, chiral fashion. Suppose we identify one of them as our starting point, say it's a circle in the XY plane. Around it we find circles that swirl around it, interlocking with it, and around them are more swirling circles, at an incrementally flatter angles to the original XY circle, wrapping themselves around it in an increasingly thick bundle. As we approach 90°, these circles start to swirl around the ZW plane, until they converge upon a single circle in the ZW plane, which is perfectly orthogonal to the original XY circle.

Now, it may not be immediately clear from this description, but all of these circles are actually transitive: pick any one of them, and you find an identical structure of swirling circles around it, just like around every other circle. So it forms a continuous symmetry group, in which all the circles are equivalent via a rotation. What kind of rotation? Precisely a Clifford double rotation — that is, a simultaneous rotation in two orthogonal planes with the same rotation rate.

If you mark out a point on the 2-sphere, say at the north pole, then the Hopf fibration maps that north pole point to some particular circle on the 3-sphere — let's say the XY circle that we marked out above (it doesn't really matter which as long as we consistently choose the circles, since the structure is symmetric). The south pole point, then, maps exactly to the orthogonal ZW circle. So you may think of it as a kind of "unzipping" of the 2-sphere, where you stretch it open at the south pole point so that it becomes an equatorial circle, and then spin the rest of the sphere around in 4D to cover the surface of the 3-sphere exactly once (under said Clifford double rotation).


The above may not adequately convey the rich structure produced by the Hopf fibration. Jonathan Bowers had the idea to examine what happens when our starting 2-sphere has a tiling imposed upon it — i.e., what happens if we "apply the Hopf function", so to speak, to a 3D polyhedron — that is, the spherical tiling corresponding with that polyhedron. When we do that to, say, a dodecahedron, we find that the top pentagonal face (say) maps to a pentagonal torus that wraps around a section of the 3-sphere, and the adjacent 5 pentagonal faces map to 5 pentagonal tori swirling around this first torus, and then the adjacent 5 other pentagons map to another 5 pentagonal tori swirling around the previous 5 tori, and the bottom pentagon maps to the last torus that wraps around the plane orthogonal to the first torus. This is what Jonathan Bowers calls a "dodecahedral regular polytwister": these 12 tori are all equivalent to each other under a certain symmetry, which is variously known as swirlprism symmetry. Other polyhedra undergo similar mappings to 4D, each Platonic solid producing a regular polytwister. All of their symmetries are subsymmetries of the Hopf fibration; just as the polyhedra themselves belong to a subsymmetry of the 2-sphere.

When we apply this mapping to the dihedral tiling of the 2-sphere (i.e., paint the top hemisphere red and the bottom hemisphere blue, for example), then what we get is none other than the duocylinder itself. So you see, duocylindrical symmetry is but a subsymmetry of the Hopf fibration — the simplest non-trivial one!

You can spot some of these subsymmetries in various 4D regular polytopes:

  • The geochoron can be decomposed into 2 orthogonal rings of 4 cubes each, corresponding with duocylindrical symmetry, which is the same as the Hopf fibration of the dihedral tiling of the 2-sphere.
  • The xylochoron can be decomposed into 4 rings of 6 octahedra joined at opposite faces, corresponding with the Hopf fibration of the tetrahedral tiling of the 2-sphere. The xylochoron can also be decomposed into 6 rings of 4 octahedra joined at opposite vertices, which corresponds with the Hopf fibration of the cubical tiling of the 2-sphere.
  • The cosmochoron can be decomposed into 12 rings of 10 dodecahedra joined at opposite faces, corresponding with the dodecahedral tiling of the 2-sphere.
  • The hydrochoron can be decomposed into 20 rings of 30 tetrahedra each, in an interesting formation that exhibits a local 3-fold twisting (known as the Boerdijk-Coxeter helix), corresponding with the icosahedral tiling of the 2-sphere.
  • None of the regular polychora correspond with the octahedral tiling of the 2-sphere, but there is a CRF polytope that does: the bixylodiminished hydrochoron, a curious non-uniform yet cell-transitive and vertex-transitive polychoron consisting of 48 tridiminished icosahedra that form 8 rings of 6 cells each.

In any case, all of these rings have that characteristic interlocking structure, of which the Hopf fibration is the continuous version. The duocylinder is just the simplest discrete subsymmetry of it; and even within the duocylinder itself you can already spot some of the larger symmetry group: if you pick a point on the duocylinder's ridge (where the two tori touch each other), then it's possible to pick out a great circle (i.e., a geodesic path on the surface of the 3-sphere) that lies entirely on this ridge: this circle would form a swirling pattern around the ridge, and it corresponds with one of the Hopf fibers that isn't exactly on one of the two planes of the duocylinder's symmetry. Put another way, you obtain duocylindrical symmetry if you color one of the circular Hopf fibers red, and the orthogonal fiber blue, and then color the other fibers according to whether they are closer to the red or blue fiber. So, duocylindrical symmetry is just where you specially mark out a pair of mutually-orthogonal fibers in the Hopf fibration of the 3-sphere. If you were to mark out other sets of fibers, say 4 fibers in tetrahedral symmetry, then this produces the tetrahedral subsymmetry of the Hopf fibration. Or mark out 20 of them in icosahedral symmetry to get an icosahedral subsymmetry.