3-Sphere notes

Rotations

A rotation in 2 dimensions leaves one single point stationary. A rotation in 3 dimensions leaves an axis stationary. A rotation in 4 dimensions leaves a plane stationary. So in 4d everything rotates around a plane. Now we can additionally choose to rotate that stationary plane, thereby doing 2 planar rotations at once. I believe this is what is referred to as an isoclinic or Clifford rotation, whereby you have a pair of rotating orthogonal 2 planes that intersect only at one point. [3]

Hopf Circles, Hopf Fibrations, Hopf Links

The intersection of a complex line in C² containing the origin, with S³  is a circle. This is true for every line passing through the origin whose equation is of the form z2=a*z1 where a is a complex number and z2 and z1 are the coordinates of C². By ‘line’ we mean one-dimensional subspace which we can think of as a plane. Thus there is a circle in S³ for each complex number a (including the case where a is infinite). The sphere S³  is therefore filled with circles, one for each point of S², that is, for each complex number a. (The as fill all of  C¹ which we can identify with the Riemann Sphere). As Thurston says “exactly one Hopf circle passes through each point of S³” and “the three-sphere is a two-sphere’s worth of circles”.[4]

(Note: A complex line in C² is a one dimensional subspace thereof. We can visualize a complex line in C² by imagining a plane in R⁴, but not all planes in R⁴ are one dimensional subspaces of C². [1])

The Hopf fibration is this map from the unit 3-sphere to the unit 2-sphere. (This is different than stereographic project to R³!).

  1. The inverse image of each point on the 2-sphere is a circle. Thus, these images decompose the 3-sphere into a continuous family of circles, and each two distinct circles form a Hopf link. The linking is what makes the fibration a non-trivial fibration.[9]
  2. The inverse image of a circle of latitude on S2 under the fiber map is a torus, and these project to nested toruses in R³ which fill space. The individual fibers link to Villarceau circles on these tori with the exception of the circle through the projection point and the one through its opposite point: 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 radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through every circle, both in R3 and in S3. Two such linking circles form a Hopf link in R3 [10]

The n-dimensional complex project space CPⁿ is the set of all complex lines on Cⁿ⁺¹ passing through the origin. [2]  Thus C² projects to CP¹, the Riemann sphere.  Homogenous coordinates represent the same point in the projective space. As we saw above a=z2/z1, and each value of a specifies a different set of homogeneous coordinates, therefore a line through the origin of C²  maps to a point in CP¹. CP¹ is the set of all complex one-dimensional subspaces of C². A fancier way to say it, as Wikipedia does is: “CP1 is the quotient of C2\{0} by the equivalence relation which identifies (z0z1) with (λ z0λ z1) for any nonzero complex number λ. On any complex line in C2 there is a circle of unit norm, and so the restriction of the quotient map to the points of unit norm is a fibration of S3 over CP1.”

Stereographic Projection from  to  R³

So the Hopf fibration can be thought of as a map from S³  to CP1, or to S². But that is not really a visualization tool at all, it is just a map. To visualize entities embedded in S³ we can stereographically project to  R³. I’m not going to summarize how to do that, because it is in a million places on the web, e.g. [7].

Under stereographic projection of S³ to R³,

  1. Radial planes in R³ correspond to 2-spheres through N (the north pole, i.e. infinity in R³) in S³.
  2. The equator of S³ projects to a sphere , S², of radius 1 centered at the origin of R³.

I initially confused Hopf Circles with Great circles and C² with R⁴. Not every  two dimensional plane in R⁴ is a one dimensional subspace of C² . Any plane in R⁴ that goes through the origin intersects S³ in a great circle. But Hopf circles are the intersections of complex lines with S³. Note that unlike for S², two great circles need not intersect in S³.

Entities embedded in

  1. Clifford Torus.

This is a flat torus that lives in R⁴ and is embedded in S³. “Flat” means it has zero Gaussian curvature everywhere. By “embedded in S³”, we mean that all points on the surface are unit distance from the origin. So both of these statements are mind-bending but also strangely intuitive. If you use the standard formula for a Clifford Torus and perform stereographic projection:

// Clifford Torus Parameterization
var f=1/Math.sqrt(2);
var xx=f*Math.cos(u);
var yy=f*Math.sin(u);
var zz=f*Math.cos(v);
var rr=f*Math.sin(v);
// Stereographic projection:
var x=xx/(1-rr);
var y=yy/(1-rr);
var z=zz/(1-rr);

you get… a donut:

However, if you rotate the projection point:

var xx = Math.cos(u+theta) *Math.cos(v+phi);
var yy = Math.cos(u+theta) *Math.sin(v+phi);
var zz = Math.sin(u+theta) *Math.cos(v+phi);
var rr = Math.sin(u+theta) *Math.sin(v+phi);

you get something that looks like a plane with handles:

I got these formulas from “Sculptures in S3” by Saul Schleimer and Henry Segerman, which is a very interesting paper with many practical (as in fun) applications.

This plane with handles has more structure than it seem, so I left some gaps in the surface which brings out the structure more:

A Clifford torus divides the 3-sphere into 2 solid tori. This was hard for me to grasp initially. But if you consider the surface of the donut above (the Clifford torus is a surface), it in-fact divides R³ into 2 solid tori.  It is clear that the inside of the donut is a solid torus, but the outside is also a solid torus in the pre-image of the projection, i.e. in S³. Perhaps a little sloppy, but that’s the intuition.

There are other fun parameterizations in the Segerman/Schleimer paper. Here are a few I coded up:

Inverted (3,11) Torus Knot from Robert Woodley on Vimeo.

 

Trefoil on S3 from Robert Woodley on Vimeo.

 

Trefoil on S3, II. from Robert Woodley on Vimeo.


[1] https://math.stackexchange.com/questions/1265551/how-to-verify-whether-r2-is-a-subspace-of-the-complex-vector-space-c2

[2] http://www.math.poly.edu/courses/projective_geometry/chapter_three/node1.html

[3] http://eusebeia.dyndns.org/4d/vis/10-rot-1

[4] Three-dimensional Geometry and Topology, Volume 1. page 103.

[5] http://emerald.tufts.edu/~gwalsh01/dissertationfinal.pdf

[6]Topology, Geometry, and Gauge Fields, Foundations – Gregory Naber, page 17.

[7] Sculptures in S3. Saul Schleimer and Henry Segerman. https://arxiv.org/pdf/1204.4952.pdf

[8] https://en.wikipedia.org/wiki/Villarceau_circles

[9] https://en.wikipedia.org/wiki/Hopf_link

[10] https://en.wikipedia.org/wiki/Hopf_fibration

 

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