The Hopf Fibration Video

https://www.youtube.com/watch?v=AKotMPGFJYk

I found this video very helpful but on closer examination realized that my understanding was superficial. Here is a guide to what I learned.

Suppose you have an ordinary mundane 2-sphere. Instead of using the traditional longitude and latitude you choose two longitudes. If you allow both longitudes to range from 0 to 360 then you will have a redundant labeling. (0,180) = (180,0), and so forth. To eliminate this it is necessary to restrict one of the longitudes to the interval [0,180) and the other to [0,360). Nothing profound about this.

Add more dimensions to get an N-sphere. All these new dimensions are similarly restricted to [0,180). The 3-sphere has two [0,180) dimensions and one [0,360) dimension.

Now look at that Hopf video. That 2-sphere is of two such [0,180) dimensions. We emphasize that those circles do NOT represent 360 degrees on the 3-sphere. Where did the [0,360) dimension go? Each point on that sphere represents a 360 degree great circle. That’s where that [0,360) dimension went.

Now given this special Hopf 2-sphere, the same argument applies. If we think of both dimensions as [0,180), we get the redundancy again. So we may arbitrarily choose one of those dimensions and restrict it to [0,90) . This avoids redundancy.

All this is why the basic Hopf parameterization has the “Swedish brick” proportions of 1x2x4, ie 90x180x360.

Going back to the video, each point on the 2-sphere represents a great circle on the 3-sphere. Great circles are all of the same size. They don't appear to be the same size and are ovals, but that is just because part of the circle has disappeared into the 4th dimension. What's more, consider a great circle on the 2-sphere. This corresponds to a great torus on the 3-sphere. All great tori are the same size and shape, though they don't look it because part of them has disappeared into the 4th dimension. Great tori also have the unique property that for each great circle in the torus, the great circle perpendicular to it is also in that torus.

How do we know these things? It is helpful to consider antipodes on the Hopf 2-sphere. These two points represent two great circles on the 3-sphere that are both perpendicular and parallel. These great circles are as far apart from another as can be on the 3-sphere. They are 90 degrees away from one another. Any smaller circle on the 2-sphere does not contain a pode-antipode pair, so it does not have this property.

Now consider two intersecting great circles on the 2-sphere. An angle between the two can be measured. The two tori they represent can be considered to be at this angle w.r.t. one another on the 3-sphere.

Kindly consider again the two intersecting great circles on the 2-sphere. They intersect at two points. These points are antipodal, so they represent a pair of mutually perpendicular parallel great circles.

Each of the tori generated by a great circle on the 2-sphere is what is called a “square flat torus.” It can be flattened out into a square 2D projective plane with no distortion whatsoever. Note further that it is not required to flatten out a torus into a flat square. One may flatten only one of the two 2-planes of the torus to get a hollow cylinder embedded in a 3D space. The length is equal to the circumference, so it could be called a “square cylinder.”

The torus on the 3-sphere generated by a great circle on the Hopf 2-sphere is a special kind of torus one may call a great torus. All these great tori are of the same size and proportion, indeed identical in every way but their location on the 3-sphere.

As the Hopf 2-sphere is partitioned into equal hemispheres by any great circle, each great torus partitions the 3-sphere into equal hemispheres.

Distinct great circles on the Hopf 2-sphere intersect at two points. Distinct great tori on the 3-sphere intersect at two great circles. These points are antipodal on the 2-sphere, so the corresponding great circles on the 3-sphere are perpendicular.

It is possible to choose a set of great circles on the 2-sphere such that their union is the 2-sphere and their intersection two points. The corresponding set of great tori on the 3-sphere have as their union is the 3-sphere and their intersection two great circles.

MAPS

Now suppose we wish to make a flat 3D map of the 3-sphere. Start with a great circle on the Hopf 2-sphere. Get the corresponding torus on the 3-sphere. Flatten it into a square. Next, tilt that great circle some tiny increment in some perpendicular direction. Get the corresponding torus on the 3-sphere. Flatten it into a square too. Stack the squares on top of one another. Do this for 180 degrees on the 2-sphere (which is 90 degrees on the 3-sphere). Done. You get a flat 3D map with certain properties. It is much like the maps we commonly use. One gets a rectilinear grid of coordinate lines on a distorted surface. The map is distorted because the squares represent great tori on the 3-sphere that are not disjoint.

One may have instead a map with cylindrical coordinates. Generate square flat tori in the same way, but don’t stack them. We have generated the square flat tori in such a way that they share two common great circles. How do these great circles appear in those square flat tori? All great circles in the great torus appear as parallel straight lines in the projective square, sloped at a 45 degree angle. The two great circles in the intersection of the great tori appear as parallel lines a maximal distance apart. The simplest thing is to arbitrarily choose one of the two. Take two squares that are tilted at an infinitesimal angle w.r.t. one another. Allow them to intersect at this common line to get a sort of butterfly shape. Continue the process for 180 degrees worth and you will get a solid cylinder. That’s a 3D object, so you have a flat 3D map of the planet. We have made a map, but it is rather a rough draft. It’s highly redundant and distorted.

It is easy to see that this map is distorted, rather severely so. Recall that second line of intersection that we didn’t choose. This process spuriously expands that line into a hollow cylinder. Regions close to that cylinder are greatly expanded. The map is also redundant. The regions past that cylinder have already appeared inside the cylinder. So simply leave all that stuff out. We lose both the redundancy and the most distorted regions. We have only 90 radial degrees, but that is enough. More is wasteful.

Is there room for improvement? Two solutions present themselves. The simplest thing is to do what we do on Earth. Arrange things so that the desirable areas are close to the axis of the cylinder and the nether regions close to the boundary. As we shall see later, there actually are such maps of our Earth.

If that map doesn’t please, one may retrace one’s steps. Recall that arbitrary choice of one of the two great circles of intersection. Simply choose the other and repeat. Now we have two redundant maps, but each is more convenient and accurate in certain regions. For any small region one may find a map on which the distortion is limited to a factor of 2.

Now we have made two complete maps. Each is not redundant, but of course the combination is. Each covers certain areas better than others. So we can simply eliminate the worst areas from each map. Instead of the radius representing 90 degrees, cut off the region with radius from 45 degrees to 90 degrees. This reduces each map to span 45 degrees. Then the redundancy is gone and the most distorted regions disappear. The resulting two maps have a total volume only half of the single map that covers the whole 3-sphere, so we have saved much. The distortion is nowhere greater than a factor of two.

What if we applied this sort of thing to our mundane Earth? What would that look like? Amazingly enough, this has been done. There are Mecca-centric maps. Eurasia and Africa come out nicely and the rest of the continents are quite recognizable, being distorted by a factor of two. The severe distortion occurs in the middle of the Pacific Ocean. The residents of Pitcairn and Easter Islands might actually like it, as it would greatly enhance the size of their tiny domains.

You may find the map here. The canonical longitude and latitude coordinates appear in orange. [url]http://3.bp.blogspot.com/-ga9M9TIGnH8/TmUYDYh_TWI/AAAAAAAAA3M/mjChkmzqKIU/w1200-h630-p-nu/mecca+center+of+world.jpg

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Not only that, in Muslim countries one may find in hotel rooms the kiblat, an arrow showing the great circle route to Mecca. So a billion people actually use a system like this.

Should anyone care to produce an ecumenical version of this sort of map, I’d suggest New Delhi and Panama City as the respective centers of the two maps.