What the 3-Sphere Looks Like

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What the 3-Sphere Looks Like

Postby Sir Puppum Hat » Fri Mar 12, 2010 2:10 am

Okay. Now, the 3-sphere, or the "glome" as some people call it has been confusing me for months now. When I search for the 3-sphere to find out what it looks like, there's either multiple representations of it (each picture being drastically different from each other) or just no info on what it looks like at all, leaving the user to imagine what it looks like by inferring mathematically.

These are the pictures I've come across looking for a "correct" picture of the 3-sphere:

http://upload.wikimedia.org/wikipedia/c ... _coord.PNG
and
http://upload.wikimedia.org/wikipedia/c ... _coord.gif
This is a big ugly monstrosity of a geometrical figure if I ever saw one. It could probably be correct but is just downright confusing. If it is indeed the 3-sphere, could someone make sense out of it for me?

http://www.redicecreations.com/ul_img/4 ... ation1.jpg
This is what people refer to as the "Hopf Fibration". The first place I saw this thing was in the Dimensions series. I knew it definitely had something to do with the 3-sphere, but I couldn't really understand what they meant with all the graphs, etc. This probably is a projection of what the 3-sphere would look like, but I have doubts. Also, the more I look at this, the more I think it could be the same figure as http://upload.wikimedia.org/wikipedia/c ... _coord.PNG, but I just don't know.

http://upload.wikimedia.org/wikipedia/c ... sphere.png
To me, this seems to be the most likely candidate for an accurate 3-sphere. Basically a sphere, but instead of being composed of circular sections, there are spherical ones.

I have zero experience in the mathematics of the fourth dimension, but I can understand the geometry quite well, so can someone tell me which one of these pictures is the "real" 3-sphere/elaborate for me what this thing would look like? (If that's possible of course)
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Re: What the 3-Sphere Looks Like

Postby wendy » Fri Mar 12, 2010 10:58 am

A 3-sphere seen in 4-space, looks pretty much like a 2-sphere in 3-space. It's a round thing, like a ball. People who regularly wrangle in 4D, rather than occasionally visit it, call it a 'glome', like a 2-circle is called a 'sphere'.

Many of the links ye show are surface projections, like atlases are full of 2-sphere (earth) onto 2-paper (atlas). You can do 3-sphere onto 3-paper, but then you have to see through the mess, which is why line drawings are used. It's pretty much what ye see in all 4D projections. First you get a 3d paper, and then you look through the 3d thing.

The stereographic projection is where one is looking at the sphere from a point opposite to a tangent plane. Great circles cross the equator at diametrically opposite points, and angles are preserved.

1. http://www.redicecreations.com/ul_img/4 ... ation1.jpg : The diagrams with lines are pretty much like showing 2-spheres with circles representing rotation. This one with rays coming out the end of it, like (1), is a stereographic projection of a 3-sphere, with great circles representing the tracks that different points would follow as the sphere rotates.

2. http://upload.wikimedia.org/wikipedia/c ... _coord.PNG : This is a fairly unnatural view of the 3-sphere, but it's meant to show sections in different axies, of some projection of it. Like (1), it ia a stereographic projection, but here the circles represent intersections of 3-flats crossing the 3-sphere. You can see 'lattitude' (red), 'longitude (blue)', of the ordinary 2-sphere cross-section, and the green one represents sections from "front to back", which appear variously as full spheres. The whole lot are joined in the main picture.

3. http://upload.wikimedia.org/wikipedia/c ... sphere.png : This view here is an isometric view, of slices of the sphere in the correct depth position. It's rather like if you took photos of slices of a tomato, and arranged them at the right distances according to the slices. Of course it would look pretty much like the original.

HOPF FIBRATION.

You could read 3-space as x,y,z, where z is height, y is forward, and x is across. A wheel would stand as a circle in the yz plane. You could ride the wheel, in the fashion of the "B.C. " cartoons, by standing on the axle. Pushing down on an axle arm would make you turn in that direction. In four dimensions, you have an extra axis w. You still stand on what is perpendicular to front/height, (y,z), that is w,x. But steering is now not just pushing down on the -x, +x, but on anywhere on the circle. Think of flying in a flight-sim, where there is no gravity. You can steer to anywhere in front of you by a point on a clockface. Well, basically the sky becomes a map of the ground, and the axies are w,x (up, left), and y (forward).

Because wx is a 2-space, you can put another circle in there, and rotate that too. When the rotation rates of wx, and yz are equal, every point on the sphere goes around the centre exactly once in a revolution. That is, the track of any point is a circle that contains the origin. These lines do not cross, and look like fibres (the hairy-ball thing says you can comb a hairy sphere if the sphere is in an even dimension).

If you are familiar with complex numbers, all of this becomes apparent when one considers a point X = w+xi, Y = y+iz. When one uses a rotation in the form of cis(wt) = exp(iwt), where w is a speed (omega), and t is a time function), then the effect is that every point will go around the centre, and that if points A, B pass through a common point C, then they pass through each other too: ie the tracks are simple circles. Combing the hairy ball is a matter of finding these circles. One sees also that it applies to the shere (ball) in 2D, 4D, 6D, etc, but not the odd dimensions.

This rotation is different to its mirror image, that is, if one reflects wx, yz in the plane z=0, then wx remains positive, but yz reverses. Points still go around but along different fibres.

Hopf fibration is fairly important, because this is the tracts followed by a single multiplication of 'quarterions'. Every rotation in four dimensions, can be made of a pair of quarterion rotations, ie any point P becomes Q by aPb, where a and b are quarterions.
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Re: What the 3-Sphere Looks Like

Postby Sir Puppum Hat » Fri Mar 12, 2010 11:44 am

Great, thank you for all the info. It's still a bit confusing considering that so many ways of looking at this thing have been made. Although I did find out early on that the 3-sphere would look much like a 2-sphere, just given 4D depth.

I still don't quite understand what the whole Hopf Fibration thing with the great circles representing tracks that different points follow when it rotates, I'll have to learn more about that sooner or later. Also, I keep wondering about if all of these are technically correct ways to view the 3-sphere, how would someone get http://upload.wikimedia.org/wikipedia/c ... _coord.PNG from http://upload.wikimedia.org/wikipedia/c ... sphere.png

A bit mind-boggling.
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Re: What the 3-Sphere Looks Like

Postby PWrong » Fri Mar 19, 2010 10:22 am

I don't understand the picture of it, but I can address some of the confusion about the Hopf fibration. The Hopf fibration is not meant to describe or help you understand the 3-sphere. It's a separate thing entirely, more like a process whose outcome is the 3-sphere. It's a type of fiber bundle, which is a complicated topological idea.

A fiber bundle is like a cartesian product locally, but something else globally. For example a cylinder is the cartesian product of a circle and a line. A Mobius strip is similar locally but it has a different global structure from a cylinder. They're both different ways of gluing line segments to a circle. The Hopf fibration is basically a special way of gluing a bunch of circles to a sphere to get a 3-sphere.
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Re: What the 3-Sphere Looks Like

Postby pat » Wed May 12, 2010 11:24 am

Ugh. I have an animation of a textured 3-sphere rotating around. The textures helped me visualize the surface. Unfortunately, I just changed webhosts yesterday and just now realized I have yet to propagate "old.nklein.com". I will do that today.
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Re: What the 3-Sphere Looks Like

Postby pat » Wed May 12, 2010 1:57 pm

Okay, here is an animation of a 3-sphere rotating in various ways. What you see are slices of the 3-sphere. The 3-sphere is textured so that (IIRC) red is longitude, green is the other longitude, and blue is latitude. But, I don't seem to have uploaded the source, so I can't quite verify it.

http://old.nklein.com/products/rt/rt2.5.2004.03.XX/hyperpolar.avi

It's DivX encoded so you might need to download a DivX decoder.
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Re: What the 3-Sphere Looks Like

Postby quickfur » Mon May 24, 2010 3:43 pm

An easy "intuitive" way of understanding what the 3-sphere looks like is to generalize from lower dimensions.

Projection by itself doesn't necessarily help very much: just as the 2-sphere projects into a circle, so a 3-sphere projects into a 2-sphere. But that doesn't really give you much help in terms of "visualizing" it.

What may help, however, is to imagine what happens to the projected image if you were to add some surface features to the 3-sphere. For example, taking a 2-sphere as our reference, the way we usually envision a 2-sphere is one with lighting from the side, such that part of its surface is illuminated (forming various crescent-shaped regions of brightness or darkness), possibly with a specular highlight on it. These little details of lighting, shade, and the shape of the specular highlight help our brains to infer a "bulge" in the middle of the circular image of the 2-sphere as seen by our eyes.

So in the same way, a 4D being looking at a 3-sphere would see it with lighting from the side, with a crescent-shaped 3D volume of its surface illuminated, and with a specular highlight on it. From this light and shading, it would infer a "bulge" in the center of the spherical image of the 3-sphere. Just as the closest point on the 2-sphere to your eye is the center of its circular image, so the closest point on the 3-sphere to the 4D being's eye is the center of the spherical image.

Now suppose we are looking at the Earth (a 2-sphere) from space, and there is a ship on the equator directly in front of us, it would appear in projection to be at the center of the circle. From that point, it can sail north or south (vertically), or east or west (horizontally): there are 2 degrees of freedom. The analogous situation with a ship on a 3-sphere would be that of the ship lying at the center of the sphere, with 3 degrees of freedom: it can sail vertically (north/south), or horizontally (east/west/ana/kata). In terms of the projected image of the 3-sphere, which is a 2-sphere, the ship lies at the center of the 2-sphere, and can travel upwards, downwards, left, right, or forwards/backwards. That's 3 pairs of directions.

Now, say you paint a large white square on a 2-sphere, and orient the sphere such that the white region faces you. In projection, the image of this white region will be (more or less) a white square at the center of the circular perimeter of the projection image. If you were to rotate the sphere, say, on its vertical axis westwards, what happens to the white region in projection is that it starts moving leftwards, and appears to become more squished as it nears the limb of the circle. It then starts to "disappear" past the edge of the circle (its leftmost part is now on the far side of the sphere), and by now it has become a crescent-shaped area in appearance.

Analogously, if you paint a large white cube on a 3-sphere, and orient the sphere such that this region faces your 4D viewpoint, you would see it as more or less a white cube inside a sphere (as projected onto your 4D eye). With a 3-sphere, there are now many ways to rotate it (while preserving vertical orientation): you could rotate it from left to right (east/west), or forwards/backwards (ana/kata), or along any horizontal line that passes through the origin. It doesn't really matter which way you choose; the situation is completely analogous. As you rotate the sphere along your chosen horizontal line, the cube travels along this line within the 2-sphere image of the 3-sphere, flattening in the process and distorting into a crescent-shaped volume. When it reaches the surface of the 2-sphere (i.e., the limb of the 3-sphere), it will start to disappear behind the far side of the 3-sphere. So in this sense, you may think of the 3-sphere as two 2-balls (the volume inside the 2-sphere) "pasted" together at the 2-sphere boundaries, and "inflated" like a balloon so that it bulges in the middle.

More analogies can be drawn this way: for example, on a 2-sphere, if you designate the top as the North Pole and the bottom as the South Pole, then the set of points equidistant from these two poles forms a line around the middle of the sphere called the equator. In projection, the North Pole is the top of the circle, the South Pole is the bottom, and the equator projects to a line that bisects the circle in the middle. On a 3-sphere, the analogous North Pole and South Poles project to the top and bottom of a 2-sphere, and the equator is actually a plane that projects to a circle bisecting the sphere in the middle. Hence, the 3-sphere has a "3-quator" that spans a 2D area. Now if you were to rotate a 2-sphere such that you're facing the poles, then they will project to the middle of the circle, and the equator will project to a circle lying on the boundary of the projection image. Analogously, looking at a 3-sphere's poles, they would project to the center of a 2-sphere, and the "3-quator" actually maps to the spherical boundary of the projection. In other words, the 3-quator is in the shape of a sphere.

There are many other things you can do to better understand the shape of a 3-sphere; these examples hopefully have given you a good idea.
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