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.