The 3-sphere looks just like ordinary 3d space except there's one extra point, the point at infinity. There's a special way to fill the 3-sphere with circles so that every point lies on exactly one circle. Some of them are drawn here. Most come in bunches that lie on tori (doughnuts). The exceptions are the red circle and the red vertical line, which is actually the circle that goes through the point at infinity! This movie may help:

This way of filling the 3-sphere with circles is called the Hopf fibration. What's special about it is that nearby circles stay near to each other all the way around, and each pair of circles is linked in the simplest possible way.

What does this have to do with the 120-cell and our #4d story? Well, if you revisit the movie I showed you last time (120-cell) you'll see the dodecahedra in the 120-cell come in bunches of 10, which form circles... and 60 of them make a nice solid doughnut. So the Hopf fibration is at work here!

There's much more to say, but it's probably best if you think, ask questions, and think some more. It's my bed-time here in Singapore, so my answers will take about 9 hours to arrive. Luckily, there are other mathematicians on G+ standing by, eager to help out!

I found this nice picture on Yongfei Ci's website:

He's a topology grad student at the University of Illinois. If you go to grad school and study topology, the Hopf fibration will become your friend just like the equation 2+2=4. It's very important in math and physics.
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