The

http://www.knotplot.com/hopf/hopf-fibration1.mpg

This way of filling the 3-sphere with circles is called the

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:

http://www.math.uiuc.edu/~ci1/

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.

**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:http://www.knotplot.com/hopf/hopf-fibration1.mpg

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:

http://www.math.uiuc.edu/~ci1/

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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- +Niles Johnson "But can anyone explain why the 12 necklaces of dodecahedra in the 120 cell can be arranged to lie precisely along Hopf fibers? Equivalently, can you explain why taking 12 evenly spaced points on the 2-sphere, and arranging dodecahedra along the corresponding Hopf fibers, yields 12 necklaces that fit exactly together, thus yielding the 120 cell?"

I know I'm late to the party here but here it goes. The dodecahedron has opposite parallel faces. They can be stacked in a straight line in 3-space, and in the 3-sphere they form a geodesic decagon. So the 12 "necklaces" each form a non-intersecting geodesic, and each of these is rotationally equivalent to the other. Can you think of another way to arrange 12 rotationally equivalent points on the 2-sphere other than in a dodecahedral arrangement? All six of the 4-D platonic polytopes can be partitioned into discrete Hopf Fibrations that map to various 3-D tiling of the sphere. 600-cell to icosahedron (20 30-tet Boerdijk–Coxeter helices), 24-cell to tetrahedron (4 6-cell face stacked chains) and to cube (6 4-cell vertex stacked chains), 16-cell to dihedron (2 8-tet Boerdijk–Coxeter helices), 8-cell to dihedron (2 4-cube chains), 5-cell to sphere (1 5-tet Boerdijk–Coxeter helix). Tom Ruen an I had been contributing to a "Discrete Hopf Fibration" section of the Wikipedia Hopf Fibration article for over a year, but the section was just recently deleted by a Wiki Pharisee due to "Lack of citations to references or sources". He also refered to the section and figures as "cruff" otherwise polluting the purity of the article in the deletion comment. But in truth, many of the figures and tables are original work because I couldn't find many citable sources so according to Wiki policies (WP:NOR) we don't have a leg to stand on to challenge him. This section is still available in back issues.Nov 15, 2014 - Maybe you could at least turn your work into a G+ post? Tag me if you do, since I'd like to see it :)Nov 17, 2014
- Yes, it sounds interesting.Nov 17, 2014
- I'll look into it. I have to come up to speed on Google+ first.Nov 17, 2014
- +Bruce Bowen - it's best to post articles with just one picture each. Click on "Share what's new...", throw a jpeg or png or gif into the box, write some interesting text on top... keep doing it.Nov 17, 2014
- +Niles Johnson +John Baez Just published a proof on the Wikipedia 600-cell talk page that the 30-cell Boerdijk–Coxeter helix circumnavigates the 600-cell.Sep 3, 2015

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