To: Clarence Wilkerson

From: Ken Shoemake

Subject: Hopf fibration picture

I just found the Hopf Topology Archive on the Web, which it appears you handle. At the top of the page I read that the logo will eventually be "a visualization of the Hopf fibration of the three sphere over the two sphere."

I have such a picture. Perhaps you would be interested in using it? My research discipline is actually 3-D computer animation, but a few years back it was my good luck to take a graduate course in algebraic topology with Gluck at U. Penn. As it happens, many of my publications have used quaternions in an essential way, so I was particularly interested in the fiber bundle structure S^1 -> S^3 -> S^2. When a computer graphics camera tracks an object, implicitly it operates within the fiber structure.

I recalled seeing pictures in "Mathematical Intelligencer", and thought I would try my hand at something similar. It turned out that using quaternions I could do it pretty easily. Because I wanted to explicitly show as much as I could of the topology in the fiber structure, I created pictures somewhat different from those I had seen. I wanted to show both S^3 and S^2, and visually connect fibers to base points, and so I gave each point on S^2 a unique RGB color by simply putting the sphere inside a color cube. Then I colored each fiber uniformly with the color of the base point. To display S^3 I punctured and flattened it to B^3 using a method that was easy for me: I took unit quaternions for my S^3 and took their logarithms (the inverse of the exponential map at 1+0i+0j+0k) for B^3.

To show structure clearly, I chose only a few arcs of lattitude as base points, giving sliced tori for fibers. Of course the coloring is continuous on S^2, so it is also continuous and delightful in the fibers. My program is interactive on an SGI workstation, but I created one image I like because it shows nesting tori, linking fibers, base-fiber relations, and so on. And, because it looks nice.