One way of understanding a set of points in space is to pick a direction, slice it by planes orthogonal to that direction, and see what’s in the planes. In the real world, this practice is called tomography. One of the more delicious and enlightening uses of this that’s been of percolating through the net this summer has been the blog Inside Insides.
Here’s a fun one. (Edit: Just linked to that blog entry directly.)
Notice how the chosen direction of slicing from end to end reveals a symmetry of the guts of the watermelon that wouldn’t have been apparent if it were sliced from side to side.
Taking an object in 3D and slicing along a direction it gives a sequence of objects in 2D, like with this watermelon. Show them one after another and you’ve got a movie. We can do the same one dimension higher. Take an object in 4D and slice it along a direction to obtain a sequence of objects in 3D. And make that into a movie.
Like our knots of 1-spheres in 3D, there are knotted 2-spheres in 4D. So what happens when we do our tomography of knots? Slicing our usual knots and links in 3D, each slice gives points in the plane. For knotted 2-spheres in 4D, our slices are knots and links in 3D. Making movies, the slices of points in the plane and the slices of knots in 3D dance around and interact.
The other day Ayumu Inoue posted on the arXiv a description of some of these tomography movies of the “n-twist spun trefoils” which are knotted 2-spheres in 4D. What makes his movies interesting is that, like that of the watermelon, they reveal a symmetry that previous movie descriptions didn’t. Moreover, he made some really neat animations.
On the right hand side is a “diagram” of the 2-twist spun trefoil. Doing the tomography trick on this diagram he is able to effectively obtain the movie of the tomography of the knotted sphere in 4D on the left.
This tomography is slightly different as the slicing rotates along an axis, but it gives a clue as to why we refer to this knotted sphere as a 2-twist spun trefoil. (Wikipedia doesn’t have an entry yet for spun knots, let alone twist spun knots. The Mathworld descriptions are lacking…)
Check out the rest of his work. Inoue used Blender, a free and open source program, to create these. I might have to give Blender another try.