Here is the spectrum of a sparse random graph. Note how there are isolated eigenvalues at the average degree and to the community structure; the bulk of the spectrum appears to be confined to a disk in the complex plane of radius sqrt(average degree).

My 1993 work was based on a simple action-minimizing approach in which I discretized the trajectory and then performed gradient descent with the position at each time. Michael pioneered a different approach, in which we perform gradient descent in the coefficients of the trajectories' Fourier transform. Using this approach, Michael and I have found a number of lovely three-dimensional orbits, including a family with cubic symmetry where 4k masses travel around 4 roughly circular interlocked orbits (where k is odd). Topologically, these orbits form the edges of a cuboctahedron. Since they have cubic symmetry, their total angular momentum is zero and their moment of inertia tensor is diagonal in the x,y,z basis.

(If you have trouble with the following movies, note that both Quicktime and RealPlayer can play .gif files. Also, I am indebted to Greg Egan for helping me loopify these movies in postproduction.)

Here is a 4-mass orbit with cubic symmetry. (Note that this is not the same as the "hip-hop", in which the two pairs of masses co-rotate around the z-axis.) Movies in Quicktime and GIF.

Here also is a very nice page by Vicki Johnson with real-time simulations of this and other orbits, which allow you to rotate the point of view with your mouse.

Here is the cubic 12-mass orbit (which has some additional symmetry), in Quicktime and GIF. Here is another movie which shows the 4 orbits from a rotating point of view, Quicktime and GIF.

Here is the cubic 20-mass orbit, in Quicktime and GIF.

And here is the 28-mass one, in Quicktime and GIF.

Here is another lovely orbit, in which 6 masses orbit each other in two intersecting (roughly Lagrange) orbits: Quicktime and GIF.

And here is another, in which 6 masses orbit each other with dihedral symmetry. We can think of this as two Lagrange orbits which co-rotate and interleave with each other each time they pass through a common plane. This is a (large) perturbation of a 6-mass Lagrange orbit; it is an example of a class of "hip-hop" orbits found by Chenciner and Venturelli. Quicktime and GIF.

The figure-8 can be perturbed in various ways. Nauenberg and Marchal independently found this version, which rotates around the x-axis and forms a continuous family of orbits connecting the figure-8 with the Lagrange orbit. Quicktime and GIF.

Here is another orbit, which I call the "criss-cross". It was first found numerically by Henon in 1976 as one of a family of orbits in rotating frames, starting with Schubart's one-dimensional orbit. I rediscovered it in 1993 by looking for orbits with a particular braid type, and its existence was proved rigorously by Kuo-Chang Chen. (This movie was made with Michael's Fourier technique.) Quicktime and GIF. It appears to be dynamically stable, and it exists for a wide range of mass ratios: here is an example Michael found where the masses are 1, 2, and 3. Quicktime and GIF.

Finally, here is a lovely applet by Gregory T. Minton of Microsoft Research, which lets you draw a choreography and then tries to minimize its action.

And here is a "mandala" produced from a stream of sand particles at the origin, also by Vishal Sanwalani. Jim Propp of the University of Wisconsin would like to know whether the limiting shape is a circle, or some kind of complex polygon. Some partial results in this direction, showing that it contains a diamond of radius r and is contained in a square of radius r, for some r ~ sqrt(n), were recently obtained by Babai and Gorodezky in SODA 2007, and Levine and Peres showed that it is bounded between two circles.

Here is a set of applets illustrating the Propp-Wilson coupling from the past algorithm.

- With a particle initially pointing to the left, after
25,
50,
100, and
200 steps
- With an equal distribution of initial directions, after
25,
50, and
100 steps
- With a different unitary operator with less symmetry, after
25,
50, and
100 steps

Copyright 2000 by Cris Moore. All rights reserved.