Visualizing the Thomson Problem
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The Thomson Problem is a famous unsolved problem that has several applications in math, chemistry, and physics. The problem is to find the optimal distribution of point charges on a spherical shell so that the total energy of the system is minimized. Conceptually, this is achieved by making all of the points as far apart from one another as possible. For some numbers of points, such as four or six, the solution is relatively simple (these solutions take the geometry of a tetrahedron and octahedron respectively). However, most cases, such as eleven or nineteen, have no accepted solution. This Demonstration empirically finds equilibrium configurations from random initial positions for a range of numbers of points and compares their total energies to the known optimal ones.
Contributed by: Mark Peterson (January 2013)
Open content licensed under CC BY-NC-SA
Although this Demonstration often finds low-energy stable configurations, these systems are not necessarily the optimal solutions. In other words, any given number of points may have more than one stable outcome, but only one of these results is the optimal solution. For example, five points generally settle in the geometry of a trigonal bipyramid, but occasionally they will form a square pyramid. In the case of five points, the more likely result has also been proven to be the optimal solution, but it is conceivable that the solutions for some numbers of points are relatively improbable, so it cannot be assumed that a common stable state is the solution. For this, the "optimal energy" that is used as the benchmark in the Demonstration is merely the lowest energy that is known to be achievable in the cases of unknown solutions, such as nine and thirteen points.
There are three viewing options for visualizing the progression of the points:
Showing just the position vectors is the simplest option and provides the most fluid movement. It is useful for watching the progress of the points as they settle into their stable positions, particularly during the first few steps when forces are generally strongest and movement is most extreme.
Showing the faces of the equivalent polyhedron is probably the most intuitive method. This also displays some interesting characteristics of the stable states. Notice that simpler polygons are generally more stable and more likely to form than higher order ones. Regular polyhedron that have exclusively triangular faces, namely, the four-pointed tetrahedron, the six-pointed octahedron, and even the twelve-pointed icosahedron, are very stable, form reliably, and are in fact the solutions to those situations. Faces any more complex than a quadrilateral are extremely rare in these polyhedron.
Showing the energies of the points should help provide an understanding for the behavior of the points. The opacity of the area that surrounds any given point is proportional to the point's energy, so solid red displays high energy and translucent red displays low energy. If high energy exists between two points because they are nearby, then they will repel each other with a strong force. If low energy exists between two points because they are far apart, then they will only repel each other with a weak force. Consequently, the points will settle into a configuration that maximizes the distance between one another in order to achieve the lowest possible energy.
Use the "random seed" slider to generate new initial configurations of the points. This feature is particularly useful if the "progression" slider is set to its max so that moving the "random seed" slider quickly scrolls through stable states for any given number of points. This provides a good opportunity to see the different geometries of stable states that can occur. Generally, as the number of points increases, so does the number of possible stable states. Consequently, it may take several attempts to achieve a specific geometry for high-order polyhedron. An example of this is the snub cube, the solution for twenty four points (and the only known solution for any number of points greater than twelve). It is the optimal solution, but there are many stable states for twenty four points that have nearly the same total energy, so the probability of forming a snub cube in only a few steps is relatively low.
Snapshot 1: The case of six points is one of the most simple and stable. Notice how close the energy of the current distribution is to the optimal.
Snapshot 2: Contrarily, the case of eleven points is unpredictable. Depending on the initial distribution, the final configuration will be one of several common results, all of which have little symmetry.
Snapshot 3: As the shell becomes more dense with points, it also becomes more dense with energy. Compare this twelve-point configuration with snapshot 4.
Snapshot 4: Compare this sparsely energized four-point configuration with snapshot 3.
"Visualizing the Thomson Problem"
Wolfram Demonstrations Project
Published: January 21 2013