Visualization of Minimum Energy Points on the Torus

Quicklinks: Parametrization | Images 1000 points | Images 4000 points | Movies

Torus with aspect ratio R1/R<sub>2</<sub>

Consider the torus around the z axis with radii R₁ and R₂. One parametrization of this torus is

x = (R₁ + R₂ cos(v)) cos(u)
y = (R₁ + R₂ cos(v)) sin(u)
z = R₂ sin(v) where u in [0, 2 Pi) is the angle about the z axis and v is in [0, 2 Pi).

If R₁ > R₂ the torus does not intersect itself. The points on the torus satisfy

( R₁ - (x² + y²)^1/2 )² + z² = R₂² The aspect ratio of the torus is R₁ / R₂.

The Riesz s-energy of a set X_m of m points x_j = (x_j, y_j, z_j), j = 1,...,m on the torus is defined by, for s > 0,

E_s(X_m) = sum_{j ~= i} 1 / | x_j - x_i |^s = 2 sum_{j > i} 1 / | x_j - x_i |^s , where | x | is the Euclidean length of the vector x. The limit as s -> 0⁺ is the logarithmic potential E_s(X_m) = sum_{j ~= i} log( 1 / | x_j - x_i |)

The circle densities are obtained by projecting the points onto the x - z plane (u = 0), calculating the number in each of around 30 equal width bins, and normalizing by the distance from the axis of ratation (the z-axis here). These discrete distributions have been approximated by the first four Fourier modes, and the resulting approximation used to colour the underlying tori.

These experiments were motivated by the remarkable result by Hardin and Saff that for s >= 2 (the dimension of the torus), the limiting distribution of points over the surface of the torus as the number of points goes to infinity is the uniform distribution.

The minimum energy points were calculated using a limited memory quasi-Newton algorithm embedded in a local perturbation method in an attempt to find a global minimum. The point sets are normalized so that the first point has u = 0, removing the rotational invariance about the z axis. As for the sphere, these problems have many local minima close to the global minima.

Caveat: These are only approximate local minima. There is no known guarantee that a global minimum has been found. Images

These images all correspond to a torus with R₁ = 3, R₂ = 1.