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

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 Xm of m points xj = (xj, yj, zj), j = 1,...,m on the torus is defined by, for s > 0,

Es(Xm) = sum j ~= i   1 / | xj - xi |s = 2   sum j > i   1 / | xj - xi |s ,
where | x | is the Euclidean length of the vector x. The limit as s -> 0+ is the logarithmic potential
Es(Xm) = sum j ~= i   log( 1 / | xj - xi |)

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.

 s Point Set Circle density s Point Set Circle density 0.00 0.10 0.50 0.80 1.00 2.00 3.00 4.00