Consider the torus around the z axis with radii R_{1}
and R_{2}. One parametrization of this torus is

y = (R

z = R

If R_{1} > R_{2} the torus does not intersect itself.
The points on the torus satisfy

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,

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.

These images all correspond to a torus with R_{1} = 3, R_{2} = 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 |

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 |

Windows .avi files (LARGE uncompressed) of minimum energy points on a torus for a range of energy parameters s.

- te1000.avi (31 MB) 1000 minimum energy points for s between 0.01 and 4.00
- te4000.avi (27 MB) 4000 minimum energy points for s between 0.01 and 3.00

D. P. Hardin and E. B. Saff,
*Discretizing manifolds via minimum energy points*,
Notices of the AMS **51**, 10 (2004) 1186 - 1194.