Introduction

This gallery presents all equilibrium configurations of 8 dipoles located at the corners of a cube. The dipoles have equal strength and are freely orientable. The full dipole-dipole interaction is considered. This system allows for 185 energy families with exactly 9536 discrete configurations (families 1 to 182 and 184) and 2 degenerate continuous equilibria (families 0 and 183, visualized by 6 exemplary configurations along the continuum). More information can be found in this paper.

You can pick a family by dragging the slider below the rendering window or click somewhere along the slider to jump to a family. You can also use the left and right keys on the keyboard or the -/+ buttons next to the slider to go through the families consecutively. The interaction with the rendering of the configurations in the right panel above is possible through the classical trackball control (mouse left to rotate, mouse right to pan, scroll too zoom).

Energy Families and Symmetry

The families are sorted by energy. All members of a family have the same energy. The members are related by the possible symmetry operations of the system: The 48 isometries of the cube and the polarity symmetry of the dipoles (flipping plus and minus). This corresponds to 96 possible symmetry operations in total. Depending on the symmetries in a family, certain operations give identical configuration. The number of members tells you, how many distinguishable configurations there are in a family. For a family which breaks all symmetries, there will be 96 members. The lowest possible number of members is 2, since the polarity flip always gives a distinguishable configuration. In general, the range of the indicator bars goes from the minimum to the maximum values (throughout all families) of the property shown.

Net and Toroid Moment

The net moment is given by the sum of all (vector-valued) dipole moments divided by the number of dipoles. The value shown is the absolute value of the net moment. It tells you how strong the dipole moment of the whole configuration is in the far field.

The toroid moment is given by the sum of all cross products between a vector which points from a fixed point in space (e.g. the center of the cube) to the positions of the dipoles and the dipole moments, again divided by the number of dipoles. The value shown is the absolute value of the toroid moment. The toroid moment is a measure for the strength of vortex-like structures in the configuration. More details can be found in the supplements of this paper.

Stability

The question of stability is concerned with the behavior of the system under small perturbations. If a slightly perturbed equilibrium configuration relaxes back to the unperturbed state, we call the configuration stable, if it drifts further away, unstable. The stability is determined by the 16 eigenvalues of the Hessian matrix, i.e. the matrix of all second order partial derivatives of the energy with respect to the 16 degrees of freedom (2 angles per dipole). The number of positive, zero and negative eigenvalues is displayed. If all eigenvalues are positive, a configuration is stable. If at least 1 eigenvalue is negative, a configuration is unstable. A degenerate continuous state has 1 zero eigenvalue and its stability is determined by the other 15 eigenvalues.