Spatial Central Configurations with Two Twisted Regular 4-Gons

Abstract

We study the configuration formed by two squares in two parallel layers separated by a distance. We picture the two layers horizontally with the $z$-axis passing through the centers of the two squares. The masses located on the vertices of each square are equal, but we do not assume that the masses of the top square are equal to the masses of the bottom square. We prove that the above configuration of two squares forms a central configuration if and only if the twist angle is equal to $k\pi /\mathrm{2}$ or ($\pi /\mathrm{4}+k\pi /\mathrm{2}$) $(k=\mathrm{1,2},\mathrm{3,4})$.