A general solution for rotating lattices of identical point vortices

The rotating equilibrium solutions of N identical point vortices are the stationary energy states of a higher-dimensional object projected to the x-y plane, in this case, an (N-1)-dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition, which contains the equilibrium states and resolves the origin of the asymmetric solutions. This novel approach uses the geometric product and the multivector derivative, partial derivative psi, of the Hamiltonian in the geometric algebra, Cl(N-1,0) to find a geometric condition that determines the equilibrium states. The resulting bivector equation is then used as the input to an optimizer, which rotates a simplex until the equilibrium condition is met, leading to a wealth of new solutions. If the vertices of the oriented simplex are projected to the x-axis, the values form the roots of the Hermite polynomials, HN(x), and obey the Stieltjes relations, capturing the collinear solutions. The vortex simplex exhibits a striking geometrical connection with the amplituhedron of quantum field theory and gives deep insight into the quantization of a classical system.