Beltrami vector fields with polyhedral symmetries

Abstract: A 3-dimensional vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective an polynomial superflows, and as an important side result, in the first paper on this topic we constructed two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have orientation-preserving icosahedral symmetry (group of order $60$). In the current paper we extend these results to the tetrahedral and octahedral cases, and (together with an icosahedral case) we calculate all simplest Beltrami fields with polyhedral symmetries arising from solutions to the Helmholtz equation of any order (the first aforementioned paper being an order 1 approach). The notion of Beltrami vector field, slightly relaxed, generalizes to any dimension. In this paper we also present 2-dimensional vector fields which have a dihedral symmetry $\mathbb{D}_{2d+1}$ of order $4d+2$. A much more detailed analysis is carried out in case $d=1$. One of these fields is particularly exceptional since it is the only case in our investigations which arises from the order $0$ approach to the Helmholtz equation, thus relating this flow to the $ABC$ flow.