# Beltrami vector fields with polyhedral symmetries

## TL;DR

This paper classifies and constructs Beltrami vector fields with polyhedral symmetries in three dimensions, extending previous work to tetrahedral and octahedral cases, and explores their connections to solutions of the Helmholtz equation and flows like the ABC flow.

## Contribution

It extends the classification of symmetric Beltrami vector fields to tetrahedral and octahedral symmetries, and analyzes their relation to Helmholtz solutions and special flows.

## Key findings

- Constructed Beltrami fields with tetrahedral and octahedral symmetries.
- Calculated all simplest polyhedral symmetric Beltrami fields from Helmholtz solutions.
- Identified a unique 2D flow related to the ABC flow.

## 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.