# Four-dimensional reflection groups and electrostatics

## TL;DR

This paper introduces a new class of exactly solvable electrostatics problems involving charges inside cavities bounded by up to four spheres, using symmetry from 4D reflection groups to determine image charges.

## Contribution

It extends classical electrostatics problems by leveraging 4D reflection group symmetries to find exact solutions with finitely many image charges.

## Key findings

- 19 families of reflection groups enable solvable electrostatics problems.
- A specific example with the group D4 requires 191 image charges.
- The method generalizes classical image charge solutions to higher symmetries.

## Abstract

We present a new class of electrostatics problems that are exactly solvable by adding finitely many image charges. Given a charge at some location inside a cavity bounded by up to four conducting grounded segments of spheres: if the spheres have a symmetry derived via a stereographic projection from a 4D finite reflection group, then this is a solvable generalization of the familiar problem of a charge inside a spherical cavity. There are 19 three-parametric families of finite groups formed by inversions relative to at most four spheres, each member of each family giving a solvable problem. We solve a sample problem which derives from the reflection group $\mathbf{D}_{4}$ and requires 191 image charges.