# Convex Regular Polychora Nanocrystals with Dipole–Dipole Interactions

## TL;DR

This paper explores how dipoles arrange in four-dimensional shapes, finding that they can achieve zero net dipole moment with specific orientations.

## Contribution

The study introduces a novel theoretical framework for analyzing dipolar systems in four-dimensional convex regular polychora.

## Key findings

- All six four-dimensional polychora structures have zero net dipole moment.
- Minimum energy configurations involve dipole orientations with angles related to π.
- Numerical methods reveal stable dipole arrangements in higher-dimensional systems.

## Abstract

Structures composed of classical dipoles in higher-dimensional space present a unique opportunity to venture beyond the conventional paradigm of few-body or cluster physics. In this work, we consider the six convex regular polychora that exist in an Euclidean four-dimensional space as a theoretical benchmark for hte investigation of dipolar systems in higher dimensions. The structures under consideration represent the four-dimensional counterparts of the well-known Platonic solids in three-dimensions. A dipole is placed in each vertex of the structure and is allowed to interact with the rest of the system via the usual dipole–dipole interaction generalized to the higher dimension. We use numerical tools to minimize the total interaction energy of the systems and observe that all six structures represent dipole clusters with a zero net dipole moment. The minimum energy is achieved for dipoles arranging themselves with orientations whose angles are commensurate or irrational fractions of the number π.