# Interpretation of crystallographic groups under Riemann's elliptic geometry

## TL;DR

This paper explores using elliptic geometry and closed spaces like the torus to model and interpret crystallographic groups and structures, offering new geometric frameworks for understanding crystal symmetry and internal structure.

## Contribution

It introduces a novel approach of representing all 230 crystallographic groups via elliptic motions in a closed space, proposing a new geometric model for real crystallographic space.

## Key findings

- Crystallographic groups can be modeled by elliptic motions in closed space.
- A new geometric model using the surface of a torus as an interpretant is proposed.
- Properties of real crystal structures are derived from the properties of the elliptic model.

## Abstract

This paper is devoted to the problem of choosing the most suitable model of a geometrical system for describing the real crystallographic space. It has been shown that all 230 crystallographic groups used to describe the crystalline structures in a Euclidean space can be presented by elliptic motions in the closed space $V^3$. Based on these results, it is stated that a special geometric system---the crystallographic space of interpretation $R_E$, determined by a form of an interpretant (the surface of a torus $T^2$ can serve as this interpretant)---can serve as a geometrical model of the real crystallographic space. The compact model of the closed structure of a crystal has been proposed and ways of its treatment for visualizing the constructions of elements of symmetry of a crystalline lattice in the Euclidean space $E^3$ have been determined. As a modeling space for describing the internal structure of a crystal, the closed space $V^3$ with the elliptic metrics and constant positive Gaussian curvature ($K=1$) has been offered. The properties of the internal space of a real crystal are naturally deduced from the properties of the modeling space.