Imaginary crystals made real

TL;DR

This paper demonstrates the creation of a stable molecular structure mimicking Lobachevsky geometry, enabling exploration of non-Euclidean surfaces and potential tests of curved spacetime physics.

Contribution

It introduces a method to realize non-Euclidean geometries in molecular structures, specifically a Beltrami pseudosphere, with implications for physics and materials science.

Findings

Successfully simulated a carbon-based Beltrami pseudosphere

Identified a non-Euclidean crystallographic group in the structure

Accounted for the unavoidable singular boundary

Abstract

We realize Lobachevsky geometry in a simulation lab, by producing a carbon-based mechanically stable molecular structure, arranged in the shape of a Beltrami pseudosphere. We find that this structure: i) corresponds to a non-Euclidean crystallographic group, namely a loxodromic subgroup of SL(2,Z); ii) has an unavoidable singular boundary, that we fully take into account. Our approach, substantiated by extensive numerical simulations of Beltrami pseudospheres of different size, might be applied to other surfaces of constant negative Gaussian curvature, and points to a general procedure to generate them. Our results also pave the way to test certain scenarios of the physics of curved spacetimes.