The Necker cube surface

TL;DR

This paper investigates the behavior of geodesics on the Necker cube surface, revealing how initial directions determine whether they close or drift, and explores the surface's symmetry and dynamical properties.

Contribution

It provides a comprehensive analysis of geodesic dynamics on the Necker cube surface, including conditions for closed geodesics, periodic drift, and the surface's symmetry group.

Findings

Geodesic closure and drift depend solely on initial direction.

Regions related to simple closed geodesics tile the plane periodically.

The surface's affine symmetry group is fully described, informing dynamical properties.

Abstract

We study geodesics on the Necker cube surface, $\mathbf N$, an infinite periodic Euclidean cone surface that is homeomorphic to the plane and is tiled by squares meeting three or six to a vertex. We ask: When does a geodesic on the surface close? When does a geodesic drift away periodically? We show that both questions can be answered only using knowledge about the initial direction of a geodesic. Further, there is a natural projection from $\mathbf N$ to the plane, and we show that regions related to simple closed geodesics tile the plane periodically. We also describe the full affine symmetry group of the half-translation cover and use this to study dynamical properties of the geodesic flow on $\mathbf N$. We prove results related to recurrence, ergodicity, and divergence rates.