Geometry of plane sections of the infinite regular skew polyhedron {4,6|4}

TL;DR

This paper investigates the complex structure of plane sections of the infinite regular skew polyhedron {4,6|4}, revealing fractal patterns and confirming that these fractals have zero measure through an elementary algorithm.

Contribution

It provides a detailed analysis of the fractal structures arising in plane sections of the skew polyhedron {4,6|4} and verifies Novikov's conjecture about their measure.

Findings

Fractal structures appear in the plane sections of {4,6|4}.

An elementary algorithm generates these fractals.

Fractals have zero measure, confirming Novikov's conjecture.

Abstract

The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail the results for the skew polyhedron {4,6|4} and in particular we show that in this case a fractal arises and that such a fractal can be generated through an elementary algorithm, which in turn allows us to verify for this case a conjecture of S.P.Novikov that such fractals have zero measure.