Fractal bodies invisible in 2 and 3 directions

TL;DR

This paper demonstrates the construction of fractal bodies that are invisible in specific directions using geometrical optics, extending previous work on bodies invisible in fewer directions.

Contribution

It introduces methods to create fractal bodies invisible in two directions in 2D and three directions in 3D, advancing the understanding of invisibility in geometrical optics.

Findings

Existence of 2D fractal bodies invisible in two directions.

Existence of 3D fractal bodies invisible in three orthogonal directions.

Extension of previous constructions to more directions.

Abstract

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [A. Aleksenko and A. Plakhov. Bodies of zero resistance and bodies invisible in one direction. Nonlinearity 22, 1247-1258 (2009)], [A Plakhov and V Roshchina. Invisibility in billiards. Nonlinearity 24, 847-854 (2011)], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.