Projected images of the Sierpinski tetrahedron and other layered fractal imaginary cubes

TL;DR

This paper develops a method to identify directions where layered fractal 3D objects, like the Sierpinski tetrahedron, project onto sets with positive measure, enhancing understanding of their geometric properties.

Contribution

It introduces a new characterization technique for projections of layered fractal objects, including the Sierpinski tetrahedron and related fractals, onto sets with positive Lebesgue measure.

Findings

Identified directions for positive measure projections of layered fractal cubes.

Provided a comprehensive classification for layered fractal imaginary cubes.

Applied the method successfully to several well-known fractals.

Abstract

The Sierpinski tetrahedron has a remarkable property: It is projected to squares in three orthogonal directions, and moreover, to sets with positive Lebesgue measures in numerous directions. This paper proposes a method for characterizing directions along which the Sierpinski tetrahedron and other similar fractal 3D objects are projected to sets with positive measures. We apply this methodology to layered fractal imaginary cubes and achieve a comprehensive characterization for them. Layered fractal imaginary cubes are defined as attractors of iterated function systems with layered structures, and they are projected to squares in three orthogonal directions. Within this class, the Sierpinski tetrahedron, T-fractal, and H-fractal stand out as exemplary cases.