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

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.