Projected images of the Sierpinski tetrahedron and other fractal imaginary cubes

Abstract

A projected image of a Sierpinski tetrahedron has a positive measure if and only if the images O, P, Q, R of the four vertices satisfy p OP + q OQ + r OR = 0 for odd numbers p, q, r. This fact was essentially obtained by Kenyon in [7]. We reformulate his proof through the notion of projection of differenced digit set so that it could be applied to projections of other fractal objects. We study projections of H fractal and T fractal as well as Sierpinski tetrahedron, which are the fractal imaginary cubes of the first two levels. We characterize the directions from which these fractals are projected to sets with positive measures.