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

TL;DR

This work characterizes the directions along which layered fractal imaginary cubes, including the Sierpinski tetrahedron, T-fractal, and H-fractal, project to sets of positive Lebesgue measure. Using a framework based on differenced radix expansion sets and self-affine tile theory, it reduces projections to lattice-tiling conditions on affine digit sets ${D}'_{k,l}$ and derives a precise criterion: a projection along a coprime vector $(a,b,c)$ yields a positive-measure image if and only if $a+b+c$ is coprime to $k$ (with a special exception for $(k,l)=(3,1)$ due to extra symmetry in the H-fractal). The authors provide complete proofs for the Sierpinski tetrahedron and extend the approach to general ${\mathcal{F}}(k,D_{k,l})$, establishing the role of hexagonal-symmetric decompositions and auxiliary sets in the argument. This advances understanding of projection properties of self-affine fractals and links them to tiling theory, with implications for shadow-like projections and dimensional analysis of fractal images.

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.

Figures (13)

• Figure 1: Projected images of the Sierpinski tetrahedron.
• Figure 2: Examples of imaginary cubes.
• Figure 4: The arrangements of cubes by (a) ${D}_{2, 0}$, (b) ${D}_{3, 0}$, and (c) ${D}_{3, 1}$, which generate $\mathrm S_\infty$, $\mathrm T_\infty$, and $\mathrm H_\infty$, respectively.
• Figure 5: The first two approximations of (a) $\mathrm S_\infty$, (b) $\mathrm T_\infty$, and (c) $\mathrm H_\infty$.
• Figure 6: Projected images of layered fractal imaginary cubes.

Theorems & Definitions (41)

• Definition 1
• Lemma 1: bridges1
• proof
• Lemma 2
• Theorem 3
• Corollary 4
• Lemma 5
• proof
• Theorem 6
• proof : Proof (Equivalence of Theorem \ref{['mainlemma']} and Theorem \ref{['mainth1']}).