Self-similar fractals related to regular tetrahedron and imaginary cubes

TL;DR

This paper explores a family of self-similar fractals related to regular tetrahedra, focusing on their geometric properties as imaginary cubes, and provides criteria for identifying such sets along with their symmetry and connectivity features.

Contribution

It introduces a criterion for self-similar sets to be imaginary cubes and analyzes their geometric properties, including symmetry and connectedness, in three-dimensional space.

Findings

Sierpiński tetrahedron is an imaginary cube.

Criteria for self-similar sets to be imaginary cubes.

Properties related to symmetry and connectedness of these sets.

Abstract

We consider self-similar sets in three-dimensional Euclidean space related to a regular tetrahedron. Sierpi${\rm \acute{n}}$ski tetrahedron is one such self-similar set. In this paper, we study the whole family of those sets. Our motivation is to obtain three-dimensional analogues of the fractal $n$-gons. In particular, we focus on the geometric properties of those sets from a viewpoint of ``imaginary cube''. An imaginary cube is a set $A$ for which there is some cube $C$ such that the projections of $A$ in the directions of the faces of $C$ equal these projections of $C$. It is already known that the Sierpi${\rm \acute{n}}$ski tetrahedron is an imaginary cube. We obtain a criterion for self-similar sets to be imaginary cubes. Furthermore, we show some properties of those sets which are imaginary cubes from a viewpoint of rotational symmetry or connectedness.