On the connected components of fractal cubes

2020/02/07 by Dmitry Drozdov, Drozdov, Dmitry, Andrei Tetenov +1
Computer Science · Mathematics · Physics and Astronomy · #28A80 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2002.02920

openalex publication_date 2020/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a fractal cube F in \mathbb R³ may have an uncountable set Q of connected components K_α neither of which is contained in any plane, whereas the set Q is a totally disconnected self-similar subset of the hyperspace C(\mathbb R³), isomorphic to a Cantor set.

Related

• Every component of a fractal square is a Peano continuum
• Quasi-fractal sets in space
• A Survey on the Topology of Fractal Squares
• On the classification of fractal squares
• A dimension drop phenomenon of fractal cubes