On fractal cubes in dimension 3 and their components

Authors

Dmitry Drozdov Novosibirsk State Technical University, Novosibirsk 630073, Russia; Novosibirsk State University, Novosibirsk 630090, Russia
Andrei Tetenov Novosibirsk State University, Novosibirsk 630090, Russia
Marina Tulina Gorno-Altaisk State University, Gorno-Altaisk 649000, Russia Corresponding Author

Keywords:

fractal cube, fractal square, Hausdorff dimension, hyperspace

Abstract

We show that a fractal cube \(F\) in \(\mathbb R^3\) may have an uncountable set \(Q\) of connected components which are not contained in any plane, and the set \(Q\) is a totally disconnected self-similar subset of the hyperspace \(C(\mathbb R^3)\), isomorphic to a Cantor set.

References

[1] Cristea L. L., Steinsky B., Curves of infinite length in 4 × 4-labyrinth fractals,Geom. Dedicata, 141(2009), 1–17.

[2] Kuratowski K., Topology, Volumes 1 and 2, Academic Press and PWN,New York, 1966.

[3] K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc.Camb.Philos.Soc. 155 (2013) 73-86.

[4] J.J. Luo, J.-C. Liu, On the classification of fractal squares, Fractals 24 (2016) 1650008.

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Published

2023-05-14

Issue

1 (2023) No. 3

Section

Articles

License

This work is licensed under a Creative Commons Attribution 4.0 International License.