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

DOI:

https://doi.org/10.66147/lnaa.20231319

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. DOI: https://doi.org/10.1007/s10711-008-9340-3

[2] Kuratowski K., Topology, Volumes 1 and 2, Academic Press and PWN,New York, 1966. DOI: https://doi.org/10.1016/B978-0-12-429201-7.50005-3

[3] K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc.Camb.Philos.Soc. 155 (2013) 73-86. DOI: https://doi.org/10.1017/S0305004112000692

[4] J.J. Luo, J.-C. Liu, On the classification of fractal squares, Fractals 24 (2016) 1650008. DOI: https://doi.org/10.1142/S0218348X16500080

Downloads

Published

2023-05-14

Issue

1 (2023) No. 3

Section

Articles

License

Copyright (c) 2023 Dmitry Drozdov, Andrei Tetenov; Marina Tulina (Corresponding Author)

Creative Commons License

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

How to Cite

On fractal cubes in dimension 3 and their components. (2023). Letters in Nonlinear Analysis and Its Applications, 1(3), 96-100. https://doi.org/10.66147/lnaa.20231319