International Journal of Mathematical Sciences and Computing(IJMSC)

ISSN: 2310-9025 (Print), ISSN: 2310-9033 (Online)

Published By: MECS Press

IJMSC Vol.3, No.4, Nov. 2017

Construction of Fractals based on Catalan Solids

Full Text (PDF, 467KB), PP.1-7

Views:62   Downloads:4


Andrzej Katunin

Index Terms

Deterministic fractals;iterated function system;Catalan solids


The deterministic fractals play an important role in computer graphics and mathematical sciences. The understanding of construction of such fractals, especially an ability of fractals construction from various types of polytopes is of crucial importance in several problems related both to the pure mathematical issues as well as some issues of theoretical physics. In the present paper the possibility of construction of fractals based on the Catalan solids is presented and discussed. The method and algorithm of construction of polyhedral strictly deterministic fractals is presented. It is shown that the fractals can be constructed only from a limited number of the Catalan solids due to the specific geometric properties of these solids. The contraction ratios and fractal dimensions are presented for existing fractals with adjacent contractions constructed based on the Catalan solids.

Cite This Paper

Andrzej Katunin,"Construction of Fractals based on Catalan Solids", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.3, No.4, pp.1-7, 2017.DOI: 10.5815/ijmsc.2017.04.01


[1]Vidya D, Parthasarathy R, Bina TC, Swaroopa NG. Architecture for fractal image compression. J Syst Arch 2000;46:1275-91.

[2]Khemis K, Lazzouni SA, Messadi M, Loudjedi S, Bessaid A. New algorithm for fractal dimension estimation based on texture measurements: Application on breast tissue characterization. Int J Image Graph Signal Process 2016;4:9-15.

[3]Nayak SR, Mishra J. On calculation of fractal dimension of color images. Int J Image Graph Signal Process 2017;3:33-40.

[4]Xu T, Moore ID, Gallant JC. Fractals, fractal dimensions and landscapes – a review. Geomorphology 1993;8:245-62.

[5]Delogu F. Icosahedral coordination and atomic rearrangements in deformed metallic glasses Acta Mater 2011; 59:5961-9.

[6]Auffray C, Nottale L. Scale relativity theory and integrative systems biology: 1 Founding principles and scale laws. Prog Biophys Mol Biol 2008;97:79-114.

[7]Katunin A. Fractal dimension-based crack identification technique of composite beams for on-line SHM systems. Mach Dyn Res 2010;34:60-69.

[8]Katunin A, Serzysko K. Detection and localization of cracks in composite beams using fractal dimension-based algorithms – a comparative study. Mach Dyn Res 2014;38:27-36.

[9]Agarwal S. Symmetric key encryption using iterated fractal functions. Int J Computer Network Inf Security 2017;4:1-9.

[10]Shareef AN, Shaalan AB. Fractal Peano antenna covered by two layers of modified ring resonator. Int J Wireless Microwave Technol 2015;2:1-11.

[11]Hart JC, Sandin DJ, Kauffman LH. Ray tracing deterministic 3D fractals. Proc 16th Annual Conf on Computer Graphics and interactive Techniques SIGGRAPH '89. New York: ACM Press; 1989:289-96.

[12]Patrikalakis NM (Ed.). Scientific Visualization in Physical Phenomena. Tokyo: Springer-Verlag; 1991.

[13]Menger K. Dimensionstheorie. Wiesbaden: Springer Fachmedien; 1928.

[14]Jones H, Campa A. Fractals based on regular polygons and polyhedra. In: Scientific Visualization of Physical Phenomena, Patrikalakis NM (Ed.). Tokyo: Springer-Verlag; 1991:299-314.

[15]Kahng B, Davis J. Maximal dimensions of uniform Sierpinski fractals. Fractals 2010;18:451-60.

[16]Kunnen A, Schlicker S. Regular Sierpinski polyhedra. Pi Mu Epsilon J 1998;10:607-19.

[17]Molina-Abril H, Real P, Nakamura A, Klette R. Connectivity calculus of fractal polyhedrons. Pattern Recogn 2015;48:1150-1160.

[18]Liu C, Panetta RL, Yang P, Macke A, Baran AJ. Modeling the scattering properties of mineral aerosols using concave fractal polyhedra. Appl Opt 2013;52:640-652.

[19]Catalan E. Mémoire sur la théorie des polyèdres. J de l'École Polytechnique 1865;41:1-71.

[20]Katunin A. Deterministic fractals based on Archimedean solids. Sci Res Inst Math Comput Sci 2011;1(10):93-100.

[21]Katunin A, Kurzyk D. General rules of fractals construction from polyhedra. J Geom Graph 2012;16:129-137.