A mathematical model for generation of interactive fractals using a fourth order fractal square: A case study

Abstract

Fractals can be interactive by using fractal squares which evolved from mathematics. Fractal square (FS) F can be deduced from the F = (F + D)/n, where # D = m. The F_n,m shows the FS having n order square boxes in the fractal square are maintained with m boxes and the rest of the boxes are to be discarded. Moreover, FS squares are a common used techniques in mathematical formulation (Pattern recognition and analytical imaging) for creating image patterns that are self-similar in nature. Nonetheless, recently there have been a number of studies produced on the 3^rd order fractals to explore the properties of different classes of fractals formed from it, however only the distinct patterns that are generated from it are being studied, along with Hausdorff dimension (HD) also been popular in order to determine the size of a fractal set. Although, all these existing techniques are limited in order to analysis of fractal set i.e., derived corner type, edge type and interior type. In this paper, we propose a novel dimension method called 4^th order FS and their dimensional analysis. As a new dimension approach, we present the Practical Grid Dimension (PGD) method, which utilizes the grids to calculate dimensions. As a last comparison, we have compared the HD approach with the PGD method in order to validate the proposed model. Researchers will be able to use this model to conduct more in-depth study for successful implementation in a wide range of fractal applications.

Subject Classification:

• Primary 93A30
• Secondary 49K15

Keywords:

• Interactive fractal
• Self-similar
• Fractal square
• Hausdorff dimension
• Practical grid dimension

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.