Analysis of Sierpinski Triangle Based on Fuzzy Triangular Numbers and Dihedral Group

Abstract

Fractals are indefinitely complex patterns such as self-similar across at different scales; for example, Sierpinski triangle is a fractal. This paper analysed in the Sierpinski triangle. It is considered as equilateral triangles such as 1 unit, k unit and k + 1 unit. Each iteration is divided as [(0, 1/4, ½, …, 1)], [(0, k/4, k/2…, k)], [(0, (k + 1)/4, (k + 1)/2,… (k + 1))], so on. It analysed this triangle which satisfies fuzzy triangular numbers and the number of the theoretical aspect of fuzzy triangular numbers (FTNs) in self-similarity set of fractal set (Sierpinski triangle) and some arithmetic operations of α-ut and discussed that this triangle satisfied the centroid and median of the normal triangle. Multiplication of fuzzy triangular numbers α-cuts is explained graphically. It also analysed that these smaller equilateral triangles form a group, and this group satisfies the property of dihedral group.