scholarly journals New Classes of Regular Symmetric Fractals

2021 ◽  
Author(s):  
Subhash Kak
Keyword(s):  
Sierpinski Carpet ◽  
Special Cases ◽  
Menger Sponge ◽  
Sierpiński Carpet

The paper introduces new fractal families with annular and checkerboard structures that include the Sierpinski carpet and the Menger sponge as special cases. The complementary mapping is defined and a notation to represent the families is proposed.

scholarly journals Two Classes of Regular Symmetric Fractals

2021 ◽  
Author(s):  
Subhash Kak
Keyword(s):  
Sierpinski Carpet ◽  
Special Cases ◽  
Menger Sponge ◽  
Sierpiński Carpet

The paper introduces new fractal families with annular and checkerboard structures that include the Sierpinski carpet and the Menger sponge as special cases. The complementary mapping is defined and a notation to represent the families is proposed.

scholarly journals Two Classes of Regular Symmetric Fractals

2021 ◽  
Author(s):  
Subhash Kak
Keyword(s):  
Sierpinski Carpet ◽  
Special Cases ◽  
Menger Sponge ◽  
Sierpiński Carpet

The paper introduces new fractal families with annular and checkerboard structures that include the Sierpinski carpet and the Menger sponge as special cases. The complementary mapping is defined and a notation to represent the families is proposed.

scholarly journals Amazing Properties of Fractal Figures

2016 ◽  
Vol 16 ◽  
pp. 37-43
Author(s):  
Oksana Mandrazhy
Keyword(s):  
High School ◽  
High School Students ◽  
Surface Area ◽  
Total Surface Area ◽  
School Students ◽  
Sierpinski Carpet ◽  
Menger Sponge ◽  
Sierpiński Carpet

This article describes the process of research of the properties of geometric fractals by high school students. The general formulas of calculating the length and the area of the Sierpinski carpet have been derived in the article. The total surface area and the volume of the Menger sponge have been calculated in the paper. The amazing facts of geometric fractals have been revealed. For instance, if n→∞, the length of the Sierpinski carpet is Ln→∞, and its area is Sn→0, similarly, if n→∞, the total surface area of the Menger sponge is Sts(n)→∞, and its volume is Vn→0.

scholarly journals THE GENERALIZATION OF SIERPINSKI CARPET AND MENGER SPONGE IN n-DIMENSIONAL SPACE

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750040 ◽  
Author(s):  
YUN YANG ◽  
YUTING FENG ◽  
YANHUA YU
Keyword(s):  
Affine Transformation ◽  
Dimensional Space ◽  
Sierpinski Carpet ◽  
Menger Sponge ◽  
Sierpiński Carpet

In this paper, we generalize Sierpinski carpet and Menger sponge in [Formula: see text]-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet and Menger sponge. Exactly, Menger sponge in [Formula: see text]-dimensional space could be drawn out clearly under an affine transformation. Furthermore, the method could be used to a much broader class in fractals.

GEODESICS IN THE SIERPINSKI CARPET AND MENGER SPONGE

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050120
Author(s):  
ETHAN BERKOVE ◽  
DEREK SMITH
Keyword(s):  
Lower Bound ◽  
Higher Dimensions ◽  
Sharp Bound ◽  
Sierpinski Carpet ◽  
Smooth Curves ◽  
Euclidean Metric ◽  
The Family ◽  
Menger Sponge ◽  
Rectifiable Curves ◽  
Sierpiński Carpet

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related to the skeletal graph approximations of the Sierpinski carpet that have been studied by many authors. We then provide a sharp bound on the ratio of the taxicab metric to the Euclidean metric, extending Cristea’s result for the Sierpinski carpet. As an application, we determine the diameter of the Sierpinski carpet taken over all rectifiable curves. For other members of the family, we provide a lower bound on the diameter taken over all piecewise smooth curves.

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