The Study on the Evaluation of Closedness and Self- Similarity of 3D Fractals

Objective: The current study was aimed to illustrate a methodology that demonstrates the closedness
and self-similarity of 3D fractals taking 3D Sierpinski Gasket as an example.
Methods/Statistical Analysis: The 2D and 3D images of the Sierpinski Gasket were perceived by
taking the initial generators as a foundation giving rise to either Sierpinski Triangle/Sierpinski Pyramid
or Sierpinski Carpet/Sierpinski Gasket. This method ensures that final 3D Sierpinski Gasket is
generated which an exact copy of the original image is taken.
Findings: This is accomplished by taking a cube as the base and apply the same algorithm in a
recursive manner by way of IFS-like transformation consisting of ((x,y) rotation, z(zoom)) and
changing the depth parameter from 3 to 2 to 1 in that order, giving rise to a new 3D Sierpinski Gasket
that has a resemblance to an exact copy of the original 3D cube-based Sierpinski Gasket. The results
depict the closedness in self-similarity aspect of fractal images and thereby give a real-time vision of
fractal images.
Application/Improvements: The results embodied from the above method can be useful for further
research and the extension of this work gives us more innovative analysis in the field of 3D fractals
and the like. The results demonstrate the closedness in self-similarity aspect of fractal images and
thereby give a realtimevision of fractal images, and how when using theory, computation and
experimentation, show that fractal geometry is an art and a science.