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Objective: 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 Study of Closedness and Self-similarity of 3D Fractals