Five-Collinear Sierpinski Gasket is Not Self-Similar

TL;DR

This paper proves that while smaller line arrangements of Sierpinski gaskets are self-similar, a line of five Gaskets is not, revealing a limit to their self-similarity property.

Contribution

It establishes the non-self-similarity of a line of five Sierpinski gaskets, extending understanding of their geometric properties.

Findings

Line of five Sierpinski gaskets is not self-similar.

Smaller line arrangements (two to four) are self-similar.

Provides a geometric proof of non-self-similarity.

Abstract

A geometrical conclusion: Sierpinski gasket, two Sierpinski gaskets in a line, three Sierpinski gaskets in a line, and four Sierpinski gaskets in a line are self-similar, but five Sierpinski gaskets in a line is not, which is proved in this paper.