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During my time with the TAMU Fractals Research Team (August 2023 - May 2024), I explored the fascinating world of fractals, with a particular focus on the Sierpiński gasket.

The Sierpiński gasket (also called the Sierpiński triangle) is one of the most famous examples of a fractal, which is a type of mathematical object that has “self-similarity” at all scales. The Sierpiński gasket is named after Polish mathematician Wacław Sierpiński, and can be generated through several different methods.

The most direct method to generate the Sierpiński gasket is through an infinite process of splitting an equilateral triangle into 4 smaller triangles. Since repeating a process infinitely is impossible, we instead approximate the Sierpiński gasket using a finite number of iterations.

Research Approach

In our research, we used C++ to implement various algorithms for generating and analyzing the Sierpiński gasket. Using matrices and techniques from linear algebra, we were able to recursively generate approximations of the Sierpiński gasket and analyze its properties as the number of iterations increased.

Applications and Connections

The Sierpiński gasket has applications in:

• Antenna design
• Computer graphics and procedural generation
• Network topology optimization
• Natural patterns (such as some leaf structures)
• Making cool shapes