Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. My first-year PhD student ( Kemizaa Huraashavy ) and I , have managed to introduce a theoretical framework that leverages isospectral reduction to effectively simplify complex fractal structures, revealing the presence of topologically protected boundary and corner states on semi-polygonal “curves” . Closures and partially-connected sets on 5_D and 6_D momentum analysis as per corrected general relativity , lead to our approach demonstrating that fractals can support topological phases, BOTH in matter AND in virtuality , even in the absence of traditional driving mechanisms such as magnetic fields or spin-orbit coupling. The Internet and Intranet resulting from such an isospectral reduction , not only elucidate the underlying topological features but also make this framework broadly applicable to a variety of fractal systems [ not accomplished elsewhere up till now ] . Furthermore, our findings suggest that these topological phases may naturally occur in materials with fractal structures found in nature , as well . This in itself opens new avenues for designing fractal-based topological materials, advancing both theoretical understanding and experimental exploration of topology in complex, self-similar geometries , and Internet fault-finding by means of AI .(( Dean Reza Sanaye ))

Relating indirectly to : Mesoscale and Nanoscale Physics