This literature review investigates the evolving field of analysis on fractals. The applications of this analysis span from the quantification of structural complexity in medical imagery to the evaluation of non-linear financial market patterns. The complexity of an infinitely self-similar geometric object necessitates novel mathematical constructions to represent these fractals, and thus, new models are required to describe how differential equations behave on these fractals. This review aims to consolidate current understandings of analysis, specifically on the Sierpinski Gasket. Building on the foundational frameworks established by preeminent researchers like Robert Strichartz, this review begins with a comprehensive construction of the Gasket. This is followed by two formulations of the Laplacian on this non-smooth surface, and an overview of how the eigenvalues and eigenfunctions of this fractal Laplacian are derived. Finally, the review examines non-linear elliptic and parabolic equations on the gasket, and recent steps to distinguish the gasket’s spatial geometry from its underlying linear connection topology. 

Mathematicians have been able to expand on the ideas of a man named Albrecht Dürer and have found nets that fold into 2 distinct boxes with the same surface area. For example, the 1x1x5 box and the 1x2x3 box share the same surface area of 22 and share multiple nets. In fact, they were able to prove that any two boxes with the same surface area can share a net. Taking the same 3-dimensional ideas of unfoldings of boxes, our goal is to move them into the 4th dimension and find a net that is able to fold into two different 4-dimensional boxes. The surface area of a 3-dimensional object is 2-dimensional, in the same way that the surface area of a 4-dimensional object is 3-dimensional. So, the unfolding of the 4-dimensional box, would be a 3-dimensional object. The smallest 4-dimensional boxes that share the same surface area are the 1x1x1x7 box and the 1x1x2x4 box, so a shared net of the two is the goal. Since we are working with boxes, we can use a lattice polytope, so we work with cubes that represent the surface area of the 4-dimensional boxes, to understand how to unfold in 4 dimensions. Everything that is learned from this project can help find patterns from 3 dimensions into 4, as the 4th dimension has not been heavily researched as it is hard to understand and imagine.