A Multidisciplinary Approach to Triangular Shapes: Philosophy, Art, Mathematical Properties, and Application Purposes for High-Frequency Signal Processing Using Sierpinski Geometry

Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems have been formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. This paper will review triangles by adopting a multidisciplinary approach, recalling ancient science and Plato's arguments in relation to their connection with philosophy. It will then consider the artistic utilization of triangles, particularly in compositions created during the medieval era, as exemplified by the Cosmati Italian family's masterpieces. Various scientific environments have explored triangular 2D and 3D shapes for different purposes, which will be briefly reviewed here. After that, the Sierpinski geometry and its properties will be introduced, focusing on the equilateral shape and its internal complexity generated by subdividing the entire triangle into smaller sub-triangles. Finally, examples of triangular planar shapes that fulfill the Sierpinski geometry will be presented as an application in signal processing for high-frequency signals in the microwave and millimeter-wave range.