A pedagogical guide into trigonometric transformations

Abstract

A tool is a device that augments an individual's ability to perform a particular task. The more specificity a tool has, the narrower its instrumentality. Tools inherently constrain the way individuals design; however, designers are often unaware of their influence and bias. Digital tools are becoming increasingly complex and filled with hierarchical symbolic heuristics, creating a black box in which designers do not understand what is "under the hood" of the tools they drive. And yet designers are becoming fascinated with engineering mentalities: optimization and automation. Simply, it gives a solution. But, this is not design! Designers need to work outside of a fixed atmosphere! The future of digital instruments is not more complex heuristics, but rather the contrary. It is imperative to go back to the most basic building blocks of these "engines:" mathematics. Within mathematics, functions can be embedded inside other functions at anytime, giving designers endless freedom to alter the computational hierarchy. By "playing" with parametric equations and tacit engagement with the algorithm, one can begin to learn explicitly how discrete operations transform shapes in a particular way. This guide embraces the thought that all shapes could potentially be described by the trigonometric functions of sine and cosine. These functions became the only fixed constraint to instrumentalize. Through the recursive "play" and learn process, a new morphological classification of topological transformations emerged, leading to the development of this guide, the first pedagogical guide into trigonometric transformations. This guide does not invent new realms within the field of mathematics, but develops a new cognitive narrative within it, emphasizing the interconnected and plastic nature of shapes.