A Journey from Curiosity to Higher-Dimensional Geometry

The first draft of this work was completed in June 2017, when I was turning 18 in my final year of high school. At the time, I was learning a lot of fundamental mathematics and had a strong desire to understand dimensions beyond what we can see with our own eyes, in other words, trying to visualize higher-dimensional objects in my mind. Motivated by this curiosity while feeling I had acquired enough knowledge on the subject, I wrote a manuscript called “Dimensions”, which explored some mathematical concepts but primarily focused on interpreting and visualizing such new objects. During that period, I regarded the connection between geometry and combinatorics as the fundamental truth of mathematics, much like the mathematicians of earlier eras [2]. With this in mind, my essay aimed to bridge the gap between abstract mathematical concepts and intuitive visualization. This current version serves as a translation, reinterpretation, and self-review of the original draft, featuring refined explanations and updated insights. Adopting a bottom-up approach, it simultaneously serves as both a complement to and detailed revision of Coxeter’s foundational work [1].