The Wallflower Fractal: A Decade-Long Mathematical Odyssey

Starting from a simple geometric doodle from middle school, the author delves into a decade-long mathematical exploration. The fractal pattern, affectionately called "the wallflower," can be generated using iterative algorithms or L-systems. The author discovers subtle differences between the patterns generated by these two methods, and reveals the connection between them through the introduction of a matrix base number system and vector digits, explaining the origin of the "approximately 27-degree" rotation angle. Further, the author explores 3D and 4D extensions of the fractal and proposes a novel number system based on quaternions. This article is full of mathematical ingenuity and surprising discoveries, showcasing the beauty of mathematics and the joy of exploration.