For centuries, Euclidean geometry was seen as the absolute framework to describe space. It’s built on a handful of simple, seemingly self-evident axioms—basic truths about points, lines, and planes. Among these, one stands out: the Fifth Axiom, or the Parallel Postulate, which essentially states that through any point not on a given line, there is exactly one line parallel to the given line.
But here’s where things get interesting. Mathematicians eventually realized that this axiom isn’t the only way to build a coherent, consistent geometry. If you tweak or reject the Fifth Axiom, you open the door to entirely new types of geometries — non-Euclidean ones. These are just as logically valid, but they describe worlds where the rules of parallel lines, angles, and distances behave differently.
To understand these "strange" geometries, it helps to think about how we represent dimensions. Our computer screens, after all, are two-dimensional, but we often view convincing three-dimensional images on them. This illusion comes from projections—rules for translating 3D objects into a 2D view, like orthographic or perspective projection. Similarly, we can visualize non-Euclidean spaces by projecting them onto familiar 2D surfaces.
One of the most famous of these non-Euclidean spaces is hyperbolic geometry, where the Parallel Postulate fails dramatically—infinitely many lines can pass through a point and never intersect a given line. There’s also elliptic geometry, but we won’t focus much on that here. Instead, we’ll dive into hyperbolic space and explore it through a powerful tool: the Poincaré disk model.
In the Poincaré disk, the entire infinite hyperbolic plane is compressed into the interior of a circle. But there’s a catch—the edge of the circle represents points at infinity. Just like how, in real life, the horizon feels endlessly far away no matter how far you travel, the boundary of the Poincaré disk gives a visual sense of infinity. It’s similar to how parallel lines in a perspective drawing appear to meet at a vanishing point: it's not that the lines actually meet, but our projection forces that behavior.
Building on this foundation of non-Euclidean exploration, this project serves as a dynamic 3D rendering engine that leverages ray-casting techniques to navigate an infinite maze set in a hyperbolic world—visualized through the Poincaré disk model. Rather than simply crafting a labyrinth to solve for amusement, the primary goal is educational. This project acts as a concise yet comprehensive tutorial, offering insights into several advanced concepts:
- Non-Euclidean Spaces: An exploration of geometries beyond the Euclidean, with a focus on hyperbolic space and its unique properties.
- The Poincaré Disk: A detailed look at how the infinite hyperbolic plane is mapped within a finite circle, providing an intuitive grasp of infinity and projection.
- Hyperbolic Tiling: An examination of the intricate patterns that emerge from tessellations in hyperbolic geometry.
- Maze Algorithms and Ray-casting: A demonstration of algorithmic techniques for maze generation and the application of ray-casting in a 3D environment, all within the context of non-Euclidean rules.
Through a series of videos and animations (using the project itself or Manim), this project is designed to clarify how these concepts interconnect, offering both a practical rendering engine and a brief documentation resource. The aim is to provide a hands-on understanding of non-Euclidean spaces, making abstract mathematical theories accessible through interactive visualizations.