I'm interested in creating software or perhaps a game set in four spatial dimensions. Although it's difficult to visualize four dimensions from a three-dimensional perspective, the mathematics is well defined and higher-dimensional objects can be projected into lower-dimensional views. I'm not sure what concepts, algorithms, or tools I should learn first, and I'd like to understand how other developers approach 4D games and visualization.
4 Answers
Start by treating a point as four coordinates, such as (x, y, z, w), and implement the usual vector and matrix operations for that space. You can then define transformations, geometry, lighting, and collisions in 4D before projecting the result into 3D or directly into a 2D image. A 4D-to-3D perspective projection works much like projecting 3D coordinates onto a screen, just with one extra stage.
You may find it helpful to study existing four-dimensional games and visualization projects. There are already examples of 4D puzzles, platforming, golf, and non-Euclidean experiments, and some developers have published explanations of how their projection and movement systems work. These can provide practical ideas for structuring your own project.
A useful first project would be a rotating tesseract or another simple 4D polytope. Build the vertices and edges, rotate them using 4D rotation matrices, then project the transformed points into 3D and draw the resulting wireframe. This lets you learn the core math without having to build a complete game engine.
Keep in mind that ordinary 3D graphics already use higher-dimensional representations in places, especially for transformations and rotations. The main new challenge is giving objects a fourth spatial coordinate and deciding how the player perceives it. You can render a 4D scene by projecting it to 3D first and then letting a normal 3D camera project that result to the screen.
Exactly. The mathematics can happen in four dimensions even though the player ultimately sees a two-dimensional image.
So the final display is still 2D, but the scene can be processed as 4D and reduced in stages for visualization?