How do I start programming a four-dimensional world?

Look at existing four-dimensional games and interactive visualizers for ideas. They demonstrate different approaches, including cross-sections, projections, portals, and movement through the extra coordinate. Developer write-ups and videos about projects such as 4D Golf or Code Parade’s four-dimensional experiments can be especially useful because they explain both the mathematics and the gameplay problems.

You can think of it as extending the usual graphics pipeline by one step. Transform 4D geometry, perform a 4D-to-3D projection, and then let a conventional engine project the resulting 3D scene onto the screen. A perspective projection might scale coordinates based on the fourth coordinate, much like ordinary perspective scales based on depth. This lets you use existing 3D tools for the final rendering while writing the higher-dimensional math yourself.

A manageable first project would be a rotating 4D hypercube. Define its 16 vertices and edges, rotate it in one of the coordinate planes, project it into 3D, and draw the resulting wireframe. After that, try slicing a 4D object with a moving 3D hyperplane, then add collision tests and player movement. Building these pieces incrementally will teach you more than starting with a complete four-dimensional game engine.

Start by representing points as four-component vectors such as (x, y, z, w). Most familiar operations—distance, translation, scaling, and interpolation—generalize naturally. The biggest differences are rotation and visualization: in four dimensions, rotations happen in planes such as XY, XW, or YW rather than around a single axis. Once you transform your 4D objects, project them down into 3D and then use an ordinary 3D renderer to display the result.