4d and n-dimensional physics engine

There are numerous research groups using relativistic codes, for all sorts of physics problems; from Relativistic Electrodynamics, Relativistic Fluid Dynamics/Magnetohydrodynamics and for gravitational based simulations etc. Astrophysical applications are the main place you would meet relativistic codes.

A 4D game engine is what you already have in games like FIFA and COD. This is just a 3 + 1 implementation, which incedently is what many relativistic codes are (they use the 3 + 1 formulation of space-time). This splitting of space-time is much easier to handle computationally for many different reasons. Of course as you go from 1D to 2D etc. you complexity increases inline with the simulated physics.

To me it makes no sense to have a physics engine in n-dimensions. We do not experience physical processes in n-diemensions, but four. To ask about hypercubes etc. is not physics but geometrical/mathematical constructs. These are separate from what you would traditionally associate with a physics engine.

An example of a 4D physics engine is 4D Toys, created by Marc ten Bosch.

The author goes into more detail about how it works on their blog (see this post about geometric algebra, for instance).

Their technical paper, N-Dimensional Rigid Body Dynamics, was also accepted to SIGGRAPH 2020.

I also explained a bit how angular velocity works in higher dimensions here.