Hyper-Dimensional Deformation Simulation

Fig. 1. Two 4D armadillo prisms brush past each other. In the top left, a 3D slice where the armadillos are most visible. Bottom left, a different 3D slice reveals the extrusion direction. In the middle are two stills from a four-dimensional rotation about their collision at frame 800. On the right, they have failed to hug and instead have passed through each other.

We present a method for simulating deformable bodies in four spatial dimensions. To accomplish this, we generalize several pieces of the traditional simulation pipeline. Starting from the meshing stage, we propose a simple method for generating a pentachoral mesh, the 4D analog of a tetrahedral mesh. Next, we show how to generalize the deformation invariants, allowing us to construct 4D hyperelastic energies that lead directly to hyperdimensional deformation forces. Finally, we formulate collision detection and response in 4D. Our eigenanalyses of the resulting deformation and collision energies generalize to arbitrarily higher dimensions. The resulting simulations display a variety of previously unseen visual phenomena.