Fun with 4D Meshes

   Over summer break I wrote a fun bit of code in Unity. It's a component for displaying a three dimensional mesh representation of a four-dimensional shape.

   How did I go about doing this? The first thing I needed to do was create a data structure for describing a “4D Mesh.” The key thing to understand about meshes, of any dimensionality, is that they describe shells, not actual shapes. A 3D mesh is a collection of 2D faces arranged in 3D space. Similarly, my 4D mesh asset is simply a collection of 3D tetrahedrons arranged in 4D space. Its data structure works on exactly the same principles; an array of Vector4s describing vertices positioned in 4D space, and an array of integers describing tetrahedrons by referencing sequences of four vertices.

4DShape.gif

   The second piece of necessary script is a process for translating a cross-section of a 4D mesh into a 3D mesh that can be read by Unity’s mesh renderer component. To do this, I need need the two key elements of a 3D mesh: an array of verts, and an array of triangles. The 3D verts need to be calculated based on the viewer’s current w-position since they describe points along the edges that connect 4D verts. These verts are added to a list, and their indices are also stored in a two-dimensional array. This is so that when I translate the tetrahedrons into triangles, I can use the indices of two 4D verts in a tetrahedron to find the index of the 3D vert that represents the edge between them.

   I plan to post the code on Github eventually, once I’ve done a little more cleanup and optimization. Currently the script makes an effort to cull unused vertices from the mesh it outputs to the mesh renderer, but I suspect that it’s faster and less memory intensive to simply process all the verts since it saves me the trouble of reconstructing a 3D vert index list whenever the mesh needs to be updated. I’m also looking into Unity’s new job system since it could allow me to do much of the calculations on a separate processor.

   The biggest roadblock in the experiment right now is actually the difficulty of inputting the data for a 4D mesh. So far I’ve been doing it by hand. For the shape in the gif above, I needed to input 16 Vector4s (64 values), and 32 tetrahedrons (a sequence of 128 integers). Now that ProBuilder is available for free in Unity 2018, I’m thinking about writing an extension that would allow me to better model and visualize a 4D shape.