OK, so I've seen many, many BSP tree tutorials. All of them were on a top-down perspective, and polygons were in fact 'walls' or lines. Then to split the polygons, all they had to do was find the intersection point of two lines. Now, how would one 'extrapolate' this into 3D space, with arbitrary sets of 3D polygons that may not have any kind of limits on placement or rotation? How could I split polygons in 3D space?

Cant you represent them as 4D plane vectors? then ray tracing can present your solution...

Actually, I did find a function online to split triangles. I must look into it and see how it works...

As for '4D plane vectors'... I'm not really familiar with all that yet. We only learn about matrices next year in school...