Thanks for your response.
Since the clipping is applied to the material in those examples, though, it constantly clips a different portion of the geometry as it is transformed. How would I go about clipping the actual geometry so that I can apply transformations without the clip changing?
I actually came across this comment of yours on operations with geometry and it seems like this library may do what I want. I think all I would need to do is subtract an appropriate plane mesh from the tetrahedron mesh to produce a new, truncated mesh.