I felt like I had to repost this question because I didn't explain it well enough in the first attempt.
Given a regular $n$-simplex find a way to cut it in half with a plane so that each half looks the same and is mirrored (not point symmetrical). You cannot rotate the halves so that they are symmetrical.
How do you do it for a $n$-simplex in higher dimensions than 3? If we look at the 3-simplex but in 3 dimensions instead of 4, you can only do it by cutting along one edge. Is it the same in 4 dimensions?
I know that the hyperplane we use to cut a simplex always has $n-1$-dimensional volume, so I cannot imagine it in my head.
Also, I'd like to minimize the points that are neither in one half or the other.
Choose any face of the $n$-simplex. That face is an $(n-1)$-simplex; bisect it using an $(n-2)$-plane. (Assume we already know how to do this... this method is inductive.). Then form the $(n-1)$-plane containing that $(n-2)$-plane and the final point of the simplex. This plane bisects your original simplex.
Note how this works for the tetrahedron: you bisect a face, with a line segment, and then the plane containing that line segment and the final point cuts the tetrahedron in half.