I guess I misunderstood or mis-scoped the problem. Does the 3-body problem state 'the general case' has no solution, but that does not preclude some configurations from having a solution?
The three-body problem only states the problem to solve, it doesn’t itself state anything about the existence or non-existence of solutions. It has been proven that there is no general closed-form solution. And there are obvious solutions for trivial special cases, such as three equal masses in an equilateral triangle rotating around each other.
There is no closed-form solution for finding the roots of >4th degree polynomials in general, but that doesn’t preclude many families of >4th degree polynomials from having closed-form solutions. As a trivial example, x^5 - 1. The exact same thing with the three-body problem.
It's also not computable, as in chaotic. Small differences in initial positions will lead to unpredictably large differences in trajectory (with small and large having specific meanings to match the formal definition of a chaotic system).
Very cool visuals and site!
Could I make a suggestion:
You show the masses (1,1,1), but not the starting positions, which alter the course of events too.
Why would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?
One of the necessary conditions here is that the three objects return to their exact initial position, and so does the centre of mass. Initial conditions with non-zero momentum must trivially be ruled out. But this doesn't stop them from having velocities perpendicular to the initial plane that cancel out perfectly, so this doesn't refute the assertion that the planes keep changing.
Further reading: https://en.wikipedia.org/wiki/Three-body_problem#Solutions
(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)
(but yes I think you might be right if we're centered on the CG)