I’m ruling them out because they’re point masses moving through perfectly empty and uniform space and unaffected by any outside force. That’s the setup of the three-body problem. It doesn’t and isn’t intended to be a system that exists in the real world; it’s a simplification, an idealization. It’s a thought experiment.
But, from the images in your post, I’m starting to think you’re talking about numerology and not math: ‘3’ isn’t just a number, it’s a symbolic reference to some special Trinity, and the “three-body problem” has Kabbalistic significance or something. Am I way off on that?
If I’m right, you should check out Period Three Implies Chaos – grist for the mill.
What force is this?
Why does the earth orbit the sun if the sun exerts no pull on the earth?