Several lines of evidence, both theoretical and observational, indicate that additional planets in the outer solar system remain to be discovered. Most recently, attention has been focused on 'Planet Nine',a few Earth mass planet with a moderate inclination at a distance of a few hundred AU (Trujillo & Sheppard 2014, Batygin & Brown 2016, Brown & Batygin 2016). I will review this evidence for the existence of this body and will present a variety of approaches that are being used to find it.
I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops. The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.
I did not manage to solve this problem, but I did get the result that at most 4 points of any Jordan loop are vertices of inscribed rectangles. I will sketch a proof of this result, mostly through visual demos, and also I will explain two other theorems about inscribed rectangles which at least bear a resemblance to theorems in symplectic geometry.