Better video (and explanation!) of this week's mathematical conspiracy; a thread. 1/n
The space of quadratic polynomials can be identified with R^3; we think of the function ax^2+bx+c as the point (a,b,c). The polynomials with integer coefficients form the cubical lattice. 2/n 
The discriminant of ax^2+bx+c, namely b^2-4ac determines whether the polynomial has two real roots (positive discriminant) or complex conjugate complex roots (negative discriminant). The discriminant=0 surface is a cone. 3/n
Thus; the space of quadratics looks like Minkowski space, with the discriminant as the norm. Standing at the origin of this space and looking "straight up inside the cone" we see the polynomials with complex conjugate roots. 4/n
We get a better view of this perspective through projectivization (which is a useful thing to do if we are interested in studying polynomials through their roots anyway, as f(x) and cf(x) have the same roots). Here I've colored and the polynomials by their discriminant. 5/n
That's the picture: green polynomials have complex roots, blue have two real roots. Where does the motion come from? Any homeo A:C->C of the plane induces a transformation of (projectivized) polynomials taking the poly f with roots z,w to the poly A.f with roots A(z), A(w). 6/n
Let f_t be the horizontal translations of C, given by z-> z+t. This induces a shearing motion on the space of quadratics. 7/n
We can see this projectively by standing inside the cone and looking upwards. One ray, on the edge of the cone, appears to be fixed. Which polynomials are these? 8/n
A better (more wide-angle) view of this comes via projectivization. This motion looks like a parabolic isometry of the Klein disk, and was induced by translation of C (a parabolic isometry of Upper Half Space). Coincidence? No! More to come another day. 9/n
