Since the space of all cubic polynomials is (complex) two-dimensional and thus too difficult to comprehend, we study a one-dimensional slice of it: the space of all cubic symmetric polynomials of the form $f(z)=z^3+\lambda^2 z$. Thurston has built a topological model for the space of quadratic polynomials $f(z)=z^2+c$ by introducing the notion of quadratic invariant laminations. In the spirit of Thurston’s work, we parametrize the space of cubic symmetric laminations and create a model for the space of cubic symmetric polynomials. This is a joint work with Alexander Blokh, Lex Oversteegen, Vladlen Timorin, and Sandeep Vejandla.