  A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if
two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called
biaccessible if two parameter rays land at $z$. In both cases, we say that the
external angles of these two rays are biaccessible as well.
  In this paper we give upper and lower bounds for the Hausdorff dimension of
biaccessible external angles of quadratic polynomials, both in the dynamical
and parameter space. In particular, explicitly describe those quadratic
polynomials where this dimension equals 1 (if and only if the Julia set is an
interval), and when it equals 0, namely, at finite direct bifurcations from the
polynomial $z^2$, as well as limit points thereof.
