  We present new examples of complete embedded self-similar surfaces under mean
curvature by gluing a sphere and a plane. These surfaces have finite genus and
are the first examples of self-shrinkers in $\mathbb R^3$ that are not
rotationally symmetric. The strategy for the construction is to start with a
family of initial surfaces by desingularizing the intersection of a sphere and
a plane, then solve a perturbation problem to obtain a one parameter family of
self-similar surfaces. Although we start with surfaces asymptotic to a plane at
infinity, the constructed self-similar surfaces are asymptotic to cones at
infinity.
