Embryos seamlessly integrate morphogenesis, the change in shape; with fate specification, the process that turns the fertilized egg to all the specific cell types in the body. Several groups have used mammalian embryonic stem cells to make approximations to fetal organs and even entire early embryos. The synthetic systems allow one to deconstruct the myriad of signaling events that are largely hidden from study after implantation in mammalian systems. Mathematics of dynamical systems can supply compact models for spatial patterning that are uniquely potent in fitting time lapse data. In the best of cases they can reduce what appear to be complex genetic interactions to linear models inside of nonlinear functions.