Life for all animals starts with a precise 3D choreography of reductive divisions of the fertilized egg, known as cleavage patterns. These patterns exhibit conserved geometrical features and striking interspecies invariance within certain animal classes. To identify the generic rules that may govern these morphogenetic events, we developed a 3D-modeling framework that iteratively infers blastomere division positions and orientations, and consequent multicellular arrangements. From a minimal set of parameters, our model predicts detailed features of cleavage patterns in the embryos of fishes, amphibians, echinoderms, and ascidians, as well as the genetic and physical perturbations that alter these patterns. This framework demonstrates that a geometrical system based on length-dependent microtubule forces that probe blastomere shape and yolk gradients, biased by cortical polarity domains, may dictate division patterns and overall embryo morphogenesis. These studies thus unravel the default self-organization rules governing early embryogenesis and how they are altered by deterministic regulatory layers.