Cortical microcircuits in a variety of brain regions express similar, highly nonrandom, synaptically-connected cell triplets. The origin of these universal network building blocks is unclear and was hypothesized to result from plasticity and learning processes. Here we combined in-silico modeling of dense cortical microcircuits with electrophysiological/anatomical studies to demonstrate that this recurring connectivity emerges primarily from the anisotropic morphology of cortical neurons and their embedding in the cortical volume. Using graph-theoretical and machine-learning tools, we developed a series of progressively more complex generative models for circuit connectivity that considers the geometry of cortical neurons. This framework provided predictions for the spatial alignment of cells composing particular cell-triplets which were directly validated via in-vitro whole-cell 12-patches recordings (7,309 triplets) in the rat somatosensory cortex. We concluded that the local geometry of cortical neurons imposes an innate, highly structured, global network structure, a skeleton upon which fine-grained structural and functional plasticity processes take place.