Results of large-scale first principle nuclear structure studies using the symmetry-adapted no-core shell model are reported. It is shown that nuclei up through the intermediate-mass region display highly regular and ubiquitous patterns of dominant nuclear shapes that vibrate and rotate. This emergent structure is tied to an approximate symplectic Sp(3, R) symmetry, and it is shown to determine dominant features of low-lying states, even in close-to-spherical nuclear states without any recognizable rotational properties.