We develop methods for constructing explicit generating elements, modulo torsion, of the K3-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3-space or on direct calculations in suitable pre-Bloch groups, and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite K3-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K3 of any infinite field, predict the precise power of 2 that should occur in the Lichtenbaum conjecture at −1, and prove that this prediction is valid for all abelian number fields.