AbstractWe develop methods for constructing explicit generators, modulo torsion, of the$K_3$-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$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for$ K_3 $of any 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.