The geometrical problem of protein folding, especially in its later stages, is composed of two types of freedom, the full torsional flexibility of loops connecting nearly rigid structural pieces (helices, beta-sheets etc), and the relative placing of such pieces. We present a method for sampling the feasible conformations of protein loops, based on Triaxial Loop Closure (TLC), a simple and highly efficient inverse kinematic (IK) method for solving the loop closure problem. TLC is easily extended to incorporate additional (i.e. position, orientation) constraints, or more general geometrical conditions. Due to its relative simplicity TLC compares favorably to more general IK robotics algorithms, both in robustness and in speed. We consider two applications: (i) An algorithm for the rapid sampling of the conformations of protein loops including three or more residues which uses quasirandom Sobol sampling of the Ramachandran regions. Ideas akin to Delauney triangulation may be employed to ensure sampling loop shapespace at a desired density. (ii) An efficient method for the sequential assembly of helical proteins via maximal hydrophobic packing. The geometrical problem of considering all possible mutual arrangements of a system of helices that are compatible with closing the corresponding loops is already too large to sample directly. We introduced a measure of hydrophobic packing by seeking to minimize the radius of gyration of the hydrophobic residues. Thus, we sequentially assemble the helices, by sampling relative orientations of pairs of them that bring specified hydrophobic residues in proximity. For the best candidates, in terms of energy and hydrophobig radius of gyration, the loops are closed using the algorithm in (i) and another helix is added to the assembly, always seeking to maximizing hydrophobic contact. We tested this iterative assembly method on 26 helical proteins each containing up to 5 helices. The method heavily samples native-like conformations. The average RMSD-to-native of the best conformations for the 18 helix bundle proteins that have 2 or 3 helices is less than 2 Angstroms with slightly worse errors for proteins containing more helices.