We have recently (1) introduced an exact mathematical description of chain molecules based on discrete analogues of the curvature and torsion concepts in differential geometry called the κ-τ framework, and applied it to the analysis of all known protein structures. We here use it to extract and analyze defined structural motifs from the Protein Data Bank. We find a class of left handed 4 residues long, open turns, tentatively called left α-turns, which do not exhibit internal backbone H-bonding and join both α and β strands at the surface of many different proteins. Further, we discuss results on helix capping substructures such as the Schellman and α_L motifs. The κ-τ methodology proves to be the most efficient methodology to handle such problems available to date, and will greatly expand our capability to decipher the information hidden in the known structures and understand the fundamental relation between sequence and structure.