AbstractThis study in projective geometry reveals that the principle of duality applies to the screw. Here, the screw is demonstrated to be an element of a projective three‐dimensional space (P3), right alongside the line. Dual elements for the screw and line are also revealed (the nut and spline). Reciprocity is demonstrated for a pair of screws, and incidence is demonstrated for screw and its dual element. Reciprocity and incidence are invariant for projective transformations of P3, but only incidence is invariant for the more general linear transformations of screws. This latter transformation is analogous to a projective transformation of a projective five‐dimensional space (P5), which is shown to induce a contact transformation of the original P3, where some points lying on a Kummer surface are directly mapped. © 2003 Wiley Periodicals, Inc.