  This paper introduces cellular sheaf theory to graphical methods and
reciprocal constructions in structural engineering. The elementary mechanics
and statics of trusses are derived from the linear algebra of sheaves and
cosheaves. Further, the homological algebra of these mathematical constructions
cleanly and concisely describes the formation of 2D reciprocal diagrams and 3D
polyhedral lifts. Additional relationships between geometric quantities of
these dual diagrams are developed, including systems of impossible edge
rotations. These constructions generalize to non-planar graphs. When a truss
embedded in a torus or higher genus surface has a sufficient degree of axial
self stress, we show non-trivial reciprocal figures and non-simply connected
polyhedral lifts are guaranteed to exist.
