The study extends existing closed‑form mathematics for symmetric waterbomb cells to cells folded with a 180° rotation symmetry, providing an exact analytical description of every stage of the folding process. This new solution enables designers to calculate the three‑dimensional shape of a tessellated origami shell directly from a single cell’s design parameters.

The authors develop an algebraic derivation that introduces both a length‑based and an angle‑based parametrisation. These equations capture the kinematics of the semi‑symmetric cell and simplify rigid‑origami modelling compared with conventional numerical approaches. By reducing algorithmic complexity, the algebraic method makes it possible to explore the full admissible range of configurations without extensive simulation.

A key illustration (Figure 7) shows how two congruent semi‑symmetric cells are joined to form a tessellation, confirming that the new equations correctly predict the geometry of paired cells. Another illustration (Figure 9) presents a parametric study of curvature in thick‑walled waterbomb shells, demonstrating that the closed‑form solution accurately predicts global curvature and validates that the algebraic approach yields exact configuration limits.