These "pleated hyperbolic paraboloid" origami demonstrate the concept of a doubly ruled surface, whose applications within architecture enable material efficiencies when forming curved surfaces.
A singly ruled surface can be created by sweeping a straight line through space. Only three particular surfaces – the plane, the hyberboloid of one sheet, and the hyperbolic paraboloid – are doubly ruled surfaces, which contain two sets of straight lines. These surfaces are especially useful within architecture, because they can form opposing sets of cross-members, all built from straight pieces, folded sheets, or cables in tension. These types of curved structures offer sustainable building solutions, being materially light and structurally efficient.
In analyzing the folding of these origami, we seek to understand more about the structural properties of folded sheets, and possible applications towards space packing and panelized modular systems. The geometric relations between the faces of the origami is useful in understanding kinetic sculptures and movable, bendable structures.
The paper origami shown here demonstrate some interesting physical properties of paper, that will differ from more rigid materials such as steel sheets. As proven in 2009 by Erik Demaine, et al., in their paper "(Non)existence of Pleated Folds: How Paper Folds Between Creases," the fold pattern from which we create these paper origami is incomplete – to model these from a less deformable material, an extra crease must exist across the quadrilateral faces. The physics of the paper will attempt to reconcile this missing crease through crumpling or bending, while a more rigid material would require it to be explicitly formed.
The resin model shown here was created using a 3D rapid prototyping machine, with a stereolithograph file created using Grasshopper generative modeling software. To approximate the impossible smooth quadrilateral faces, the surfaces were triangulated into polygon meshes before prototyping.