Hyper Folds

The right sequence of origami folds turns a square sheet of paper into the saddlelike form of a hyperbolic paraboloid.  Photo by I. Peterson

Traditional origami allows no cutting or gluing. If the rules are relaxed a little, even more designs and sculptures can emerge from nimble fingers.

In one painstaking effort, Erik D. Demaine, Martin L. Demaine, and Anna Lubiw created polyhedral sculptures based on the hyperbolic paraboloid, an infinite surface discovered in the 17th century. The central portion of a hyperbolic paraboloid resembles a saddle shape.

Demaine and his coworkers started with a traditional method of folding a hyperbolic paraboloid from a square sheet and showed how it is possible to glue these components edge to edge in different ways to form paper sculptures they describe as hyparhedra.

Their algorithm allows you to construct the hyparhedron version of any given polyhedron. It typically takes several hours to complete a model.

Gluing together partial hyperbolic paraboloids, or hypars, along their edges produces a cubic form—one of a number of closed, curved surfaces called hyparhedra that can be constructed from paper hypar units.

"The richness of these forms excites us visually and presents us with interesting mathematical problems," the team reported in their paper "Polyhedral Sculptures with Hyperbolic Paraboloids.".