Hyperbolic Paraboloids

Erik Demaine, Martin Demaine, and Anna Lubiw

A hyperbolic paraboloid is an infinite surface in three dimensions with hyperbolic and parabolic cross-sections. A couple of ways to parameterize it and write an equation are as follows:

z = x2 - y2
or
x = y z

The plots shown to the right use the first equation. (They were generated using Maple.) The second plot shows that the xy cross-sections (i.e., cross-sections parallel to the xy plane) are hyperbolas. The yz cross-sections are copies of the same parabola, and the zx cross-sections are the same parabola but upside-down.

Hypars

We use the term hypar to mean a hyperbolic paraboloid shape, or more formally a partial hyperbolic paraboloid, cut from the full infinite surface. The term hypar was introduced by the architect Heinrich Engel in his 1967 book Structure Systems (page 215).

We are particularly interested in how hypars can be joined together at their edges to make interesting sculpture.

Hypars in Architecture

Hypars and joining hypars in a few special ways have been used extensively in architecture. For example, Curt Siegel's 1962 book Structure and Form in Modern Architecture (page 256) illustrates the roof of the Girls' Grammar School in London (designed by Chamberlin, Powell, and Bonn) which is what we call a “5-hat” with five hypars spread apart slightly. Later (page 264) the idea of joining two 5-hats is suggested, although the two hats are cut to have a curved boundary, making them easy to join. Page 260 shows a photo of the Philips pavilion at the 1958 Brussels exhibition (designed by Le Corbusier) which is a beautiful surface made of eight or so hypars that rests on the ground. A few more wonderful examples with straight boundaries are illustrated by Heinrich Engel in his book mentioned above (pages 228-229), each involving between five and twelve hypars. Finally, a grid of connecting “4-hats” is illustrated and analyzed by Frei Otto in the 1969 book Tensile Structures (volume 2, page 64).

Folding Pleated Hypars

It is amazingly easy to fold a hypar from a square piece of paper. (And this is how we got started playing with joining hypars together.) If you fold the diagonals of a square, and several concentric squares in alternating direction (a square of mountain folds, then a square of valley folds, and so on), then the piece of paper naturally forms a pleated hyperbolic paraboloid shape.

More detailed diagrams are available.

Hats

There are many ways to glue hypars together edge-to-edge. One method, which we call a k-hat, is to glue k hypars together in a kind of ring. For example, here is a photograph of a 4-hat:

In particular, you can close a k-hat up onto itself to make a k-star. Here is a photograph of a 5-star, which I've successfully used on a Christmas tree:

Hyparhedra

We have developed algorithms for building hypar "sculptures" based on polyhedra. We call these structures hyparhedra. Please refer to the paper below for details on the algorithms. For now, this page just contains photographs of a few examples. The first five arising from the Platonic solids:
• Tetrahedron

• Cube

• Octahedron

• Dodecahedron

• Icosahedron
The last one arises from a (degenerate) doubly covered triangle:

References

This material is based on the paper “Polyhedral Sculptures with Hyperbolic Paraboloids”, which was presented at the 2nd Annual Conference of BRIDGES: Mathematical Connections in Art, Music, and Science, 1999. Please refer to this paper for more details on the above subject.

This work is also briefly mentioned in Ivars Peterson's Science News article “Unlocking Puzzling Polygons” (September 23, 2000, volume 158, number 13, pages 200-201).

Last updated May 28, 2014 by Erik Demaine.Accessibility