# 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*^{2} = *x*^{2} - *y*^{2}

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).