## Saturday, August 21, 2010

### Hyperbolic Paraboloid Buildings

Ever since I first stumbled upon them in "A Visual Dictionary of Architecture" I've been fascinated with the concept of hyperbolic paraboloid buildings.

Essentially a hyperbolic paraboloid is a structure which, in some frame, is described by the equation

z=x*y

If you slice it along constant z you get

1~ x*y or y~1/x or x~1/y

If you slice it along a line y=k*x you get

z=k*x*x or z=k*x2

But the part that makes it useful as a surface for buildings is that if you slice it along constant x or constant y you get

z~x or z~y

in other words, a straight line.

For this reason, it is called a "ruled surface", and a framework for it can be made entirely out of straight beams, as in the pictures from this site

http://www.savetrees.org/Hyperbolic%20Paraboloid%20roof%20shelter.htm

One problem with such a surface is that, while it is easy to construct the frame out of straight beams (and the resulting structure is known for its strength), constructing the roof to fit into the frame is non-trivial, especially if the surface must be hard.

It is *possible* to construct something that mostly fits snugly over it by cutting pieces out of a cloth tarp (as shown in the above link), but any patch of the true surface of the object is curved (as well as any line which is not constant in either x or y)

One solution might be a "concrete tent" similar to what is proposed in this article
http://www.wired.com/science/discoveries/news/2005/03/66872 (start with a "tarp with sections cut out" and brush wet concrete over it. Then let the structure harden)

Another alternative might be to layer a fine mesh of straight wires along constant x and constant y, and take some goopy filler material, such as plaster, and brush it over the wire mesh. I wonder if something like this is how they make permanent buildings with hyperbolic paraboloid roofs. Some of the images of such structures certainly look like a wet material was painted on top of a mesh and then allowed to dry, although larger hyperbolic paraboloid structures, such as the Catholic Cathedral on Gough Street in San Francisco, are often clearly made from piecewise surfaces (float or otherwise). I think to have a piecewise surface of flat segments perfectly conform to a frame of straight segments in such a roof, the segments have to be triangles (and highly irregular ones at that!)