The hyperbolic paraboloid is not a minimal surface, but several people (including Chuck Hoberman) make the mistake of saying it is

Can you see any difference between these 2 curves? (apart from the colour!)
hyper.jpgcat.jpg

When they are superimposed it becomes easier to see the slight difference :

cat-hyper.jpg
Visually they are almost indistinguishable, but mathematically they are completely different objects.

The blue one is a hyperbola, a conic section, defined by the parametric equations :

x = ± a cosh t + h

y = b sinh t + k

While the red one is a catenary, the shape of a hanging chain, defined by the function :

y = a cosh(x/a )

A similar visual confusion arises between the hyperboloid and the catenoid, the surfaces of revolution of these 2 curves

hyperboloid-lines.jpgcatenoid.jpg

The hyperboloid is ruled (ie it can be swept out by a straight line) but not minimal.

The catenoid is minimal (it is the shape of an ideal soap film between two rings) but not ruled.

Which brings me to the hyperbolic paraboloid, a saddle shaped surface sometimes known as a hypar.

The hypar is ruled, but it is most definitely not minimal. However, its superficial resemblance to a minimal surface sometimes leads to confusion.

The essay Membrane Organisation by Pavlos Sideris in the latest AA Files publication makes this error, repeatedly referring to form-found membranes as hypars.

The page on Chuck Hoberman’s website about one of his deployable structures makes this error most explicitly when it says : “A hypar (short for hyperbolic paraboloid) is an example of a minimal surface”

When he lectured at the AA a few years back he repeated this error. I approached him about this and he suggested I was confused, saying if he was wrong then everyone else was wrong with him. Well Chuck, here’s the proof.

The hyperbolic paraboloid (or hypar) can be simply parametrized as :

x = u

y = v

z = uv

and its mean curvature H is given by :

H = uv/(( 1 + u² + v² )³/²)

From which it should be obvious that H is not always equal to zero, but just to make it explicit I’ll show it anyway:

If we ignore the case where either u or v are zero :

It is obvious that if u and v are both non-zero then uv is also non-zero

The square of any real non-zero number is positive, so u² + v² > 0

Therefore 1 + u² + v² > 0

A positive real number to any real power is positive so ( 1 + u² + v² )³/² > 0

A non-zero number divided by a positive number is also non-zero

Therefore H is not zero and the surface is not minimal.

To reinforce the point heres a hypar and a helicoid, both coloured according to mean curvature (where dark blue corresponds to zero mean curvature) :

curvature.jpghelicoid.jpg

The hypar’s mean curvature varies across its surface (and is zero only along the lines where u or v = 0) as opposed to the helicoid which has zero mean curvature everywhere. The helicoid is in fact the only surface which is both minimal and ruled (apart from the plane).

There does exist a minimal surface bounded by the skew quadrilateral edges of the hypar which is visually similar but mathematically distinct. Its definition(found by Schwarz in 1890) is considerably more complicated than that of the hypar, involving Abelian integrals (whatever the hell they are!).

I realise that for practical purposes a hypar may often be a reasonable approximation to a minimal surface, but that is not the same as actually calling it one. The term hyperbolic paraboloid has a precise mathematical meaning, and if this is not what is meant then the less rigorous term ‘saddle surface’ should be used instead.

Also, mathematical pedantry aside, I do think Chuck Hoberman designs some wonderful structures

References:

Weisstein, Eric W. “Hyperbolic Paraboloid.” http://mathworld.wolfram.com/HyperbolicParaboloid.html