HOW A 19TH CENTURY MATH GENIUS TAUGHT US THE BEST WAY TO HOLD A PIZZA SLICE, Part 2

Perhaps the most mundane example of strength through curvature are the ubiquitous corrugated building materials (corrugate comes from ruga, Latin for wrinkle). You could hardly get more bland than a corrugated cardboard box. Tear one of these boxes apart, and you’ll find a familiar, undulating wave of cardboard inside the walls. The wrinkles aren’t there for any aesthetic reasons. They’re an ingenious way to keep a material thin and lightweight, yet stiff enough to resist bending under considerable loads.

Corrugated metal sheets use the same idea. These humble, unpretentious materials are a manifestation of pure utility, their form perfectly matched with their function. Their high strength and relatively low cost has blended them into the background of our modern world.

Today, we hardly give these wrinkled sheets of metal a second thought. But when it was first introduced, many saw corrugated iron as a wonder material. It was patented in 1829 by Henry Palmer, an English engineer in charge of the construction of the London Docks. Palmer built the world’s first corrugated iron structure, the Turpentine Shed at the London Docks, and although it might not seem remarkable to modern eyes, just listen to how an an architectural magazine of the time described it.

“On passing through the London Docks a short time ago, we were much gratified in meeting with a practical application of Mr Palmer’s newly invented roofing. [...] Every observing person, on passing by it, cannot fail being struck (considering it as a shed) with its elegance and simplicity, and a little reflection will, we think, convince them of its effectiveness and economy. It is, we should think, the lightest and strongest roof (for its weight), that has been constructed by man, since the time of Adam. The total thickness of this said roof, appeared to us from a close inspection (and we climbed over sundry casks of sticky turpentine for that purpose,) to be, certainly not more, than a tenth of an inch!” [1]

They just don’t write architectural magazines like they used to.

While corrugated materials and soda cans are pretty strong, there’s a way to make materials even stronger. To discover it for yourself, go to your fridge and take out an egg. Put it in the palm of your hand, wrap your fingers around the egg, and squeeze. (Make sure you aren't wearing a ring if you attempt this.) You’ll be amazed at its strength. I wasn’t able to crush the egg, and I gave it everything I had. (Seriously, you need to try this to believe it.)

What makes eggs so strong? Well, soda cans and corrugated metal sheets are curved in one direction but flat in the other. This curvature buys them some stiffness, but they can still potentially be flattened out into the flat sheets that they came from.

In contrast, egg shells are curved in both directions. This is the key to an egg's strength. Expressed in math terms, these doubly curved surface have non-zero Gaussian curvature. Like the orange peel we encountered earlier, this means that they can never be flattened without tearing or stretching – Gauss’s theorem assures us of this fact. To crack an egg open, you first need to dent it. When the egg loses its curvature, it loses its strength.

The iconic shape for a nuclear power plant cooling tower also incorporates curvature in both directions. This shape, called a hyperboloid, minimizes the amount of material required to build it. Regular chimneys are a lot like giant soda cans - they're strong, but they can also buckle easily. A hyperboloid shaped chimney solves this problem by curving in both directions. This double curvature locks the shape into place, giving it extra rigidity that a regular chimney lacks.

Another shape that gets its strength from double curvature is the Pringles potato chip*, or as mathematicians tend to call it, a hyperbolic paraboloid (say that three times fast).

Nature exploits the strength of this shape in a mind-blowingly impressive way. The mantis shrimp is infamous for having one of the fastest punches in the animal kingdom, a punch so strong that it vaporizes water, creating a shock wave and a flash of light. To deliver its impressive death blow, the mantis shrimp uses a hyperbolic paraboloid shaped spring. It compresses this spring to store up this immense energy, which it releases in one lethal blow.

You can watch biologist Sheila Patek describe her discoveryof this amazing phenomenon. Or have Destin explain it to you in his brilliant Youtube channel Smarter Every Day.

The strength of this Pringles shape was well understood by the Spanish-Mexican architect and engineer Félix Candela. Candela was one of Eduardo Torroja’s students, and he built structures that took the hyperbolic paraboloid to new heights (literally). When you hear the word concrete, you might think of dreary, boxy constructions. Yet Candela was able to use the hyperbolic paraboloid shape to build huge structures that expressed the incredible thinness that concrete can provide. A true master of his medium, he was equal parts an innovative builder and a structural artist. His lightweight, graceful structures might seem delicate, but in fact they’re immensely strong, and built to last.

So what makes this Pringles shape so strong? It has to do with how it balances pushes and pulls. All structures have to support weight, and ultimately transfer this weight down to the ground. They can do this in two different ways. There’s compression, where the weight squeezes an object by pushing inwards. An arch is an example of a structure that exists in pure compression. And then there’s tension, where the weight pulls at the ends of an object, stretching it apart. Dangle a chain from its ends, and every part of it will be in pure tension. The hyperbolic paraboloid combines the best of both worlds. The concave U-shaped part is stretched in tension (shown in black) while the convex arch-shaped part is squeezed in compression (shown in red). Through double curvature, this shape strikes a delicate balance between these push and pull forces, allowing it to remain thin yet surprisingly strong.

Strength through curvature is an idea that shapes our world, and it has its roots in geometry. So the next time that you grab a slice, take a moment to look around, and appreciate the vast legacy behind this simple pizza trick.