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Curvy Pizza Math

There are plenty of pizza places that would love nothing more than to lay claim to discovering or better yet, deeply understanding the trick of bending the edges of pizza slice towards each other to achieve a straight line from crust through the tip to your mouth. But it ends up that a 19th century mathematician went there almost 100 years ago.

At the time of his discovery, Carl Friedrich Gauss wasn’t thinking about tossing dough in the air, but he did come up with the Theorema Egregium (Latin for “remarkable theorem”) that mathematically defines the curvature of a particular surface in a way that won’t change when you bend that same surface.

If you’ve ever held a piece of pizza by bending the two edges of crust in towards each other and “folding” what was a flat triangle into itself to stabilize the piece, rather than having it flopped all over the place, you’ve practiced this same principle without even realizing it.

To further illustrate the principle itself, think about taking a blank 8.5” x 11” piece of paper, then roll it up along its shorter width into a cylinder and place it so it sits 11” high on a flat table. The formerly flat surface has now become a cylindrical surface, but its properties really haven’t changed. If you envision microscopic little ants marching up and down on the surface of the cylinder, perpendicular to the table, you’d be able to clearly delineate an infinite amount of “flat surfaces” they could walk across – even while it was in its cylindrical shape.

Gauss had the mind and the insight to define the curvature of the surface that takes all of those infinite “flat surfaces” into account. Starting at any point – literally any point – an individual could find the two most extreme paths along that surface (the most concave and the most convex pathways) and then multiply the two curvatures of those paths together. The number you would arrive at, in this case with a cylinder, zero, is the very definition of that curvature as defined by the principle pioneered by Carl Friedrich Gauss, or its Gaussian curvature used in differential geometry. The sphere pictured above has positive curvature and the curved cylinder has negative curvature. Gaussian curvatures remains the same no matter how objects are positioned in 3-D space. Enjoy your pizza…