Tuesday, September 9, 2014

Deep math of the folded pizza slice

from Boing Boing:

Why does a flat pizza slice flop over unless you bend it into a curve? Thank Gaussian curvature, the 19th century mathematical principle that underpins everything from corrugated cardboard to eggshells to Pringles chips.

Wired's Aatish Bhatia uses the pizza-slice as a jumping-off point to explain one of the most elegant and fascinating parts of geometry, and once you read his work, you'll never be able to look at a curved surface again:
Well, the pizza slice was flat before you picked it up (in math speak, it has zero Gaussian curvature). Gauss’s remarkable theorem assures us that one direction of the slice must always remain flat — no matter how you bend it, the pizza must retain a trace of its original flatness. When the slice flops over, the flat direction (shown in red below) is pointed sideways, which isn’t helpful for eating it. But by folding the pizza slice sideways, you’re forcing it to become flat in the other direction – the one that points towards your mouth. Theorema egregium, indeed.

By curving a sheet in one direction, you force it to become stiff in the other direction. Once you recognize this idea, you start seeing it everywhere. Look closely at a blade of grass. It’s often folded along its central vein, which adds stiffness and prevents it from flopping over. Engineers frequently use curvature to add strength to structures. In the Zarzuela race track in Madrid, the Spanish structural engineer Eduardo Torroja designed an innovative concrete roof that stretches out from the stadium, covering a large area while remaining just a few inches thick. It’s the pizza trick in disguise.
How a 19th Century Math Genius Taught Us the Best Way to Hold a Pizza Slice [Aatish Bhatia/Wired]

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