Originally published at: Mathematicians analyze the geometries of the Pringle | Boing Boing

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And yet, no mention of how two hyperbolic paraboloids can be used to make a cheeky little duck bill. Some mathematicians these people are.

The three mathematicians didn’t calculate the shape of a Pringle, they simply repeated what they were taught in math school, which is that a Pringle chip is a hyperbolic paraboloid. I’d like to see them do some math, please.

Yes, I have a suspicion that it may be closer to a section of a hyperboloid of one sheet.

Here are a couple of things to measure about a surface, at a point, and what they look like on a Pringles chip.

The first is: consider all the points at some small distance r, i.e. the boundary of a disc with radius r. What is its circumference? On the flat plane this will of course be 2 pi r for any point and any r. On a sphere it will always be < 2 pi r. On the Pringles chip it will always be > 2 pi r. If you consider the difference, divide by r (or maybe r^2, I forget), and take the limit, you get a number indicating the “curvature” of the surface at that point. A sphere is very different from a Tic Tac, in that the latter is differently curved at different points. A sphere has the same, positive, curvature at every point. A Pringles chip has negative curvature at every point; possibly the same negative curvature? I’m not sure.

(Off-topic: consider small spheres vs. big. The big ones are less curved, but are… bigger. If you integrate the curvature, you always get 2, even if you stretch your sphere into some other shape like a Tic Tac. Things are different with a doughnut: the integral is always 2*(1 - the number of holes).)

Another way to measure curvature at a point is to slice the surface with a perpendicular plane through that point. If you try this at a point on, say, a straw you’ll notice that the choice of plane matters. Also, is it curving up, or down? In fact there will be a plane giving most curvature and another giving least, and they’ll be orthogonal. Multiply those two numbers together.

If you imagine doing this from the inside of a sphere, you get two curves both curving up, so pos * pos = positive curvature. If you imagine it from the outside, you get two curves both curving down, so neg * neg = positive curvature again. On the Pringles chip you get pos * neg = negative curvature.

If you’ve ever wondered how a rattleback works – how it breaks symmetry – look at the point where it balances on the table. You’d guess that the two principal axes of curvature would be aligned with the whole rattleback, but they’re *not*, they’re 45 degrees off, key to its very weird behavior.

can any of this explain why they deal death by 1000 blows to the corners of my mouth ?

That’s a materials science question, not a math question.

Insert “that’s what she said” GIF

I love the little plastic boxes that let you carry about a dozen or so Pringle chips in a lunch box, without them shattering into tiny fragments. I’ve got three of them, allowing the choice of three different flavours.

It’s the little things that mean so much!

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