Comment by dxbydt | Hacker News Reader

Comment by dxbydt | Hacker News Reader

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This is silly. Every math textbook that teaches Theorema Egregium includes the same pizza example. That's how I learnt it as well. In my case we had an animated math professor who chose to bring a slice of pineapple pizza with canadian bacon to class, but during his demonstration the pineapples combined with the bacon and turned all gooey and started dripping on his shirt, so Theorema Egregium had to take a backseat to the practical realities of maintaining spotless formal attire in the classroom in front of a hundred giggling freshmen.

But seriously, this Theorema Egregium => Eating Pizza example is straight out of recreational math[1] & is very popular.

standard numerical geom text [2]:"In our everyday life we encounter the Theorema Egregium in a pizzeria..."

another riemann geom text[3]: "There is an interesting real-life application of Theorema Egregium...Notice that when you hold the pizza in one hand, the principal curvature of the crust is much smaller than along the direction of falling toppings."

third complex analysis text[4]: "Gauss defined Theorema Egregium in 1828. He defined principal curvatures to be maximum and minumum values k1 and k2...He then defined Gaussian Curvature K = k1*k2. k1 & k2 are not intrinsic but Gauss discovered K is intrinsic. Pizza has K=0 so we introduce a non-zero k1 forcing k2 to be 0 in order to preserve K because K is locally isometric. For this reason we bend the sides of the pizza to stop the free end from drooping"

[1]http://mathoverflow.net/questions/5450/cocktail-party-math [2]http://tosca.cs.technion.ac.il/book/index.html [3]http://www.damtp.cam.ac.uk/user/pz229/Teaching_files/GR.pdf [4]http://www.amazon.com/Lectures-Complex-Analysis-Contemporary...

You can always roll up the slice into a cylinder with the crust on the straight edge, and that also is an example of the theorem. It says nothing about the mechanics of the problem, i.e. how much will the pizza deform. It is quite possible to fold up the pizza as recommended and still have the tip sag - this depends on the material of the pizza and the self-weight, i.e. the mechanics rather than only the geometry.

Tomato tomaato. You formulate equations of motion s = ut + gt^2/2 by essentially ignoring air friction. You formulate kirchoff's voltage law L(di/dt)+1/C(integral(i)dt) + iR = V, by ignoring voltage losses across the rest of the circuit. You formulate the heat equation du/dt = laplacian(u) by assuming no lateral heat loss across the rod. Almost all equations in stochastic calculus in finance make the assumption that trading fees are zero & there's an unlimited pool of equity derivatives so you won't move the market when you buy & sell. Including real-life considerations like weight of the pizza & the specific toppings it has & so forth only leads us away from the beautiful math that underlies this problem. As you know, Gauss was so thoroughly impressed by the theorem he called it "Theorema Egregium" - the Remarkable Theorem! It is consistently voted one of the ten most beautiful theorems in geometry[1].

[1]http://www.reddit.com/r/math/comments/1eoo1p/q_what_are_the_...

It's perfectly possible for it to be a Remarkable Theorem which does not actually explain what is going on with the way people hold slices of pizza. And this can be true even if people are fond of saying it does provide such an explanation; it's not uncommon for this sort of thing to be frequently repeated without critical examination.

(I don't know anything about physics, but I felt like making the above comment nonetheless)

Except the beautiful math of the theorum only holds if distances between points on the pizza remain constant, which is manifestly not true for real pizza in particular and real materials in general. Force applied to a pizza curved width-wise will cause it to curve length-wise, changing the value of K. The remarkable theorum predicts that the pizza will not bend regardless of the radius of curvature and length of the slice, however, in practice, insufficient curvature relative to length will result in failure. Thus, the remarkable theorum hypothesis of pizza strength is falsified.

Except this is over simplifying. The tip of the slice won't bend down, but it quite easily folds up, contrary to the claim that it must remain flat.

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