How Gauss Taught Us the Best Way to Hold a Pizza Slice | Hacker News Reader

How Gauss Taught Us the Best Way to Hold a Pizza Slice

wired.com

182 points_mayo33 comments

The author's point that "curvature imputes stiffness" conflates several different and distinct mechanisms, and offers an inadequate explanation.

For the examples of the pizza, the leaf, and the corrugated sheets, the stiffness is due to the fact that the bending moment of inertia of the cross-section increases when we fold the pizza or the sheet in a particular way [1]. The Theorema Egregium shows that such a structure can be made from a flat sheet of material, not that this construction imparts stiffness to the structure.

The example of arches show the well-known arch action in mechanics, where forces are carried through pure compression without any tensile stresses, which makes it appropriate for using stones to make the arch [2]. In principle, one could make a triangular "arch", i.e. part of a truss structure, where we use two straight rods joined together at the top [3]. This shows that its not really the curvature that is giving the stiffness.

The example of hyperbolic paraboloids shows arch action in one direction and beam bending in the other.

The examples of the egg and the can show that it is hard to break a surface when it does not have stress concentrations [4].

So the point is that there's a lot of classical solid mechanics at play here, of which the author seems to be unaware.

[1] http://en.wikipedia.org/wiki/Bending

[2] http://en.wikipedia.org/wiki/Arch

[3] http://en.wikipedia.org/wiki/Truss

[4] http://en.wikipedia.org/wiki/Stress_concentration

Arguably the dependence of bending moment on shape is intuitive, but the geometry of developable surfaces is not.

Just like in Maxwell's theory of hills and dales: the location of topographic peaks, saddles etc. is "obvious" but the constraints on where you get saddles and how many, etc. are not. (http://en.wikipedia.org/wiki/Morse_theory or http://www.maths.ed.ac.uk/~aar/surgery/hilldale.pdf)

Or the similar territory of the Euler characteristic. You could know polyhedra very well from a physical, practical point of view and never notice it. (http://en.wikipedia.org/wiki/Euler_characteristic#Polyhedra)

Maybe?

Anyway, I am commenting because I'd be interested to hear your reaction to the problem stated here: https://www.youtube.com/watch?v=36gOx3dguWs#t=17m35s

Seems like an interesting way to cast a structural shape. Possibly, it transfers stresses efficiently because it follows the deformed configuration of the fabric.

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.

The point is that stiffness is provided by reducing the degrees of freedom that would cause flopping to those that would require you to "stretch, shrink or tear" the piece of pizza. As a result, in cases where your stresses are negligible compared to the yield strength of your material this approximation accurately predicts the behavior without resorting to FEM or in depth analysis.

While I agree that there are more complicated theories that are correct for more diverse circumstances, I think it's tremendously valuable to find the simplest models that describe the easiest situations if only for the purposes of developing intuition. I must admit that this is very much a physicist's perspective, though.

This has nothing to do with yield strength, which is relevant only where the materials "yields" or plastifies. This is just linear elastic beam bending theory - you have two different beam cross sections in either case with two different moments of inertia. See also my other comment: https://news.ycombinator.com/item?id=8276173

You could fold a piece of fabric like you do the pizza, and it will not keep its shape.

Sorry, I meant it depends on the elastic modulus. Mechanics was a while ago.

If the model matches the prediction, the model works. The argument is only over what regime. In this regime it matches.

If you read the article it specifically mentions it applies to paper. I expect it would apply to many fabrics as well. When it doesn't it's because it's outside the regime of the model because stress enables significant "stretching".

You could use beam theory as well, and I would be surprised if the author hasn't heard of it, but that doesn't mean it's the only technique available.

Depending on the fabric, you most certainly can.

Yeah... well summarised. I kind of enjoyed the article, but while reading it I kept thinking, "This isn't really the reason..."

The Oatmeal provides a very nice illustrated pics about the Mantis Shrimp and its killer claws http://theoatmeal.com/comics/mantis_shrimp

I totally call bullshit on not being able to crack an egg. Next time I see an egg, I'm trying it.

YHBT

I've done it before and found it surprisingly difficult but not impossible to break the egg. In order to break it, I had to put more pressure on the egg with my fingertips, which should probably be considered cheating.

I've seen this tested with an uncooked egg. A lot of pressure was required and when it finally broke the mess was unbelievable.

I went straight to the fridge and found I hurt my hand before I cracked the egg. I was as surprised as you are.

If that was a troll sir, it was a good one. I wonder how many HNers had to clean their kitchen today ;).

More of an engineering problem that uses a bit of math. Unless you can state what the weight would be to cause the surface to collapse, you're just wanking off in math space.

See Second moment of Inertia (or area depending on who you talk to) https://en.wikipedia.org/wiki/List_of_area_moments_of_inerti...

Dont the hyperboiloid chimneys have something with maximizing its surface? I think i remember sth like that from a course, but it is too far.... can someone confirm/reject?

A couple of years ago I got really interested in the shape of cooling towers after hanging out inside a couple of derelict ones. I couldn't find a solid answer as to why they are hyperboloids, and in fact not all of them are, but the most common explanations were:

1) the throat at the top could be the optimum shape for creating cooling via the Venturi effect

2) they can be built entirely with straight diagonal structural members, as each section of the Shukhov tower illustrates, but only the very earliest ones would have been made this way and they're certainly not any more

3) they were the only suitable shape that could be analysed on paper, before the advent of computer-based structural analysis

4) uniform structural stiffness with no particular points of failure, as in the above article

Even a thorough literature review from the period after some collapsed in storms was inconclusive... from the proceedings of the 5th International Symposium on Natural Draught Cooling Towers: http://books.google.com/books?id=6j5nuvAd44QC&pg=PA3

I highly recommend a look inside one, the acoustics and general enormity are quite something. Being inside an active one looks to be even more of something from these pictures: http://www.foantje.com/active-cooling-tower/

why oh why do I keep falling for wired linkbait. I didn't learn this from a mathmatician, the pizza just sluffed in my had the right way one time and it stayed.

Huh, why does the original Wired article say "19th century math genius" instead of "Gauss"? Is Gauss really that unknown to Wired's audience?

Yes.

Rules of Clickbait Headlines: never use one specific word where four vague ones, with at least one indicator each of antiquity, exulted discipline, and grandiosity.

Next week in Wired: How Newton taught my dog to catch rubber balls.

you know how i know you all are not from the northeast?

right?

"Best way" implies there are other correct ways. There is only one way to hold a slice - you fold it.

He used a knife and fork?

I found the article very interesting.

I've noticed this article is amongst a growing trend of articles that purport to solve 'non problems'

From wired, no less.

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