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

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

#21

Post by TTPrograms »

ubasu wrote: post #7

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 structu…

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.

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

#22

Post by ubasu »

TTPrograms wrote: post #21
ubasu wrote: post #7

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 structu…

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…

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.

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

#23

Post by ubasu »

theoh wrote: post #12
ubasu wrote: post #7

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 structu…

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 territ…

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

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

#24

Post by dxbydt »

ubasu wrote: post #20
dxbydt wrote: post #16

Earlier quoted context omitted.

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…

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_...

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

#25

Post by TTPrograms »

ubasu wrote: post #22
TTPrograms wrote: post #21

Earlier quoted context omitted.

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…

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.

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

#26

Post by lcrs »

raphinou wrote: post #17

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/

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

#27

Post by Chinjut »

dxbydt wrote: post #24
ubasu wrote: post #20

Earlier quoted context omitted.

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…

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)

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

#28

Post by Nacraile »

dxbydt wrote: post #24
ubasu wrote: post #20

Earlier quoted context omitted.

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…

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.

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

#29

Post by tedunangst »

dxbydt wrote: post #16
ubasu wrote: post #7

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 structu…

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…

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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