5 ms·

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

by ubasu 12y ago

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 http://en.wikipedia.org/wiki/Bending [2] http://en.wikipedia.org/wiki/Arch http://en.wikipedia.org/wiki/Arch [3] http://en.wikipedia.org/wiki/Truss http://en.wikipedia.org/wiki/Truss [4] http://en.wikipedia.org/wiki/Stress_concentration http://en.wikipedia.org/wiki/Stress_concentration

theoh 12y ago
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 http://en.wikipedia.org/wiki/Morse_theory or http://www.maths.ed.ac.uk/~aar/surgery/hilldale.pdf 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 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 https://www.youtube.com/watch?v=36gOx3dguWs#t=17m35s
- ubasu 12y ago
  Seems like an interesting way to cast a structural shape. Possibly, it transfers stresses efficiently because it follows the deformed configuration of the fabric.
dxbydt 12y ago
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 http://mathoverflow.net/questions/5450/cocktail-party-math [2]http://tosca.cs.technion.ac.il/book/index.html http://tosca.cs.technion.ac.il/book/index.html [3]http://www.damtp.cam.ac.uk/user/pz229/Teaching_files/GR.pdf http://www.damtp.cam.ac.uk/user/pz229/Teaching_files/GR.pdf [4]http://www.amazon.com/Lectures-Complex-Analysis-Contemporary-Mathematics/dp/0821848097 http://www.amazon.com/Lectures-Complex-Analysis-Contemporary...
- ubasu 12y ago
  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.
  - dxbydt 12y ago
TTPrograms 12y ago
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.
- ubasu 12y ago
  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 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.
  - TTPrograms 12y ago
    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.
  - Rapzid 12y ago
    Depending on the fabric, you most certainly can.
irremediable 12y ago
Yeah... well summarised. I kind of enjoyed the article, but while reading it I kept thinking, "This isn't really the reason..."

by ubasu 12y ago

theoh 12y ago

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 http://en.wikipedia.org/wiki/Morse_theory or http://www.maths.ed.ac.uk/~aar/surgery/hilldale.pdf 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 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 https://www.youtube.com/watch?v=36gOx3dguWs#t=17m35s

ubasu 12y ago
Seems like an interesting way to cast a structural shape. Possibly, it transfers stresses efficiently because it follows the deformed configuration of the fabric.

dxbydt 12y ago

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 http://mathoverflow.net/questions/5450/cocktail-party-math [2]http://tosca.cs.technion.ac.il/book/index.html http://tosca.cs.technion.ac.il/book/index.html [3]http://www.damtp.cam.ac.uk/user/pz229/Teaching_files/GR.pdf http://www.damtp.cam.ac.uk/user/pz229/Teaching_files/GR.pdf [4]http://www.amazon.com/Lectures-Complex-Analysis-Contemporary-Mathematics/dp/0821848097 http://www.amazon.com/Lectures-Complex-Analysis-Contemporary...

ubasu 12y ago
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.
- dxbydt 12y ago

TTPrograms 12y ago

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.

ubasu 12y ago
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 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.
- TTPrograms 12y ago
  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.
- Rapzid 12y ago
  Depending on the fabric, you most certainly can.

irremediable 12y ago

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