9 ms·

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

182 points by _mayo 12y ago

hugotox 12y ago
I already knew that from all my years as a pizza eater. You don't have to be a genius to know that
mhurron 12y ago
He used a knife and fork? I found the article very interesting.
suprgeek 12y ago
The Oatmeal provides a very nice illustrated pics about the Mantis Shrimp and its killer claws http://theoatmeal.com/comics/mantis_shrimp http://theoatmeal.com/comics/mantis_shrimp
DINKDINK 12y ago
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_inertia https://en.wikipedia.org/wiki/List_of_area_moments_of_inerti...
hayksaakian 12y ago
I've noticed this article is amongst a growing trend of articles that purport to solve 'non problems' From wired, no less.
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...
mach5 12y ago
you know how i know you all are not from the northeast?
- Neff 12y ago
  right? "Best way" implies there are other correct ways. There is only one way to hold a slice - you fold it.
StavrosK 12y ago
I totally call bullshit on not being able to crack an egg. Next time I see an egg, I'm trying it.
- DmitriRavinoff 12y ago
  YHBT
- Crito 12y ago
  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.
- sengstrom 12y ago
  I've seen this tested with an uncooked egg. A lot of pressure was required and when it finally broke the mess was unbelievable.
- maxiepoo 12y ago
  I went straight to the fridge and found I hurt my hand before I cracked the egg. I was as surprised as you are.
- TeMPOraL 12y ago
  If that was a troll sir, it was a good one. I wonder how many HNers had to clean their kitchen today ;).
anigbrowl 12y ago
Next week in Wired: How Newton taught my dog to catch rubber balls.
raphinou 12y ago
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?
- lcrs 12y ago
  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 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/ http://www.foantje.com/active-cooling-tower/
jordigh 12y ago
Huh, why does the original Wired article say "19th century math genius" instead of "Gauss"? Is Gauss really that unknown to Wired's audience?
- aatishb 12y ago
  Yes.
- dredmorbius 12y ago
  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.
grogenaut 12y ago
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.

182 points by _mayo 12y ago

hugotox 12y ago

I already knew that from all my years as a pizza eater. You don't have to be a genius to know that

mhurron 12y ago

He used a knife and fork? I found the article very interesting.

suprgeek 12y ago

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

DINKDINK 12y ago

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_inertia https://en.wikipedia.org/wiki/List_of_area_moments_of_inerti...

hayksaakian 12y ago

I've noticed this article is amongst a growing trend of articles that purport to solve 'non problems' From wired, no less.

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...

mach5 12y ago

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

Neff 12y ago
right? "Best way" implies there are other correct ways. There is only one way to hold a slice - you fold it.

StavrosK 12y ago

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

DmitriRavinoff 12y ago
YHBT
Crito 12y ago
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.
sengstrom 12y ago
I've seen this tested with an uncooked egg. A lot of pressure was required and when it finally broke the mess was unbelievable.
maxiepoo 12y ago
I went straight to the fridge and found I hurt my hand before I cracked the egg. I was as surprised as you are.
TeMPOraL 12y ago
If that was a troll sir, it was a good one. I wonder how many HNers had to clean their kitchen today ;).

anigbrowl 12y ago

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

raphinou 12y ago

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?

lcrs 12y ago
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 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/ http://www.foantje.com/active-cooling-tower/

jordigh 12y ago

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

aatishb 12y ago
Yes.
dredmorbius 12y ago
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.

grogenaut 12y ago

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.