Next week in Wired: How Newton taught my dog to catch rubber balls.
How Gauss Taught Us the Best Way to Hold a Pizza Slice
11–20 of 35 posts
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#12The 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 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
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#13I totally call bullshit on not being able to crack an egg. Next time I see an egg, I'm trying it.
YHBT
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#14I totally call bullshit on not being able to crack an egg. Next time I see an egg, I'm trying it.
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.
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#15I totally call bullshit on not being able to crack an egg. Next time I see an egg, I'm trying it.
I've seen this tested with an uncooked egg. A lot of pressure was required and when it finally broke the mess was unbelievable.
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#16The 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 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...
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#17Dont 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?
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#18Huh, why does the original Wired article say "19th century math genius" instead of "Gauss"? Is Gauss really that unknown to Wired's audience?
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#19Huh, why does the original Wired article say "19th century math genius" instead of "Gauss"? Is Gauss really that unknown to Wired's audience?
Yes.
Re: How Gauss Taught Us the Best Way to Hold a Pizza Slice
#20The 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…
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.