
How Gauss Taught Us the Best Way to Hold a Pizza Slice - _mayo
http://www.wired.com/2014/09/curvature-and-strength-empzeal/
======
ubasu
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)

~~~
dxbydt
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...](http://www.amazon.com/Lectures-Complex-Analysis-Contemporary-
Mathematics/dp/0821848097)

~~~
ubasu
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
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_...](http://www.reddit.com/r/math/comments/1eoo1p/q_what_are_the_10_most_beautiful_theorems_in/)

~~~
Nacraile
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.

------
suprgeek
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)

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

~~~
Crito
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.

------
DINKDINK
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...](https://en.wikipedia.org/wiki/List_of_area_moments_of_inertia)

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raphinou
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
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/)

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grogenaut
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.

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

~~~
dredmorbius
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.

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

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mach5
you know how i know you all are not from the northeast?

~~~
Neff
right?

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

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mhurron
He used a knife and fork?

I found the article very interesting.

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

From wired, no less.

