Are food-borne pathogen survival times really exponentially distributed?

From Scientific American, an excerpt from Modernist Cuisine: The Art and Science of Cooking on the complex origins of food safety rules.

This is the six-volume, six-hundred-dollar magnum opus of Nathan Myhrvold (former chief technology officer at Microsoft, and chefs Chris Young and Maxime Bilet; you can read more about it at the Wall Street Journal.

In particular I noticed the following:

If a 1D reduction requires 18 minutes at 54.4 degrees C / 130 degrees F , then a 5D reduction would take five times as long, or 90 minutes, and a 6.5D reduction would take 6.5 times as long, or 117 minutes.

A "nD" reduction is one that kills all but 10^-n of the foodborne pathogens.

What struck me here is that the distribution of the pathogen lifetimes, assuming these numbers are actually correct, is exponential. And, therefore, memoryless -- if you're a bacterium under these conditions, your chances of dying in the first eighteen minutes are ninety percent, and if you're still alive at ninety minutes, your chances of dying in the next eighteen minutes are still ninety percent. This surprised me. The decay of radioactive atoms can be described in this way -- but are bacteria really so simple?

The excerpt as a whole is quite interesting -- apparently a lot more than just science is going into recommendations of how long food should be cooked.

(Myhrvold has a bachelor's degree in math and a PhD in mathematical economics, among other degrees; Young has a bachelor's degree in math and was working on a doctoral degree before he left for the culinary world. So perhaps it is fair of me to think that they would get this right.)

Plats diviseurs, or how the French cut cakes

Apparently in France they have an interesting solution to cake-cutting problems -- a plate with markings on the rim for the proper place to cut into 3, 5, 6, 7, or 9 slices, called the plat diviseur. See also here. You can buy them here; the site is in French. Some especially ornate examples are due to Paul Urfer, who appears to be the original inventor.

I found out about these from The Number Warrior, Jason Dyer. Unfortunately I have no use for one of these, because I live alone, in a small apartment where I couldn't reasonably have enough people over to need a whole cake, and so I do not buy a whole cake at once.

I suspect I have some French readers. Have you seen these before?

Brownies and space-filling curves

The Baker's Edge brownie pans, which are pans constructed in such a way that everybody gets an edge piece and nobody gets a piece from the middle, remind me of space-filling curves.

The isoperimetric inequality suggests that the only way to do the reverse -- to have pans where nearly everybody gets the middle and nearly nobody gets the edge -- is to have really big pans.

On foods of genus one

It seems that some people describe the torus as the shape of a bagel, and others as the shape of a doughnut.

I wonder if this is somehow correlated with geography; bagels are more common in some places, doughnuts in others.

Fractal cookies

Fractal cookies, from Evil Mad Scientist Laboratories

Take nine "square cylinders" (i. e. rectangular solids which are much longer in one direction than the other two) of dough, one of which has chocolate in it.

Arrange the nine sticks in a three-by-three grid with the chocolate one in the center; squish them together so that they are one big piece of dough.

Stretch the whole thing to eight times its current length; cut into eight pieces of equal length (the length of the original piece), each of which will have a chocolate center. (This can be done by stretching to twice the length, cutting in half, and repeating twice more.)

Add a piece of chocolate dough of the same size; again arrange in a three-by-three grid with the chocolate one in the center, stretch, and cut. Then do it again. Then cut the whole thing into slices and cook.

Of course, you get the Sierpinski carpet in cookie form.

However, at the level of iteration given here, (8/9)³, or about seventy percent, of the cookie will consist of non-chocolate dough! This is sad. I recommend interchanging the chocolate and non-chocolate doughs.

See also the Sierpinski gaskets made from polymer clay, which are made by a similar process. These are inferior, because they cannot be eaten.

Hoagies and permutations (yes, really!)

About once a week, I buy a hoagie from Wawa for lunch. (Apparently people not from the Philadelphia area think "Wawa" is a very funny word. Here's an explanation; basically Wawa the store is named after Wawa the town, which in turn is the name for the Canada Goose in the language of the people who lived there before Europeans did.) Today was such a day.

Now, when you go in and buy a hoagie, you order on a touch screen, and the touch screen prints out a receipt with a number on it. These numbers are assigned sequentially -- but for various reasons, people's orders don't get filled in the same order that people put in their orders. This is immensely frustrating, and gives one a visceral sense of why permutations with a lot of inversions are kind of annoying. (A permutation is just a reordering of some totally ordered set, say {1, 2, ..., n}; an inversion, informally, is what we have when a larger number comes before a smaller number. For example, 1 5 3 2 4 is a permutation of {1, 2, 3, 4, 5}; it has four inversions, namely the pairs (5, 3), (5, 2), (5, 4), and (3, 2). I know that some people define inversions a bit differently, looking at the position in which the offending numbers appear instead of the offending numbers themselves.) The reasons why orders don't come out in the same order they come in have to do with the fact that there are multiple people preparing orders in parallel (at least during the lunch rush; the dinner rush is much less intense, both because there are less people buying dinner than lunch on a university campus and because the times at which peopel eat dinner are more spread out than the times at which they eat lunch) and that some orders take longer to prepare than others. To a first approximation, there seem to be three classes of order: food which is already prepared, cold made-to-order things, and hot made-to-order things.)

The wait was long today, so I got to thinking -- basically this procedure is generating a permutation of the integers. But it's a permutation of the integers with only Θ(1) inversions per element; that is, for any given customer, the number of people who came in after them and yet get served before them is on average some constant number. (I have no idea what this number is.) If we consider, say, the permutation that's generated in this way over the course of an hour during the lunch rush, it might be a permutation of [100] with a few hundred inversions. A "typical" permutation of 100 has about 2,500 inversions -- as n gets large, the number of inversions of a permutation of n selected uniformly at random is approximately normally distributed with mean n(n-1)/4 and standard deviation on the order of n^3/2/6. Efficiently generating a random permutation with, say, less than 10n inversions (for large n) is not just a matter of generating random permutations (which is easy) and throwing out the ones which you don't like; there are sampling algorithms which work this way, to sample uniformly at random from some subset of a set which it's easy to sample u. a. r. from -- but as far as I know they don't throw away all but a vanishingly small proportion of all the elements in the large set! I don't know if anybody's thought about this problem.

A sighting of mathematics on Jeopardy!

Today's Final Jeopardy question (alas, I didn't write down the category, but it was something like "brands"):
"Each unit in this brand, introduced in 1968, is a hyperbolic paraboloid, & they fir together for perfect storage."

The answer: What are Pringles? Two of the three contestants got it right; the third answered "What is Orville popcorn?"

I wonder if the inclusion of the "hyperbolic paraboloid" makes it harder or easier. I think it made it harder for me, because I got confused and was picturing a hyperboloid of one sheet instead. Fortunately I realized that those wouldn't fit together in any reasonable way, at least if they were all the same size. I don't know the background of the contestants tonight; I suspect most people, even most people who know enough random things to appear on Jeopardy!, would probably just filter out those words, or at best replace them with "funny shape" (because they've heard of hyperbolas and parabolas). "Funny shape" is probably the right way to think of it here; the fact that Pringles are oddly shaped is a much more salient fact about them than the precise shape, unless of course you work in a Pringle factory.

(Before you ask: No, I've never tried out for Jeopardy. Yes, I probably should.)

Boiling eggs and dimensional analysis

A formula for boiling an egg. We learn that the time it takes to cook an egg is

where ρ is density, c specific heat capacity, and K thermal conductivity of egg. T_egg is the initial temperature of the egg, T_water is the temperature of the cooking water, and T_yolk is the temperature of the yolk-white boundary when the egg is done. (The egg is modeled as a homogeneous, spherical object; the yolk-white boundary is just a proxy for a certain distance from the center of the egg.)

What I keep noticing about formulas that purport to model real-world situations is that no matter how complicated they are, when numerical exponents appear they are always rational numbers. This is also true for purely mathematical phenomena. If someone told me that some statistic of some combinatorial object, for large n, had mean an and standard deviation bn^7/22, for some bizarre, probably-transcendental constants a and b, I would think "ah, that must be a complicated proof, to generate such a large denominator." But if they said that this statistic had mean n and standard deviation n^1/π, I would know they were pulling my leg. Of course, these would be fairly difficult to distinguish "experimentally". I suspect the reason that such exponents are rational in physics problems is for reasons having to do with dimensional analysis. For example, in the case of this problem, someone with sufficient physical intuition would probably guess that, fixing the temperatures, the important variables are the mass, density, thermal conductivity, and specific heat capacity of the egg. How can we multiply these together in some way to get time? Well:

• mass (m) has units of kg


• density (ρ) has units of kg m^-3


• thermal conductivity (K) has units W/(m × K), or kg m K^-1 s^-3


• specific heat capacity (c) has units J/(kg × K), or m^2 s^-2 K^-1

Thus m^α ρ^β K^γ c^δ has units kg^α+β+γ m^-3β+γ+2δ K^-γ-δ s^-3γ-2δ. But we know this is a time, so it must have units of seconds; thus we have the system of equations

α + β + γ = 0, -3β + γ + 2δ = 0, -γ - δ = 0, -3γ-2δ=1

which has the solution α = 2/3, β = 1/3, γ = -1, δ = 1, exactly the exponents we see. Similar analyses are possible for many physics problems, and in the end one gets a system in which integer combinations of the exponents are integers; thus the exponents themselves must be rational. And that's the really important part of the problem. In fact, all that really matters is that α=2/3. In practice, if you're cooking eggs, the thermal properties will be the same from egg to egg, but some will be larger than others; the exponent of mass tells you how to adjust for that, and everything else can be calibrated experimentally.

Mathematicians don't worry about this so much... but lots of these sorts of problems have some sort of physical model, which probably explains why rational exponents are common in combinatorial problems but I can't ever remember seeing an irrational exponent. (That's not to say they don't exist -- just that the solution to any such problem would be quite unconventional.)

(I learned about this from this post at Backreaction, about the phase diagram of water.)

pizza pie are square(d)

Geometry Saved Me Money, from Binary Dollar, via Grey Matters. Which is more: a twelve-inch pizza or two eight-inch pizzas?

The twelve-inch pizza, of course; it is more square inches of pizza. (I'm assuming all pizzas are equally thick.)

However, if you really like crust, the two eight-inch pizzas might actually be the better deal. One twelve-inch pizza contains 36π square inches of interior and 12π inches of crust; two eight-inch pizzas contain 32π square inches of interior and 16π inches of crust. So if you're willing to trade 4π square inches of interior for 4π inches of crust, take the smaller pizzas. That is, if you'd rather have a third of the crust of the pizza than a ninth of the interior, or if you'd prefer three crusts to one crustless slice.

I like the crust, so I might.

(This analysis assumes, of course, that the thickness of the crust is negligible, so that "an inch of crust" actually means something.)

The waitress in the restaurant where this question came up thought the customers would prefer the two eight-inch pizzas because it was more "slices of pizza". Maybe it's just me, but a "slice of pizza" is a meaningless unit, because it's not standard. I would have at least expected the argument that eight plus eight is more than twelve.

I suspect that pizza is the food that is most often "illogically" priced. I've seen, say, chicken wings sold at "10 for $5, 15 for $8" but you don't see that too often, because most people can do the math and realize that buying 15 is a bad deal. (Think of it this way: what if I want thirty? I can get three 10-packs for $15, or two 15-packs for $16.) But with pizza people will throw up their hands. Also, I have seen places where a larger size of pizza costs more per square inch than a smaller pizza (I can't find any right now); I was once told that this was because for whatever reason the large size was more inconvenient to make (it fit in the oven funny, for example). That at least seems like a plausible economic reason; it's clearly not the cost of the ingredients and almost certainly not the labor.