How big can a pizza slice be? [from a 1993 sci.math discussion]



baloglou@oswego.Oswego.edu wrote:

[snip]

: -How big is your "small" pizza?

: ... to be bolted by:

: -Well, it has six slices!

[snip]

Actually, I could argue that this was a reasonable practical response, though strictly speaking, it makes no mathematical sense whatsoever and could apply to a pizza of any size and shape. I understand that you're pointing out a common misconception, but I think I have a good idea roughly how big their small pizza pie is (I would guess between 10" and 16" in diameter), though mathematically there is no basis for this assumption and there's also a good chance even a 'real' pizzeria (not just a mathematically postulated one) might still cut up much larger or smaller pies into six pieces.

Perhaps, though, on a more conceptual level, the statement 'Well, it has six slices' actually tells you more about how much food you're going to receive than 'Well, it is 16 inches across.' You'll see what I mean as we go. In that regard, your countergirl may have been smarter than you in communicating to you how much pizza you'd be getting!!

Besides poking some fun at the whole problem, I hope I'll be able to make a couple of 'real' points along the way about how math-oriented folks often interact (or misinteract) with the 'real world.' It is often the case that what is commonly understood is not mathematically sound, but we still manage to communicate, by and large. In some cases, the mathematics might actually get in the way of expressing ourselves ("Just how upset were you that I forgot to call you last night? In standard units of 'car trouble', can you tell me how exactly how upset you were, my dear?").

OK -- so this is going to be a long treatise, but I can't resist. When you talk about pizza, you're talking about something near and dear to the core of my being.


WHAT IS A SLICE?

Although there can be tremendous variation on what is considered a reasonable 'slice' depending on where you go and whether you're paying for an individual slice or not, the whole point of cutting larger pizzas into more slices is usually to preserve some constant quantity of comestibility per slice -- be it the area, volume, weight or elusive 'stomach function' of the slice.

Professional chefs often refer to a 'portion' as though this were some glass- encased unit kept at the National Bureau of Standards. What is really meant by a 'portion' is what you would typically expect someone to eat during the course of a meal, and from that, the cook can determine how much to prepare to feed N people with no waste nor want. Having been the steward at a large fraternity at MIT (regularly feeding 40, but often as many as 80-90 on weekends or parties counting all the guests, girlfriends, family, etc), I can attest to the fact that this simple technique works remarkably well since there really is a 'mean' portion size that people tend to consume and that you can guess at beforehand, despite tremendous individual variation in consumption. We tended to experience more a local depletion or surplus of food within arm's or table's reach (depending on who-sat-where) rather than an overall positive or negative error. [The exceptions, of course, being ice cream or jello -- any finite quantity of ice cream was never enough yet *any* quantity of jello was way too much.] But even when I cooked my famous 'Thermonuclear Pizza' for 60 on several occasions, I always came remarkably close to what people actually consumed despite the fact that my general 'two to three slices per person' translated to the right amount regardless of how many slices there were in a pan (about 24 -- Sicilian style), and god-knows how many pounds of flour, cans of tomato sauce, etc., would need to be consumed.

The whole point I am trying to make is that there really is a certain 'pervasive' notion about serving size in the general population, despite the fact that the portion for an 'individual' might consume has relatively little significance. I guess there's an aspect of probability to the whole idea.

[Incidentally, one of the best cookbooks in existence is "The Professional Chef" which is used as the standard textbook reference for many culinary schools. Except for the fact that you have to convert all their recipes like

whip two dozen eggs

quarter 15 chickens

(or some other ungodly number of animal souls)

puree 2.5 gallons of sauce

into more reasonable units (I live alone), it's still a great cookbook. I can highly recommend it to anyone who's really into quality cuisine. The great thing is that each recipe tells you how many people it will serve (usually 30 to 60). Through experimentation with a number of recipes, I have thus been able to determine with certainty that I am 2.2 people in actuality. :-)


Again, what is a slice, exactly, or within reasonable bounds?

I think most people would agree that 2-4 slices is typically what you'd want to consume at a sitting, although I know some folks who will always order a whole pie for themselves (and consume it too), and others whom I've never seen finish their second slice.

It's quite possible that it is *you* who were locked into the concept of 'size' (diameter or area in particular) and the countergirl may have been trying to communicate more the intangible concept of 'satiation' to describe what you'd get. [Isn't that typical -- the male is concerned only with size and the female is concerned more with how much overall satisfaction will be delivered! :-) ]

So what we really need to do is agree on what a slice (a portion of pizza) really amounts to.

If Domino's (as an example) were to define an ANSI slice as 1/2 square foot (not a bad approximation), then you would immediately know the dimensions of your pizza by how many slices it contained, as the total area would be n/2 = pi*r^2, where n is the number of slices and r is the radius of the pizza, in feet. Thus, the diameter is d = 2r = 2 sqrt(n/2pi). So in this 'ideal' world, the phrase 'six slices' immediately tells you what the dimensions of your pie are. [Assuming, of course, that you ordered Neopolitan (round) pizza, and not Sicilian (rectangular) pizza. :-) ]

Unfortunately, you still wouldn't know if you could eat even one slice. Ever been to a Pizzeria Uno? Have you ever been able to consume more than three of their slices? They're only about 1/3 square feet apiece, and therefore roughly to equivalent to a small portion of two ANSI pieces (3 * 1/3 square feet == 2 * 1/2 square feet) but (true to their advertising claims) -- Pizzeria Uno serves *deep-dish* pizza, and they ain't kidding.

OK, so lets say Pizzeria Uno (as our alternate standard) were to describe instead that their deepdish Sicilian pizza slices are defined by a measurement of *volume* and not area, you'd have a similar general idea of how much food you were getting, but you still wouldn't know whether or not your pizza pie would come in the form of a flat pie, a perfect 'Dyson' sphere of crust with the toppings and sauce within, an extremely tall 'column' several hundred feet high, or in the form of a powder scattered over the state of Michigan. But would you care? Sure, it might be a challenge to eat it, but you'd still have the same overall 'amount' of pizza, wouldn't you?

Perhaps not. Suppose Prof. Phillip Morrison opened up "Phil's House 'O' Cosmological Pizza." Suppose further that the specialty of the house was pizza with plasma sauce and black hole topping. Would you be able to consume even a single slice, much less pick it up or survive getting within arm's reach of it? Perhaps the pizza would end up consuming you!! So the whole notion of size of the pizza becomes truly meaningless and a *much* more useful thing to know would be how many portions you might want to consume (or be consumed by).

And of course, if you happen to step into J. Christ's Desert Pizz-a-rama shop, then three small pizzas would be enough to feed any size crowd. Order three and you'll never go hungry!! :-)

Perhaps the only thing your countergirl neglected to tell you was some additional pieces of information besides just the number of slices. Why don't you go back and see if they'll supply the density, the rate of expansion of the pie based on changes in temperature (after all, you did not specify whether you meant the size of the pizza as it was cooking or the reduced size you'd have when you opened the box at home), and so on? Also, you might want to check on what they meant by 'slice', since most pizzas that are cut into six 'pieces' are actually produced by just three 'slices' of the knife. Perhaps when she said six slices she meant a whole lotta pieces!! :-) :-) :-)


BUT, SERIOUSLY, FOLKS

All kidding aside -- your counter girl could have simply pointed to the pan your pizza would be cooked in and your question would have been quickly and easily answered. On the other hand, a slice *does* connote a certain undefinable, but commonly understood concept of 'quantity' of pizza, and therefore her response was not completely absurd or uninformed, though you *did* specifically ask about the 'size' of the pizza, and not the general quantity of food you'd receive.

Then again, my local pizza shop (none of the above chains, by the way) is notorious for cutting up a perfectly good 18" pizza into something ridiculous like 30 slices (I'm serious). I have to eat about three of their slices to get the sensation of having consumed 'a slice' -- which leads me to believe that the rest of their customers are mice *or* that their cooks are Freddy Kruger and Norman Bates wannabes.

But the point is that a 'slice,' though not defined by the National Bureau of Standards yet as far as I know, does *imply* a certain amount of pizza, and therefore six slices does *imply* a certain size to the pie, though I agree, there's tremendous potential for variation out there.


THE POINT

I am of the opinion that it is very easy to condescend to those who have not had a whole lot of math or science training, but that does not mean they are any more or less intelligent for it (though their SAT and IQ scores may suffer). I am not saying you were at all being condescending to the woman -- I think your point that her response was totally ambiguous from a mathematical standpoint is perfectly valid, and humorous enough to warrant a posting to sci.math.

But isn't it just possible that she may have been the more intelligent one, since she did 'imply' exactly how much food you'd be getting in terms of comestibility? I have tried to prove that the size of the dish may perhaps be a secondary issue to the number of portions (ANSI or otherwise) to a food service proffesional. Perhaps your mathematical training temporarily clouded your innate judgement of real world compromise?

Isn't it just possible?

Then again, she may have been a ditz working in a pizza joint for minimum wage and not the genius of our time -- what do I know??

I'll bet you anything that a hundred years ago the same answer would not have come as any surprise -- but in this age of technology, I think it's easy to forget how human we all are and how we each come away with a different set of skills (or lack or skills) from our educational system. I am not convinced that having everyone in the world understand what a diameter is and how it's different from a radius would make the world a better place to live in one iota. While I'd like to see primary schools teach math and science in a much more stimulating manner so that more people would take pleasure in it and *use* it in their daily lives, I'd still like to see the world remain as diverse and interesting as it is, with the possibility of speaking to people who have diametrically different backgrounds from my own.

My 2 cents. This was intended predominantly in fun, but I guess I managed to sneak a couple of 'real comments' on the state of education and 'intelligence' in there. Please don't take offense by anything I said, and I pardon the fact that I couldn't express myself in a few thousand words less. I can write five pages of material in an hour and convey the same ideas in a page after five more hours of work.


#BEGIN FLAME

A suggestion for an end to world hunger:

Finish that pizza crust!! They're highly nutritious and represent a lot of wasted grain!

#END FLAME



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