Bertrand the Shape-Shifter's Natural But Not So Obvious Pizza

Imagine that we have the owner of a pizzeria, let's call him Bertrand, who wants to break with tradition.  For centuries the pizzeria that he is the current owner of has made round pizzas (what Americans call "pizza pies" - thus allowing some sense to be made of Dean Martin's "That's Amore": When the moon hits your eye, like a big pizza pie - which always sounded to me like When the moon hits your eye, like a big piece o' pie … who throws around bits of pie, let alone big bits?)

But Bertrand is now heartily sick of circles and he wants to make the transition to rectangular pizzas.  However, he has a minor problem.  His customer base is accustomed to a pizza base based pricing scheme - and they they don't want their pizzas to shrink (or grow) as a consequence of this shape change.  Bertrand already has a range of boxes which fit his circular pizzas perfectly, so he knows how long his rectangular pizzas will be … all he needs to do is work out how wide they have to be to keep his loyal customers happy.

Here's a graphic to illustrate his conundrum:

 

This is reasonably easy to work out.  The area of the rectangle is 2R times the width (w) while the area of the circle is πR², so we make those areas equal:

w.2R = πR²

w = πR/2

Thus we could say that the "average width" of a circle of length 2R (which is true of all circles of radius R) is πR/2.  All Bertrand needs to do is plug his values of R (10cm (bambino), 15cm (piccolo), 20cm (medio), 40cm (grandi), 60cm (ridicolo)) and Roberto's his uncle.

But let's say that Bertrand did not have a mathematician handy and he was casting around for another way to work out the area of his pizza in rectangular form.  How could he do it?

One way would be to use a form of integration.  Being a very precise person, and skilful in the ways of pizza, Bertrand could slice his pizza up into 1mm wide slivers, use those slivers to reassemble the pizza in rectangular form and then measure the resultant rectangle.  This is equivalent to how we find the area under a line (using Reimann sums).  By arranging the slivers in a 2R.w rectangle, Bertrand is effectively "adding them up".

Essentially, if not practically, Bertrand could do this with infinitesimally narrow slivers and doing so would only make his result more accurate (- see Wikipedia's article on Reimann sums which has animations that show something similar to my arbitrarily large value of N approaching infinity).  The infinitesimally narrow slivers would be equivalent to chords and as a consequence, the "average" length (or even width) of these chords would be πR/2 - where, by "average" I mean "mean", and this "average" would be the same as the "average" (mean) width of a circle of radius R … but not the "mean width" which means something else.

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I have to go into an aside here.  Or rather two asides.  Or maybe three (and three asides do not atriangle make … oops that'd be four asides now).

The existence of the mathematical term "mean width" provides us with an example of how English could, at least occasionally, benefit from more concatenation.  Such concatenation would allow us to clearly distinguish between mean width in a general sense and what would become meanwidth … in much the same way as we can distinguish between Donald Trump's wetback (I think it's the second from the left) and Donald Trump's wet back (fortunately, no image was available).

Even so, I've effectively used the concept of "mean width" without referring to it.  The length of Bertrand's new rectangular pizzas is the mean width of a circle of radius R.  Because I am not a professional mathematician, just a person who uses the more useful aspects of mathematics on a daily basis, I tend to think of three dimensional objects having three features: length, width (or breadth) and height.  Height relates to the object's orientation with respect to gravity, width can be used for both the other dimensions under many circumstances (think of a tower with a rectangular base, we know how high it is, but it seems wrong to think of its longer base as defining its "length").  However, for things which are not particularly high (like rectangular pizzas), it certainly feels like there is a convention such that the longer side gives the length and the shorter side gives the width.

I'm not saying that people who think differently are necessarily wrong, but I would surely be forgiven for thinking of them as being as thick as two short planks.

Another little aside, I tend to over complicate this allusion by thinking that someone who is talking about short planks doesn't know how to use planks properly, say we have a 10 foot 1x6 plank.  To me that is a plank that is 10 foot long, 6 inches wide and 1 inch thick.  We could make these "short" planks by considering the 1 inch to be its length, but such a plank is at least 6 inches thick.  That's thick for a plank, right, and so is the guy who thinks you can legitimately think of a 10 foot 1x6 plank as being 1 inch long …

I would be tempted to challenge such a person with a tale about a chicken who crossed a road, on one side of which was London while Dover was on the other side - so the obvious answer to the ancient riddle would thus be "to go on holidays".  It just happens to be a fact that, in length, the A2 is ridiculously short (perhaps even less than 10m in places), but it makes up for this by virtue of its incredible width (in the order of 115,000 metres).

Then I'd snort contemptuously and go back to arranging my pencils.

Anyway … I did use the concept of mean width, but I used it as if it were "mean length".  Oops.

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Bertrand the shape-shifting pizzeria owner has now been able to find out the necessary width of his new rectangular pizzas, so let's leave him behind now and look again the other Bertrand and his problem with chords.

It has been argued that the problem arises from the fact that there is no single obviously natural method to select (identify or get) chords and (either consequently or on the basis that) there is no single obviously natural probability measure.

I suggest that this mean chord length might be another useful if not exactly obvious (except perhaps in retrospect) was to arrive at a natural probability measure.  That is, if you arrive at a mean chord length which is not equal to πR/2, then you have a problem.

More specifically, I am suggesting that a method that arrives at a set of chords the mean length of which is not πR/2 then as a consequence, we have discovered that there may be something unnatural or skewed about that set, even if, prior to the discovery, the method appeared to be natural and unbiased.

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I did do some modelling and proved (to my own satisfaction) that chords selected "at random" using the 1/2 method have a length, on average, of πR/2.

Chords selected using the 1/3 method have a length, on average, of 4R/π while chords selected using the 1/4 method have a length, on average, of what appears to be πR/3.

Note that these averages were calculated on the same basis - I generated a large number of random chords using each of the methods and then obtained the arithmetic mean of the resultant chords.