In

*Bertrand the Shape-Shifter's Natural But Not So Obvious Pizza*, I wrote about we could use a set of (ALL) chords in a meaningful and natural way, by slicing up a circular pizza into arbitrarily narrow slivers and determining their average length, which would then give us the width of a pizza of length 2R with the same area as the circular pizza.
Mathematician responded by saying that we could envisage a
physician called Bertrand who lives on a circular atoll and uses his boat to
attend to emergencies which occur at random locations on the atoll. Each trip is notionally a chord (eliminating
wind effects, any strange current effects and curvature of the earth) and this
Bertrand can use data from trips over a sufficiently long period of time to
arrive at an average trip length (we’d also have to assume that he would
stubbornly use his boat even when walking would be more appropriate).

This is an intuitively appealing scenario. The problem, to my mind, is that while we most
certainly do get the average trip length I am not convinced that we get
the average length of a chord within the circle defined by the atoll. For example, as I pointed out to Mathematician,
we could conceptually slice up the disc of water surface surrounded by the atoll into
slivers/chords and arrive at the area of that disc using a similar process as
with the pizza reshaping scenario (longest sliver/chord length x average
sliver/chord length). And the result
would not be the same as the physician’s average trip length.

We could do something similar with the pizza slicing. Imagine that Bertrand the pizzeria owner had
a few padawans and a ridiculously large number of circular pizzas that he is
willing to devote to the resizing research effort.

Each padawan chooses a different method to slice the pizzas
to obtain representative samples of chords.
Because they are not as skilled as Bertrand himself, they must split
each pizza with one cut and then use the length of the cut as a chord
length. Then they add up the lengths,
divide by the number of pizzas and, voila, average length.

The first one thinks “a chord is the intersection of a line
and a disc, the pizza represents the disc and I will therefore find a way to
randomly intersect my pizzas with lines”.
He decides that what he will do is create a surface with a large blade
which can be randomly set to one of 3,600 million orientations (each with a
likelihood of 1/360,000,000) - think 3,600 different angles and 1 million parallel lines for each angle. He then
spins each pizza into position, randomises the position of the blade, engages
the blade and measures the slice.
Eventually he will arrive at an average length of πR/2.

The second one thinks “all chords of length greater than
zero cross the rim of the disc in two locations, so I can just pick two random
locations and slice between them”. She’s
not particularly well trained, so she doesn’t see any problem with using vermin
in her method and so decides to use her pet mice.
First she spins each pizza into position and then releases a mouse which
then wanders over to the pizza, then onto it in a cartoonish search for cheese
and eventually off the pizza again (at a random location). Then she brings out a samurai sword, a la
Kill Bill, and slices the pizza between the points at which the mouse mounted
and dismounted the pizza. Eventually,
she will obtain an average distance between mount points and dismount points –
each of which is a chord. However, this
average distance will be 4R/π.

The third one thinks “all chords pass through points within
a disc and each point on the disc has a shortest distance between intersections
with the circumference, making that point the midpoint of the resultant chord,
so I only need to pick points and produce the shortest slice through each
point”. Being even less aware of hygiene
considerations, this padawan brings a pet fly which is dipped in paint and
allowed to fly around until it lands randomly on a pizza, leaving a dab of
paint. The pizza is then sliced to make
the shortest chord through that point and the slice is measured. Eventually, this poor excuse for a human being will arrive at an
average length for a mid-point generated chord of something close to 4R/3. Note that this is pure observation on my part, I ran a simulation and the figure I got seems to hover around 1.33 after 4000 iterations. I don’t
have an actual equation to explain the value, it could just as well be 21R/5π
or 17πR/40 – in any event, the value lies between 4R/π and πR/2, but is closer
to the former.

My point here is that we can use all three of the standard
methods to arrive at chords, and thus average chord lengths, but only one method
produces the same result as slicing the entirety of one single circular pizza into arbitrarily
thin parallel slivers. Hopefully the reader
will grant that a single circular pizza sliced into arbitrarily thin parallel slivers
is representative of all possible orientations of the arbitrarily thin parallel
slivers – if not, consider an arbitrarily large number of circular pizzas which
are sliced the same way, but with arbitrarily small increments of rotation. The average length thus determined will not be
different from that arrived at via the slivering on one circular pizza.

Similarly, it is hoped that this method can be understood as obtaining a representative sample of all intersections of a disc (the circular pizza) with all lines that pass through that disc.

I have absolutely no problem with the average lengths arrived
at via the mount and dismount points or via the midpoints (although I’d be loathe
to eat any of the pizzas), but I cannot see them as representative of the
average length of all chords. They are merely the average lengths between the points at which the mouse mounted and dismounted
the pizzas and the average length of the shortest slices passing through random
flyspecks which, to me, seem to be different things.

And, in my opinion, Bertrand should sack two of his apprentices
with immediate effect.

---

Hopefully, it can be seen that the mouse scenario is
effectively the same as the physician scenario – it just seems a lot less
impressive, because it’s a mouse scampering around a pizza rather than a servant
of the people heroically heading off to save a life.

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