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.