There's More than One Way to Slice a Pizza

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.

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.

---

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.

---

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.

---

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.

My Problems with Mathematician's Circular Argument

In the comments to Rectangular Circles - Yet Another Responseto Mathematician, I wrote what I thought (with some hubris) was a great argument, a killer argument:

I get the point that you are making here (or at least I think I do). It's why I've talked about the "set of ALL chords".

When you say (paraphrased) "there could be far more chords with c in [R/2,R] than in [0,R/2]" you are, I presume, assuming a "proper" mathematical circle/disc - we are talking about Euclidean space and not talking about curved space, or anything tricky like that. If so, I'd have to ask, on what basis, other than your selection process for problems like this, can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length? The circle/disc under consideration is essentially undefined as far as location, size and rotation go, so we should (reasonably) be able to change the locus and not have our answer change on us - but what you are suggesting is that if we shift the locus up by R, and rotate the circle/disc by π, then we'll change the number of lines passing through the intervals [0,R/2] and [R/2,R]. Ditto if we expand our circle/disc by a factor of 2 while retaining the locus at the notional (0,0).

We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc) and more lines passing through [0,R/2] (as defined by chords in the second).

This seems odd to me. Does it not seem odd to you?

Mathematician replied (with my current responses interspersed):

> "set of ALL chords"

Wow, maybe I understand what you mean, but it would be odd.  We have a given circle, right?  When you say the "set of ALL chords", are you including the chords that are NOT inside the given circle (but inside another circle somewhere else ...)?

Was that your point all along for repeating "ALL chords" all the time? It would make sense with the rest of the argument:

I first talked about "ALL chords" in a response to a comment on Triangular Circles.  In Mea Culpa, I put some effort in to explain what I meant by "ALL chords" – there is hopefully no indication whatsoever that I had any thought about considering all chords in all circles, and thus including chords that are not in the circle being considered.  No, I meant "ALL chords" in the circle being considered.

My apologies for not making that sufficiently clear.

Sadly, the banks rarely have this sort of confusion, so when I go and tell them that I want to take out "all the money", they don’t pop out the back and give me every single dollar from everyone’s account … they just give me the money that that was in my account.

> on what basis can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length?

Since the beginning, you are thinking of chords as the intersection of a straight line with the disc. That's a great characterization and a good way to get chords. (but not the only one, as we both know)

So if I'm not mistaken, for your point of view, there is an existing set of all straight lines on the entire plane (like an infinite net), and you are just taking the intersection of this existing set of straight lines with a given disc. And you say that if you move the disc around, it will not cross the same straight lines, but the answer to Bertrand question should remain the same. Am I correct to assume that this is more or less your reasoning?

With this interpretation you are absolutely correct and agree with Jaynes argument. This is a mathematically correct argument.

At least I have that right!

But that's not the only natural point of view on this problem.

See, I'm taking another characterization for chords. For me a chord is a segment between two points on the circle. So there is nothing "outside" my circle. I have no reason to extend a chord out to infinity. The chords are not intersection of lines with the disc, they are segments inside the disc! There is no reason to consider objects (lines) that are not chords on the given circle, don't you think?

So, If I change the locus of the circle in the plane, the chords are moving with it. If I double the size of my circle, then the chords inside it will double their size. If I rotate the circle, the chords will rotate with it. So the final answer to Bertrand question will not change at all.
And with that point of view, it's perfectly natural to have more chords close to the rim than close to the locus. You only think it's odd because you are thinking of an existing "net" of straight lines on the plane, and you place your circle on that existing net of lines. But from my point of view, there is no "net" of existing straight lines.

I note that you write here "The chords are not (the) intersection of lines with the disc".  In one of your comments at Triangular Circles, you wrote (my emphasis):

A circle is a 1-dimensional curve in the plane. A disc is the 2-dimensional surface that is enclosed by the circle. So it's important to make the distinction, whether you talk about the endpoints (which are on the circle) or the midpoint (which is in the disc) of a chord (which is the intersection of a line and a disc).

Perhaps you can see why someone might get confused.

So my problem with this approach, and I’ve hinted pretty strongly at it (that is by writing it), is that as you have said yourself a chord is the intersection of a line and a disc – irrespective of how you select that chord – and a chord is thus also a segment of a line – again irrespective of how you select the chord (it just happens to be the segment of the line that intersects with the disc).

If you could have a circle and nothing else, then I guess I would have to agree with you, at least on the basis of my ignorance with respect to the implications and also in recognition of your authority as a Doctor of Mathematics.  However, immediately after invoking this free-floating circle you refer to something outside the circle – specifically when talking about changing the locus, altering the size and rotating it.  You have thus invoked an external reference plane on which the circle/disc rests.  Would you not agree that it is meaningless to talk about translational, rotational and scalar invariance if the circle is all there is?

It seems, therefore, to me, that you are attempting to have your cake and eat it too.

Perhaps there was something in Bertrand's original phrasing that leads you to think that we can talk about a free floating circle, rather than one embedded in a plane.  I just don't know.  Unfortunately, my French is that of a rather forgetful schoolboy, I remember bits and pieces, I get the general gist of a menu or a wine list, but I never got to the stage at which I might have deciphered Bertrand's original text.  If only there were some French speaking mathematician I could call on to interpret …

This paper, interesting, explicitly refers to a plane, or rather the plane: "Consider a disk on the plane with an inscribed equilateral triangle."  In the afterword, the author writes (rather pleasingly from my point of view):

In his pointedly titled paper The Well-Posed Problem, (Jaynes) applies this principle to the Paradox of the Chord with success, uniquely identifying the uniform distribution over the distance between the midpoint of the chord and the center of the disk as the correct choice of measure, which he then proceeds to verify experimentally.

The use of the term "correct" is particularly satisfying, although I'm sure that there's some reason why my understanding of the term "correct" is limited and that that lack of understanding will be swiftly addressed.

> We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc) and more lines passing through [0,R/2] (as defined by chords in the second).

If I understand correctly, with your point of view, if you have two circles in the position you gave, there is as much chords in the interval [0,R] than in the interval [R,2R], right? With your point of view, the fact that we have one, two or seven circles, do not change the "density" of chords at all, is that correct? This seems odd to me.

With my point of view, each circle has its own set of chords, so the "density" of chords will be higher in the intersection of both disc. So there will be "twice as much" chords in the interval [0,R] than in the interval [R, 2R], because there are two set of distinct chords.

If we permit a single mathematical line to "carry" multiple, overlapping, distinct chords, then I don’t see a major problem with the density of chords changing as you overlay circles (ie you could conceivably have ten identical circles on top of each other, with an infinite number of chords, each of which is replicated ten times).  But in the context that you quoted, I was not talking of chords per se, nor was I really thinking about the chords in multiple circles (except obliquely).  Perhaps I should not have even mentioned overlapping circles at all, because this has only served to confuse.

You seemed to understand what I was saying earlier, when you talked of a "net of straight lines on the plane" (my recollection is that all lines are straight, but perhaps you were just clarifying this for my benefit).

There will be no line that passes through the circle (across the disc) which is not also coincident with a chord, right?  (By this I mean that there will be a segment of the line which directly corresponds with a chord, the ends of which lie on the circumference of the circle/disc - the segment shares the same length, gradient and endpoints as the chord such that one could almost consider them to have the same identity.  When I said "carry" above, I mean to imply that such a chord is, in a sense, lying on the line with which it shares a segment.  It is possible, with multiple circles, for there to be multiple segments of the line, perhaps overlapping, lying on top of each other … perhaps entirely identical and overlapping. This might not be standard phrasing, but I hope that you can understand my intention.)

There is an infinite array of lines that pass through the circle, and a(n infinite) subset of those will intersect with a notional y-axis.  With your preferred method of selecting (or identifying) chords, via their endpoints, it seems to me (and even to you, apparently) that there will be fewer lines that intersect the y-axis at the locus of the circle than at the circumference – this is what does not appear to be justified.  Spiriting the circle out of this universe does not appear to be justified either, but perhaps the problem as originally phrased does demand it and I am simply unaware of that aspect of it.

---

Would it help to move away from circles for a moment?  I understand that there are certain things about circles that might lead you to favour endpoint generation of chords.

I was thinking of a similar problem, but involving a square (squared circles!)  What is the probability that, on selecting at random an s-chord (my term for the equivalent of a chord within a square) the endpoints of which do not share the same side of a square, the selected s-chord is greater than the length of the longer sides of an isosceles triangle which has one of the sides of the square as its base (√5L/2 where L is the length of the sides of the square)?

This does, to me, seem a more complex question to answer.  I'm tempted to say that the endpoint approach will give, once again, a result of 1/3.  We don't seem have an equivalent of the other two approaches, because s-chords are not as constrained as chords - but perhaps there is a way of thinking about them using one side of the square as a reference.

But that's not what I was thinking about specifically.  What I was thinking about is the density of s-chords.  Would you expect to see them clustered around the edges of the square (or sides), or at the corners (or vertices), and more sparsely represented in the middle of the square?

---

Why did I call your argument "circular"?  I was going to weave that into my response above, but there was no obvious or natural place for it.  So my explanation will have to stand on its own.

My argument is that a circle/disc doesn't exist outside of the plane on which it rests.  This in turn means that the "net" of mathematical lines of which the plane (conceptually) consists cannot be naturally separated from the disc, and hence from the chords (each of which is the intersection of a line and a disc).  If you ignore this, and consider chords as merely what you get if you connect two points on the circumference of a circle, a circle which (in my view) is strangely divorced from mathematical reality, then sure, you can float this circle around, squeeze or stretch it and spin it around.  With such an independent circle there is no need for invariance of any kind (translational, scalar or rotational) - because the circle is the circle is the circle.  If this is not circular …

(Please do note that this is, at least in part, a joke - a poor excuse for a pun.  My more serious efforts lie above this section.)

Rectangular Circles - Yet Another Response to Mathematician

This is yet another response to Mathematician.  Here's what he wrote, interspersed with my responses (minor editing for format, and I've excised the final part that I've already addressed back in the comments section which can be read in its entirety here):

> At N=100, the 1/2 method does not have gaps or clumping.

The whole point of my "subintervals in intervals" example, was to show you that the problem of "gaps or clumping" is only a problem if you think it is. In the interval example, if you require that there is no "gaps or clumping" when N is finite, then the only possible answer is 0.

I think we agree that this is not reasonable at all. And that the most obvious way to select a subinterval will give gaps and clumping.

So how do you choose A PRIORI, in which contexts "gaps or clumping" are problematic, and in which contexts they are not?

It seems to me that you are blending the discussion about the disc and the discussion about the interval.

The "1/2 method" that I am talking about refers to the disc (and only the disc).  I am pretty certain that you are clear on this, but I want to be as totally certain as I can be.

I'm not as convinced as you are that the only possible proportion of subintervals greater than L/2 is zero when using a method that eliminates gaps and clumping.  What I am pretty sure of, however, is that if we looked at the distribution of subintervals and found that they were clustered around the ends of the interval and their lengths clustered around what could be described as "very very short" (much less than L/2), then we'd have reason to doubt how fair this distribution was.

Perhaps there's a good mathematical reason to not care about such clumping (and the implied "gap" between the ends of the interval in which the density of subintervals would dip), but don't you agree that using such a distribution would not meet the general understanding of "at random" - perhaps not even your own understanding of what "at random" would mean in this context?

> if distribution continued towards an as yet unknown value, or whether it still approaches zero

As far as I understand what you are trying to do, I'm pretty sure that it will approach 0.

I'm not sure that what you are doing proves anything at all, but that's another problem.

I wasn't trying to prove anything.  I was just pondering the puzzle that you presented.

> Perhaps it was not clear to you, but the "corrected" 1/3 method, ends up being the 1/2 method

No it was clear.

> And, no, I don't agree that it's the same thing

Let me repeat something for sake of clarity:

For any (c,θ) in [-1,1]x[0,pi], there exists a unique chord that is at distance c from the center, in direction θ.

The "1/2 method of selecting a chord", amounts to pick a couple (c,θ) uniformly in the rectangle [-1,1]x[0,pi]. Do we agree on that?

When you draw your picture to show "granularity", what you are doing is that you choose a θ, once and for all, and then you take 100 values of c that are evenly spaced in [-1,1].

What I'm suggesting is that you do the opposite: Choose a c, once and for all, and then take 100 values of θ that are evenly spaced in [0,pi].

In the end, this is exactly the same method, but you're not drawing the same picture. (In mathematical terms, you are just doing a projection on one of the coordinates)

This is possibly where the meat of the issue is.

I agree that for any (c,θ) in [-1,1]x[0,π], there exists a unique chord that is at distance c from the centre, in direction θ.  To be absolutely clear, I am interpreting this to mean that you are talking about a chord that is offset from the locus by c at its midpoint and that, therefore, the direction mentioned is the direction from the locus to that midpoint.

This is not what I thought you meant before.  I thought you meant to pick a point at c from the locus (direction irrelevant), and then consider the chords that pass through that point with gradients defined by θ.  You'd agree that such a scheme, picking a single value of c, won't give you ALL the chords (certainly not if you pick any value of c less than R, being the radius of the disc), right?

However, you seem to misunderstand my intention.  I made clear (somewhere, I can dig it up if you insist) that I was notionally selecting a single value of θ (direction from locus to midpoint) only because that single value can represent all possible values of θ.  The same applies when selecting a single Point 1 on the circumference in the 1/3 method.

I fully expect that, to get the ALL the chords, you’d have to consider all possible values of θ - in no way was I suggesting that I should "choose a value of θ, once and for all".

So, I understand that if someone foolishly suggested that we select a value of c, "once and for all", and then look at the chords at c given all possible values of θ (as a direction from the locus to the midpoint of a chord), then you'll never get ALL chords.  You'll get an infinite number of chords with precisely the same length but different gradients.

Perhaps I am still misunderstanding your point.  I think I must be, because I do not believe that you are this foolish (insert smile here to minimise any unintended offense).

I want to step back a bit to your question:

The "1/2 method of selecting a chord", amounts to pick a couple (c,θ) uniformly in the rectangle [-1,1]x[0,pi]. Do we agree on that?

I agree, with a minor reservation.  I'm a bit uncomfortable calling [-1,1]x[0,π] a "rectangle": that space represents a circle (hence my little joke in the title of this article).  However, I think I get what you mean - it's a useful way to visualise things for the purpose of considering a uniform distribution of values of c and θ.

What occurs to me is that this can be used in association with the 1/4 method.

My "fix" involved selecting a midpoint from this space (precisely like you seem to be suggesting), while the standard 1/4 method involves selecting from a reduced space.  I think it might be, notionally, a bit like this (think density rather than direct correspondence with values of θ):

I'm not suggesting that these are accurate representations of the shapes corresponding to the 1/3 and 1/4 methods, I just used a triangle for 1/3 and cut out circular chunks for 1/4 because it was convenient.  However, the concept does point towards the notion that the 1/2 and 1/4 methods are missing chords - and where they are missing from.

> because I am not focussed on how we select chords, I am focussed on ensuring that we have ALL chords (and where N is less than infinity, a representative sample of ALL chords).

Can you provide a single example of a chord that you can get with the 1/2 method, but that you cannot get with the 1/3 method?

See above.  Of course I can't point to a single example, which you would clearly realise, but I can (at least conceptually) show that there are fewer chords near the locus with the standard 1/3 and 1/4 methods than there are with the 1/2 method.

> Between -R and -R/2 and R/2 and R, there will be a decrease in the proportion of chords greater than sqrt(3)

You are apparently thinking that "c" should be taken in a predefined direction, and then choose another direction θ. It's not what I said. Just fix some c, once and for all, and then choose a bunch of θ, and then draw the chords corresponding to (c,θ).

So, when c is between, R/2 and R (and between -R and -R/2), the proportion of chords greater than sqrt(3) is 0. So the final answer is 1/2. (Which is absolutely not surprising because it's exactly the same method)

See above.  I think I've already addressed your "once and for all" objection, perhaps once and for all (but I am not holding my breath).

I don't know how you end up with 1/2 with what you've said here, but I do agree that all my methods - the standard 1/2 method, the "corrected" 1/3 method and the " corrected " 1/4 method - are effectively (and exactly) the same method.

---

I note that there might have been confusion about my use of the word "fixed" when I mean "corrected".  When I used "fixed" previously, I did not mean "never to be changed" as in "fixed in stone".  I meant "fixed" as in "my keyboard is broken, I am going to have to get it either fixed or replaced".

Mea Culpa - Another Response to Mathematician

When responding to Mathematician in Triangular Circles (a little play on words, I know circles can't actually be triangular), I wrote this:

On parameterisation, I did some thinking about this along the lines of saying that if you have a 1/3 answer, then it seems (to me) that your selection method must simply have missed some of the chords.  In my way of thinking (standard caveat about the possibility of being wrong), if we are asked to select a chord "at random" then it follows that we would be selecting from a set of ALL chords, rather than from a specific subset, unless advised otherwise.  Thought from this perspective, our first concern is making sure that we have ALL chords available to select from.  The question then is how to express this properly.  I'm probably going to mess this up in some obscure way, but if you can at least try to understand what I am saying (and criticise the best formulation of my argument, rather than the worst), it would be appreciated.

I suggest that an expression for ALL chords in a circle defined by x²+y²=1 (in units of R where R is the radius of the circle) goes something like this:

The infinite set S of all unique sets S_i of points that fulfil the following criteria:

S:

-1 > c > 1 (defining the y axis intercept of the chord)

0 > θ > 2π (defining the gradient of the chord)

         S_i:

-√((-cosθ)²+(c-sinθ)²) > r > √((cosθ)²+(c+sinθ)²)

(x,y) = (r.cosθ,r.sinθ+c)

Note: the combined effect of these two conditions is (or is intended) to include all and only points between intercepts of the line defined by (x,y) = (r.cosθ,r.sinθ+c) and the circle defined by x² + y² = 1, thus defining a chord.  In other words a unique set S_i is intended to define a unique chord.

When corrected in terms of mathematical terminology, etc, is this a parameterisation and, if so, does it establish or define a structure (per u/Vietoris) for which there is a defined probability measure (per u/Vietoris) or probability distribution (per u/overconvergent)?  And, if so, what Bertrand Paradox related answer would be expected from this parameterisation and associated probability measure/distribution?

---

Well, I was certainly right.  I did mess it up.

First, I've doubled the number of chords by using the intervals c:[-1,1] and θ:[0,2π] (hopefully this terminology is clear, it's slightly more convenient than using the -1 > c > 1 and 0 > θ > 2π structure.

I should have used either c:[0,1] and θ:[0,2π] OR c:[-1,1] and θ:[0,π].  Mathematician, in his response, went with the latter, so I'll use that to explain the second, more egregious stuff up.

Note that I said that "our first concern is making sure that we have ALL chords available to select from".  The whole purpose my sets was to achieve this and they don't.

For any value of θ<>0 (assuming that θ=0 is standard and aligns with the positive x-axis, the notional horizontal axis and that c is the point at which the resultant chord intersects the y-axis or notional vertical axis (I did use the word "intercept" before, which is apparently right in some cases but there may be some subtlety that I am missing - or perhaps my wording was just clumsy), there are chords are missed in my schema.

We've agreed that the interval [-1,1] is (or can be) uniform, so imagine 11 equally spaced points on the y-axis in that range and say we look at θ=π/4:

This cannot produce the set of ALL chords.  Additionally, there is a "skewing" of chords towards those that are longer, so it should come as no surprise that (as Mathematician intuited) there would be substantially more chords of length greater than √3.R.  For this reason, I don't think the following comment was nearly as silly as Mathematician later thought it was:

Ok, actually I'm not sure that I am computing the correct probability here. Tell me if this is your idea :

First you pick a number between -1 and 1, uniformly on the interval [-1,1]. And then you pick an angle between 0 and pi, uniformly on the interval [0,pi]. The chord corresponding to the couple (r,θ) is the unique chord that has slope θ and that cuts the horizontal axis at r. Is that okay ?

So this defines a probability on the set of chords. And with this, the probability that a random chord is longer than sqrt(3) is given by the following formula :

P= 1/3 + ln(7+4*sqrt(3))/2pi = 0.7525...

I might be wrong here, but it seems reasonable.

To Mathematician, in answer to the embedded question " … Is that okay?"  Yes, I am reasonably happy with that, once I get over my confusion about the use of r (which I normally think of as the length of a vector from (0,0) to some other point).  If forced to pick something similar, I'd have gone with (c,θ) since we already have c defined - this would be the unique chord with slope θ that is offset from the x-axis by c when x=0.  But I get what you mean,

I'll try to define a set of ALL chords again (this requires more than just a minor shuffle, I suspect).

An expression for ALL chords on a disc defined by x²+y²=1 (in units of R where R is the radius of the disc) goes something like this:

The infinite set S of all unique sets S_i of all unique sets S_j of points that fulfil the following criteria:

S:

0 > θ > π (defining the gradient of the chord)

locus defined as (0,0)

S_i:

-1/cosθ > c > 1/cosθ (defining the y-intercept of the chord)

S_j:

-√((-cosθ)2+(c-sinθ)2) > r > √((cosθ)2+(c+sinθ)2)

(x,y) = (r.cosθ,r.sinθ+c)

Note: the combined effect of these two conditions is (or is intended) to include all and only points on the intersection of the lines (x,y) = (r.cosθ,r.sinθ+c) and the disc defined by x²+y²=1, thus defining a chord.

Defining the locus as (0,0) removes some complications to the equations that would otherwise be required to achieve invariance in terms of translation (by which I mean movement of the circle to another location).  Setting the radius of the disc to R and making R the units of length in all considerations addresses the question of invariance in terms of scale.  Defining the set of y-intercepts such that c:[-1/cosθ,1/cosθ] goes only part of the way to addressing invariance in terms of rotation.

If we revisit the image above but extend out the range of c, we will get:

The gap has gone, but we've now got more chords at θ=π/2 than we had at θ=0, so we no longer have rotational invariance.  To get it back, we need to introduce a concept that probably has another proper term to it, but I call "granularity".

Say we select an arbitrarily large number (N+1) of evenly spaced samples over the interval from which we take c.  If c:[-1,1] because θ=0, then there would be N/2 samples above the locus and N/2 below the locus and one on the locus.  If we generalise this, for c:[-c_i,c_i], then there would be still N/2 samples above the locus and N/2 below the locus, but with a different separation - rather than the samples being 2/N apart, they would be 2/N/c_i apart.  I refer to this figure, 2/N/c_i, as the "granularity".

In order to maintain invariance in terms of rotation, we need to set the granularity of the sets to 2.cosθ/N with N->∞.  If there is a better way to word this, please let me know.

If there is an iron-clad rule that says that I cannot parameterise my chord selection with anything akin to this concept of granularity, then I guess I have to graciously concede defeat, albeit with the residue of the itchy feeling that maths shouldn't be like this.  But if it is possible, without necessarily being conventional, then I think my selection of chords makes sense, is invariant in terms of scale, translation and rotation and results in the 1/2 answer.  And while it does not seem quite as elegant as my first (incorrect) version, it is more general and I don’t know that an attempt to do something similar with the 1/3 and 1/4 methods can be done as elegantly.  Perhaps it can be done, perhaps there are even more elegant ways to do it, I'm in absolutely no way certain of this.