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".
