Quote: Original post by JoeCooper
Is any of this relevant or am I barking up the wrong tree?

Actually, I'm now leaning towards an explanation based on many of the ideas you've put out there (the idea of one "approximation" being "better" than another, the idea of the "approximation" "doing something" to improve during convergence, etc), but I feel it must be set upon a firm and concise axiomatic base.

Quote: Original post by JoeCooper
Can we assume that x^2+y^2=r^2 is a valid test for whether or not a vertex lies on a circle? Is that a given?

This seems like a reasonable start. More concisely, I'd like to define a circle in the following manner.

First of all, I'd like to establish a "context", so to speak (I cannot for the life of me recall a better term, so bear with), from which to establish the idea of a circle. I'm going to work from the set of ordered pairs of real numbers, in other words ; but I'm also going to need a bit of extra structure, specifically a metric, which is a function which assigns a notion of distance to all points in it's domain. In particular, we are obviously interested in the Euclidean metric, which I'll define here:



Another example of a metric, which I mentioned earlier, and would like to return to later to observe some interesting points, is the Manhattan metric:



This metric is obviously based on the idea of movement constricted to "orthogonal" directions. A Manhattan distance between two points is the sum of the "vertical" distance and the "horizontal" distance between two points.

Anyway, back on course, putting together the set of two-tuples of real numbers, with the Euclidean metric, gives us a metric space,



ie. a two-dimensional Euclidean metric space.

From here, I propose the following definition: a circle of radius r around a central point p, is the set given by



In English, the set of points in two-dimensional Euclidean space which are exactly a distance of r away from a central point p. Is this definition acceptable? Based on what little I can gather from the Wikipedia definition, I feel relatively justified in the use of this definition.

Also, there are some interesting images of unit circles which can be arrived at by using a different metrics here.

Now, I'm not wholly sure how to formalize a notion of "circumference", but I'll give this a stab anyway. Firstly, I'm interested in the objects "enclosed" by a circle, and I'll specify them as follows. I define the line segment between two points:



...er, I'm assuming vector space axioms here. Hopefully the appropriate notions of addition and multiplication are clear, so can we please let this one slide? :P

Now, I say that a point is "enclosed" in a circle if the line segment between it and the circle's centre doesn't contain any points in the circle. Formally, the set of all points "enclosed" by a circle:



Hopefully it is clear that this is equivalent to the "open ball" of radius r around p:



In this context, the open ball is in essence an "open disc".

Now, since I'm not really familiar with any standard formal definitions of a perimeter or circumference, this bit's going to get really sketchy. I'm going to define a "circumnavigating sequence" of an open ball, as a sequence of points of length n,



and the "circumnavigating path" as the set,



I require that every true circumnavigating path have the following property:



ie. the "path" traced out by following the sequence does not "enter" the open ball enclosed by the circle. Observe that the path may touch the circle itself. The next bit's pretty ugly: I need a notion of "reachability" of a point from another, which I'm going to give roughly as



which basically means that P(p, q, S) for any two points p and q, and any set S, is the condition that there exists a path between p and q that doesn't coincide with the set S at all. I use this to give the second property I require of any circumnavigatory sequence:



where p is the centre of our circle. This is to say that, together with the first property I stated, X bounds a superset of the open ball Br(p). It's a pretty sloppy way of specifying it, you could specify a sequence that overlaps itself and does all kinds of crazy crap yet still bounds the ball; but hopefully it will suffice anyway.

Obviously, the length of the path given by X is



Now I can finally get to what I've been angling for!
Obviously, there is a "minimum circumnavigatory sequence" of size n, which I'll formalize like so:



That is to say, that Min Xn is the minimal possble circumnavigatory distance specifiable in n points. I suspect that the regular polygons of Archimedes' method of exhaustion may actually correspond to these vertices. Nevertheless, I'm actually going to offer



as the circumference of the circle .

So, my conclusion is that the troll has defined a path around the circle, but that it is not minimal under the Euclidean metric, and hence not the circumference.


Hmm, at about this time I'm thinking that I've probably gone to too much effort towards the end of this construction to reach what sounds like a trivial conclusion in my head. The troll is concerned with enclosing the circle in a shape and gauging the perimeter of that shape a it contorts to fit the edge of the circle, and I think I've formalized an understanding of the meaningfulness of such a construction to circumference; but the same point can probably be made reasonably concisely without all this Hilbertesque clutter. >_>

Nevertheless, I think the idea that the convergence need be designed to minimise perimeter in order to home in on the circumference, rather than merely fit points to the surface, has become very, very obvious. That is to say, of the set of enclosing shapes of n vertices, some have smaller perimeters than others, and the one with the smallest perimeter is the "most correct", or closest approximation.

Quote: Actually, I'm now leaning towards an explanation based on many of the ideas you've put out there

Well that made my day!

Then everything after made my head explode.

Thanks for the write-up.

I can't add anything so I better bail from this thread.

Quote: Original post by JoeCooper

Quote: But why does this metric mean anything?

It means something because knowing that these extreme-hypotenuse-vertices always alternate with the valid vertices, this metric shows whether or not the shape smooths into a circle as n approaches infinity.

Since at the end we take the sum of the distances from each vertex to the next, I take the sum of the deviation.

Maybe I misunderstood the metric you proposed. What's the "deviation"? Is it the (perpendicular) distance from the circle to the point, or the distance from the point of contact to the point of furthest deviation? I was assuming the former (and will for the rest of this post). However, if it's the latter, then I don't think it tells us, at best, anything more than we already knew: the approximation does not decrease. In particular, it doesn't tell us whether or not this is value is correct.

Quote:

Quote: The same n isn't really comparable between the two, but the way they behave as n increases is comparable.

The comparison doesn't really matter. In fact, nevermind the regular polygon method. It doesn't matter how they compare; only whether or not the procedure in question does as it says on the tin.

It matters because it gives us insight into why the value isn't decreasing. What it shows is that the proposed metric depends just as much on the number of worst points as it does on how those points converge. The thing is, there's no reason the number of worst points should matter at all.

Also, it's useful to consider the regular polygon method since any proposed metric should reject the excavation method but not reject the regular polygon method (or the three methods I suggested in my previous post).

Quote:
Since we know there are valid points, and between two valid points are extreme-hypotenuse-points, we can assess if these extremes go away as n approaches infinity.

Since the value goes up instead of down, we see that it does not, thus failing at its own goal, failing to approximate a circle' shape and therefore giving us no reason to accept its perimeter as an approximation of a circle's.

But I gave three examples that fail the given metric but still approximate the circle's shape in the sense that the limiting curve is a circle and the limiting perimeter is the circumference of that circle. One has no smoothing out to do since it's already a circle. I gave two examples that have the same performance as the excavation method but produce the correct perimeter. So, the metric fails at its own goal: rejecting models that don't produce the correct perimeter.

Quote:
The fact that its area does is interesting, but that doesn't mean anything to its perimeter. Both the perimeter and area stem from the geometry. Therefore we should not consider its area.

The reason I keep mentioning the enclosed area is because it's what you get when you iron out some of the issues with the proposed metric.

Yeah, I don't know what I'm doing.

As I said,

Quote: everything after made my head explode ... I can't add anything so I better bail

I'll write a little more just to answer, but I'm exhausted now, let's stop firing please.

Quote: What's the "deviation"? Is it the (perpendicular) distance from the circle to the point

Yessir; for a given vertex, d = r^2 - x^2 + y^2

Quote: the proposed metric depends just as much on the number of worst points as it does on how those points converge. The thing is, there's no reason the number of worst points should matter at all

It intentionally depended on the number and was intended to reflect the sum of the deviation between the valid points and extreme points.

I wanted the sum because the perimeter (what we're measuring at the end) is the sum of the distance from each vertex to the next...

That makes more problems though. For example:

Quote: take the circle circumscribing the regular n-gon. The metric is always infinite but, again, it produces the correct result.

Wrong; it's actually worse. The above deviation gives 0 for any individual vertex (ideal!), but since there's an infinite number of them, the sum of them would be 0 * infinity, which is undefined, so the test cannot be conducted.

Just thinking about predicting whether or not a shape's perimeter can be accepted as an approximation of a circle's.

Obviously my math skills are far short of letting me accomplish that.

[Edited by - JoeCooper on December 4, 2010 3:15:14 PM]

Quote: Original post by JoeCooper

Quote: everything after made my head explode ... I can't add anything so I better bail

I'll write a little more just to answer, but I'm exhausted now, let's stop firing please.

Sorry for playing the devil's advocate.

I stopped halfway through Fenrisulvur's dense math, too.

Fenrisulvur: For someone who accuses others of writing like philosophy students, your math looks like a philospher's or a new student's. Reading it is slow going because you spend so much time defining common terms and notation (you even got fed up with it and just said you would be using "vector space axioms", which usually goes without saying). It would also help if you didn't use mathematical notation to cram so much on a single line. It's like complaints that Perl can look like line noise. (I often get complaints about being too heavy on the math, so this is a bit of the pot calling the kettle black. [smile])

Quote:

Quote: What's the "deviation"? Is it the (perpendicular) distance from the circle to the point

Yessir; for a given vertex, d = r^2 - x^2 + y^2

I was thinking d = |r - sqrt(x2 + y2)|, but it doesn't really affect anything I said.

Quote:

Quote: take the circle circumscribing the regular n-gon. The metric is always infinite but, again, it produces the correct result.

Wrong; it's actually worse. The above deviation gives 0 for any individual vertex (ideal!), but since there's an infinite number of them, the sum of them would be 0 * infinity, which is undefined, so the test cannot be conducted.

I think this is the upper vs. lower bound thing again. I was going by the upper bound (i.e. the n-gon circumscribes the circle) so the deviation would be a positive number. For the lower bound, just inscribe the circle within the n-gon instead of circumscribing the n-gon.

This is driving me crazy. I'll just wait till I have enough time to try to formulate some semi-rigorous argument.

Quote: Original post by Way Walker
Fenrisulvur: For someone who accuses others of writing like philosophy students, your math looks like a philospher's or a new student's. Reading it is slow going because you spend so much time defining common terms and notation

Eh, I already more-or-less conceded that I was stumbling around in the dark for the latter half. A lot of these concepts are pretty new for me.

I'm not exactly sure which terms and notation you're taking as "common", here, or in what sense. Are you rejecting structure like metric spaces and definitions like that of the circle I gave in favour of some unstated common-sense interpretation, or are you familiar with such structural abstractions and objecting to having it all rehashed?

I maintain that an answer to the troll's paradox is going to have to be structural, it's going to have to illuminate criteria which govern whether a sequence of bounding shapes do or do not coverge to give the circumference of the circle, and we're probably going to need a formal definition of what a circumference actually is.

Quote: Original post by Way Walker
(you even got fed up with it and just said you would be using "vector space axioms", which usually goes without saying)

Oh, no it doesn't. That line would've been torn apart in a more formal setting, I hadn't defined any structure other than a metric at that point. Do you have any idea how many vector spaces can be constructed on R*R?

Quote: Original post by Way Walker
It would also help if you didn't use mathematical notation to cram so much on a single line. It's like complaints that Perl can look like line noise. (I often get complaints about being too heavy on the math, so this is a bit of the pot calling the kettle black. [smile])

Haha, point taken.

Quote: Original post by FenrisulvurDo you have any idea how many vector spaces can be constructed on R*R?

One--the two-dimensional real vector space or the one-dimensional complex space (i.e., the complex numbers C), depending on how you want to interpret it. A bijective linear map between bases extends to a bijective linear map between the entire spaces by linearity.

Quote: I was thinking d = |r - sqrt(x2 + y2)|, but it doesn't really affect anything I said.

Which is why I said "yessir"!

Quote: I stopped halfway through Fenrisulvur's dense math, too.

I don't think we could possibly have a procedure that can validate any and all possible shapes you could throw at it, for all manner of procedures, with my philosophy-student level math.

(Though I did get that those may be fightin' words on a programmer's forum, it might as well be true; the last four courses I took involved speech communications, literary non-fiction, literature including poetry, and geology for folks who aren't studying to be geologists.)

As I understood the specifications originally, all I felt I needed to do was to show that this particular gizmo isn't approximating a circle.

Dealing with infinite sets of vertices and arbitrary methods of assessing the shape in question is something I don't have the intellectual toolkit for.

I don't mean this to sound hostile in any way, but you seem confident that you have a superior handle on the maths to everyone else - why not pitch a fitness test?

Quote: Original post by nilkn

Quote: Original post by szecs
I'm pretty sure it should be easy to explain with mathematics (convergence, "lim" stuff and formulas), but I want to be convinced with simple GEOMETRY. Just like Archimedes would. So that I can IMAGINE it.

I feel it has to so something with fractal geometry, but this thinking ("we can construct an object that looks like a line blah blah") seems to be too "reverse thinking" to me. I don't feel that warm "I'm convinced" feeling.

But maybe I'm trolling myself.

It seems to me the most evident geometric reason is that the tangents to the limit curve--the circle--exhibit all possible directions while the tangents to the zig-zag curves are always either horizontal or vertical. This is the basic geometric issue which prevents the limit machinery from working out as you would expect.

I pointed it out earlier in this thread, but since you mentioned Archimedes I'll say it again. The issue I just described does not occur when you use circumscribed n-gons like Archimedes did.

In the present example, we have uniform convergence of curves to a limit without convergence of the derivatives. In Archimedes' examples, we have both convergence of curves to a limit and convergence of the derivatives.

'the derivatives'? Only the zero'th and first, actually. That still leaves the nagging question as to why it is only those two that matter.