# Pi = 4. Discuss.

This topic is 2134 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Quote:
Original post by BitMaster
Quote:
 Original post by HodgmanI'm assuming that "1/inf" is the smallest value that is still greater than zero.

Such a value does not exist. Whenever you believe you have found such a value x, then x/2 is smaller, positive and not zero either (this works regardless of whether we are working with real or rational numbers).
In actual textbook math (not school-math, university-level math) there are preciously few points were an "infinity" can be. There are some more specialized branches of mathematics which allow actually using infinity as a more regular symbol, but even there you have to be extremely careful what you are allowed to do with it. There is a Wikipedia article about that.
The rest of the post kinda dies with that.

Maybe I should not have skimmed the thread like that but so far it seems arguments are on the "drawing sketches, waving hands"-level largely.

The only proper definition for the convergence of a sequence S(n) to the limit C I know is
$\forall \epsilon > 0: \exists n_0 \in \mathbb N: \forall n > n_0: |S(n) - C| < \epsilon$
and I don't think I have seen anyone trying to apply this to the problem. That would of course first require us to put the happy sketch into an actual formula you can work with but I feel a bit lazy today.
My personal guess is the result will be that the happy sketch idea simply does not converge because for every finite value the circumference is 4 while the limit would have to be pi. So even for $\epsilon = \frac 1 2$ it's not possible to find an $n_0$ as required.

Exactly, you can't define an infinitesimal as the smallest number that is still greater than zero, because you can always generate a real number between that number and zero, and so on. The set (0,n) is open on the lower bound. It makes no sense to ask what the 'first' element of the set is.

You can however define different types of infinities. You can construct ordinal numbers by defining them in terms of sets. We start with defining zero as the empty set. Where the number x is the set that contains all subsets corresponding to the numbers less then x.Then you can define omega as the set that contains all finite ordinals (It's bigger than any finite ordinal number). However now you are able to construct omega+1 which is bigger than that all the way to 2*omega ... omega^2 ...

Similarly you can define an infinitesimal as 1/omega. Its larger than zero but smaller than any 1/n where n is a finite ordinal. But now you can construct 1/omega+1 ... 1/2*omega ... 1/omega^2 ...

This type of infinity is different than a cardinal infinity, such as the 'amount' of things we have in a set. Aleph Null is defined to be the size of a countable set(Natural numbers, Integers, Rationals) The infinity of the continuum (The real numbers) is uncountable and larger than Alpha Null (The continuum hypothesis says that it is Aleph One) You can always create a larger cardinal by taking the power set of a set (The set that contains all of the subsets of the specified set)

Things then can get really weird from here on out as you can define 'large cardinals' that are inaccessible by a succession of Alephs or even a limit of Alephs. Then you can create hyper-inaccessible cardinals, then hyper-hyper-inaccessible cardinals ...

Sorry I get carried away.

##### Share on other sites
Quote:
Original post by Talroth
Quote:
 Original post by Platinum314We created a shape with the same area as a circle, but not a circle.

But, you didn't do that. You always have a saw tooth edge that comes near the circle, but must always include an area that exists outside the boundary of the circle. It therefore must have an area Larger than the circle itself, not equal to.

The problem is, as soon as limits are involved an argumentation like that just does not work anymore. ;) Just take a look at a simple sequence like S(n) = 1/n. Whichever n you put in, it's always bigger than 0. But the sequence is the example of a sequence converging to 0.

##### Share on other sites
Quote:
 Original post by JoeCooperIf so, in a regular polygon (the method this spoofs that - I think - we're trying to differentiate it from), every specified vertex lies on a circle.

Actually, that would produce a lower bound. The upper bound is found by placing the midpoints of the edges, not the vertices, of a regular polygon on the circle. I think your analysis wouldn't be changed too much, but I wouldn't be surprised if the equivalent lower bound "spoof" would, like the Koch snowflake, have an infinte perimeter (so infinty < pi < 4 [smile]).

Quote:
 Thus we can quantify that squares are an inferior approximation of a circle compared to a regular polygon.

It's not so simple as showing that it is inferior. What we need is to show that it's inferior in a meaningful way. You can approximate the volume of a cone as a stack of cylinders or as a stack of truncated cones. You can quantify how much better the latter is, but both will produce the same result in the limit as the number of sections in the stack approaches infinity.

One thing that bothers me about this analysis is that it singles out the worst points. If we just look at the best points, the number of points with zero deviation increases linearly for the regular polygons but exponentially for the square excavation. By this metric, the square excavation is quantifiably better.

Perhaps the area enclosed between the two would be a better metric (since it's the sum total deviation for all points normalized by the arc length), but that would seem to be a better measure of how quickly the approximations converge and not how well they converge.

As I said before, I think the best criterion will be related to the convergence of the tangents. See nilkn's post for a good explanation of that issue.

Quote:
 Original post by AntheusI'd say that the problem here comes from implication that value of Pi follows directly from perimeter, whereas even when using the curve it must be derived from area.

It's maybe more robust to derive it from the area, but the most basic definition of pi is a relation between the perimeter and diameter.

##### Share on other sites
Quote:
 Original post by Way Walkerbut the most basic definition of pi is a relation between the perimeter and diameter.

The geometric definition is either that of circumference or that of area.

Quote:
 It's maybe more robust to derive it from the area,
Except that in this case the curve does not define Pi, if my assumption about difference between perimeter of curve and area holds.

##### Share on other sites
the problem really boils down to the square/polygon being turned into a concave polygon rather than a convex polygon. Geometrically a lot of things no longer apply that make the op wrong.

##### Share on other sites
Quote:
Original post by BitMaster
Quote:
Original post by Talroth
Quote:
 Original post by Platinum314We created a shape with the same area as a circle, but not a circle.

But, you didn't do that. You always have a saw tooth edge that comes near the circle, but must always include an area that exists outside the boundary of the circle. It therefore must have an area Larger than the circle itself, not equal to.

The problem is, as soon as limits are involved an argumentation like that just does not work anymore. ;) Just take a look at a simple sequence like S(n) = 1/n. Whichever n you put in, it's always bigger than 0. But the sequence is the example of a sequence converging to 0.

Like you said, 1/n, where n is any positive value, never actually equals zero. The area enclosed by the example shown in szecs's first post is that of one that neither equals the area of the circle, nor is it one who's shape is similar to that of a true circle.

I don't know what kind of kindergarten you went to, but back in my day circles didn't have a series of right angles on them.

##### Share on other sites
Quote:
 Actually, that would produce a lower bound. The upper bound is found by placing the midpoints of the edges, not the vertices, of a regular polygon on the circle.

Err, right. Still, the analysis should yield the same result either way.

Quote:
 One thing that bothers me about this analysis is that it singles out the worst points. If we just look at the best points, the number of points with zero deviation increases linearly for the regular polygons but exponentially for the square excavation.

That bothered me too at first and I actually thought about it for a while before going ahead and posting.

Lemme try to explain why I discounted it, but I'm not very articulate so I apologize in advance.

First, singling out the worst points. I don't necessarily do that, actually. If I add in the valid vertices' deviation, all I do in either analysis is add 0 (and half it, I think. I did this earlier and now I can't remember exactly.)

A focus on the bad points is a natural outgrowth of the fact that they're the only ones that differ.

Secondly, the iteration # means nothing outside of a given procedure.

We're only looking to see if a given procedure converges on 0; where they're at when n=2 isn't important. What's important is what happens when we go from n=2 to n=3 and onward toward infinity.

Therefore, the fact that the rate at which a count of valid points increases with n isn't important.

Third, it's also counterbalanced by the fact that in the square'd circle, there is an exactly equal exponential increase in bad points. In fact, that is how the sum of the deviation increases even though the per vertex deviation increases.

Quote:
 Perhaps the area enclosed between the two would be a better metric

I disagree; the area doesn't predict the shape's perimeter.

Measuring the perimeter is an integral part of the procedure.

We can't divorce the way we plan to use the shape from how we assess its fitness for purpose.

If the procedure were based on using its area and diameter to find Pi, instead of its perimeter and diameter, than the fact that its area does approximate a circle of the given diameter suggests it is a valid approach.

<EDIT>It just occurs to me I might've misunderstood that, but uhh, I'll just let you reply.</EDIT>

Quote:
 It's not so simple as showing that it is inferior. What we need is to show that it's inferior in a meaningful way.

Let me try this differently then.

Nevermind relative descriptions like inferior to.

To assess its fitness for purpose, we must show that it approximates a circle. We must do this because we're trying to find Pi, which is defined as "ratio of the circumference of a circle to its diameter". In other words, we are trying to find a property of a circle. Therefore, we try to approximate a circle, then accept that approximation's circumference as an approximation of a circle's circumference.

Key point; we can assess if the procedure is actually measuring a shape that is an actual approximation of a circle.

This procedure iterates over a square, cutting parts away, so that the vertices are closer and closer to a circle.

My analysis is to examine their hypotenuse, and the deviation from a circle, and see if it actually converges on 0 or not.

It does not, so, it is not an approximation of a circle, and that makes it inappropriate the only meaningful way; if it isn't a circle, than any resemblance between its properties and those of a circle is purely coincidental.

[Edited by - JoeCooper on December 3, 2010 5:49:36 PM]

##### Share on other sites
Quote:
 Original post by JoeCooperFirst, singling out the worst points. I don't necessarily do that, actually. If I add in the valid vertices' deviation, all I do in either analysis is add 0 (and half it, I think. I did this earlier and now I can't remember exactly.)

It's easy enough to create a construction that exploits the fact that it singles out the worst points.

For example, let f(n) be this metric for the nth iteration of the square excavation. Now, create a star whose inner vertices are along the limiting circle (radius rinfinity), outer vertices along the circle running through the vertices of the circumscribing regular n-gon (radius rn), and ceil(f(n)/(rn - rinfinty)). This creates a concave figure where the given metric will always be greater than or equal to f(n) but produces the correct result. Modifying this slightly, you could control the metric to behave like any function that doesn't go to infinity.

You could also create a sequence that is the limiting circle at all points except for the worst points of the excavation sequence. This is a discontinuous figure with metric f(n) that also produces the correct result.

For a third example, take the circle circumscribing the regular n-gon. The metric is always infinite but, again, it produces the correct result.

Quote:
 Secondly, the iteration # means nothing outside of a given procedure.

It's as meaningful as comparing an O(n) algorithm to an O(2n) algorithm. The same n isn't really comparable between the two, but the way they behave as n increases is comparable.

Quote:

Quote:
 Perhaps the area enclosed between the two would be a better metric

I disagree; the area doesn't predict the shape's perimeter.

As I said, in this case, the enclosed area is just the weighted deviation of all points normalized by the arc length. It's a generalization of the metric you proposed that takes care of the fact that it singles out the worst points (by taking the weighted contribution of all points) and the fact that arcs of the worst deviation would make it infinite (by normalizing by arc length).

Quote:

Quote:
 It's not so simple as showing that it is inferior. What we need is to show that it's inferior in a meaningful way.

This procedure iterates over a square, cutting parts away, so that the vertices are closer and closer to a circle.

My analysis is to examine their hypotenuse, and the deviation from a circle, and see if it actually converges on 0 or not.

It does not, so, it is not an approximation of a circle, and that makes it inappropriate the only meaningful way; if it isn't a circle, than any resemblance between its properties and those of a circle is purely coincidental.

But why does this metric mean anything? Another metric is how close the limiting figure is to a circle, which it passes. You could also consider whether the enclosed area goes to zero, which it also passes. I gave three examples of false negatives by the metric you proposed, so it's not a necessary condition. It may be sufficient, but it cannot reject an approximation.

##### Share on other sites
Quote:
Original post by Antheus
Quote:
 Original post by Way Walkerbut the most basic definition of pi is a relation between the perimeter and diameter.

The geometric definition is either that of circumference or that of area.

Quote:
 It's maybe more robust to derive it from the area,
Except that in this case the curve does not define Pi, if my assumption about difference between perimeter of curve and area holds.

I think I maybe misunderstood what you were saying. There's nothing wrong with deriving pi from the perimeter of a circle, the problem comes in assuming that, if the limit of a sequence of curves is a circle, then the limit of the perimeter of that sequence will be the same as the perimeter of the circle. However, you can still (my confusion is perhaps from the word "must"?) derive pi from the limit of the area of that sequence (since the area enclosed between the circle and the curve must go to 0 so the areas will be identical).

##### Share on other sites
Quote:
Original post by Talroth
Quote:
Original post by BitMaster
Quote:
Original post by Talroth
Quote:
 Original post by Platinum314We created a shape with the same area as a circle, but not a circle.

But, you didn't do that. You always have a saw tooth edge that comes near the circle, but must always include an area that exists outside the boundary of the circle. It therefore must have an area Larger than the circle itself, not equal to.

The problem is, as soon as limits are involved an argumentation like that just does not work anymore. ;) Just take a look at a simple sequence like S(n) = 1/n. Whichever n you put in, it's always bigger than 0. But the sequence is the example of a sequence converging to 0.

Like you said, 1/n, where n is any positive value, never actually equals zero. The area enclosed by the example shown in szecs's first post is that of one that neither equals the area of the circle, nor is it one who's shape is similar to that of a true circle.

I don't know what kind of kindergarten you went to, but back in my day circles didn't have a series of right angles on them.

That was exactly my point. As soon as a limit is involved you do not have a guarantee there are still any right angles in there. For example one 'correct' method for calculating pi is fitting an n-gon to the circle (either fitting just inside or just outside) and calculating the area of that n-gon, then letting n go towards infinity. Every single finite n-gon is angular but the limit is a circle.
Similarly, if we pick any single point on the circumference of the happy squares method we can keep exact track of it through the iterations (no problem since the circumference remains constant and 0° would be a fixed start point). Now, watching this single point P(n) through the iterations, it is seems obvious that for a sufficiently high n_0, P(n) with n > n_0 will be as close to the real circle as desired (without ever reaching it in the general case, just like 1/n never reaches 0).
I currently see no flaw in saying that for every single point P(n) on the surface, the limit of that point is exactly on the circle. The error creeps in because we have an infinite number of points on the circumference and I expect the cardinality of infinities (not sure this is the right expression, it's been a long time) on the squares and the circles don't fit. That's why pi using the circumference of the happy squares does not work but should work using the area (again, my guess; proof of that bit to whoever is interested enough).

##### Share on other sites
Just a couple of thoughts:

• Even as we flip the corners of the square for infinite times, the only point that touch the circle are only the tangential points.

• Square is not a continuous function, thus we can't flip its corner to inifinity and believe that it becomes an arc

##### Share on other sites
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.

It basically tells us if the points on the shape are becoming more equidistant from the center as n approaches infinity or not. Regardless of how it compares to any other procedure.

Quote:
 Another metric is how close the limiting figure is to a circle, which it passes.

Quote:
 You could also consider whether the enclosed area goes to zero, which it also passes.

There are many, many metrics (unlimited), and I did consider that one - the area.

Which would suggest that it is an approximation of a circle, and therefore useful to find Pi if you test the area.

And I stand by that; because its area should approach that of a circle as n approaches infinity, its area is an approximation for that of a circle, therefore anything we can infer from that can be accepted.

But...

Quote:
 As I said, in this case, the enclosed area is just the weighted deviation of all points normalized by the arc length.

It's always the case that you can predict the area from the geometry of an object; that the properties of its geometry will be reflected in its area.

But the reverse is not also true.

Area and perimeter both stem from the geometry, but the perimeter doesn't stem from the area. Since our goal is to use the perimeter at the end, we only need to examine what influences it.

Again, we cannot divorce how the figure is used from how we assess its fitness for purpose.

So for a metric to matter, it must make some prediction about the perimeter.

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.

In showing if this one does, it doesn't matter at all what the regular polygon method does at any value of n, nor does it matter whether or not the procedure exists at all!

So let's focus on this one.

In this method, we're trying to approximate a circle for the purpose of measuring its perimeter.

This method suggests that as n approaches infinity, the difference between it and a circle should approach 0, making it an approximation, and we can pick as high a value of n as we need accuracy.

Since the procedure uses the perimeter, to assess it, we need to ignore things that don't contribute to the perimeter and focus on whether or not the shape it uses approximates a circle's perimeter.

It starts with square edges, and appears to smooth out... But does it? We can assess whether or not it really does; we can measure whether it smooths out.

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.

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.

Any better?

[Edited by - JoeCooper on December 4, 2010 5:59:21 AM]

##### Share on other sites
Quote:
 Original post by ConcentrateJust a couple of thoughts: Even as we flip the corners of the square for infinite times, the only point that touch the circle are only the tangential points. Square is not a continuous function, thus we can't flip its corner to inifinity and believe that it becomes an arc

By your first argument, calculating the area of the circle using an n-gon is impossible too (either the edges just touch the circle or the vertices touch the circle). However, that is exactly how it is done.
I don't know why a square is not a continuous function though. You can have a go at it not being differentiable but I still do not see the point.

The thing is, as soon as you have things happening until infinity, a lot of things we believe to be common sense go out of the window. That's why it is so important to stay with the strict mathematical definitions of these processes, otherwise you always end up in hell's kitchen.

Just to make it clear again, I'm not arguing that the happy squares method is correct (it might be for area, but definitely not for circumference). But we are talking about math. The way is the goal. Having the correct solution with the wrong rationale is as bad as having the wrong solution.
I have written down one criteria for limits up there in the thread, I think Wikipedia has a slightly different definition posted (looked similar by using different means, but I haven't looked at it too closely). Go ahead, just don't use handwavium with limits. That never works out well.

##### Share on other sites
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.

##### Share on other sites
Maybe we just need the definition of Pi.

I remain clueless.

##### Share on other sites
Since this thread got started I spent quite a lot time reading on the subject, I even dared to post a completely uninformed opinion in this very same thread.

What I got so far is that the only way to get the famous Pi is to use inner polygons, the area of the circle or even the volume of a sphere. And that all this effort to approach Pi from a quad is useless.

I even remember my dad telling me (when I was just a kid) to take a string and measure a circle to see how much it's length was, and, if I'm not wrong, measuring with physical objects still gives a values closer to 3.14etc than 4.

##### Share on other sites

(I rewrite and rewrite things a lot...)

##### Share on other sites
Quote:
 Original post by szecsI'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.

##### Share on other sites
Quote:
Original post by nilkn
Quote:
 Original post by szecsI'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.

Well, to be honest, I don't see the connection. That's just a basic idea but no explanation. Okay, I should think about it.

##### Share on other sites
Write down the integrals for the lengths in the two cases. On the one hand, you have the length of the limit curve. On the other hand, you have the limit of the lengths of the individual curve. There is in general no reason to suppose these two quantities should coincide. For instance, if you try to go from the second integral to the first, you will need to first pull the limit outside of the derivative operation which appears inside the integrand, but the derivatives, as far as I can tell, of the zig-zag curves don't even have a limit, let alone converge uniformly.

##### Share on other sites
Quote:
 Original post by JoeCooperIs 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 JoeCooperCan 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 $\mathbb{R}^2 \overset{\textup{def}}{=} \mathbb{R}\times \mathbb{R}$; 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:

\begin{align*}d_{E} : \mathbb{R}^2 \times \mathbb{R}^2 &\to \mathbb{R}_{+}\\ \left(p, q \right ) &\mapsto \sqrt{\left(p_1 - q_1 \right )^2 + \left(p_2 - q_2 \right )^2}\end{align*}

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:

\begin{align*}d_{M} : \mathbb{R}^2 \times \mathbb{R}^2 &\to \mathbb{R}_{+}\\ \left(p, q \right ) &\mapsto \left|p_1 - q_1 \right | + \left|p_2 - q_2 \right |\end{align*}

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,

$\left(\mathbb{R}^2,d_{E} \right )$

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

$C_r(p) = \{ q \in \mathbb{R}^2 : d_E(p,q) = r\}.$

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:

$L(p,q) = \{r \in \mathbb{R}^2 : \left(\exists k \in [0,1] \right ) \left(r = p + k(q - p) \right ) \}$

...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:

$B_r(p) = \{ q \in \mathbb{R}^2 : \left(\forall a \in \mathbb{R}^2 \right )\left(a \in L(p,q)\Rightarrow a\notin C_r(p) \right )\}$

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

$B_r(p) = \{ q \in \mathbb{R}^2 : d_e(p,q) < r \}$

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,

$X_n=\left(x_1, x_2, \cdots , x_n \right ),\; x_i\in\mathbb{R}^2 \;\forall\, i\in\{1 .. n\}$

and the "circumnavigating path" as the set,

$\overline{X}_n = \bigcup_{i=1}^{n - 1}L(x_i, x_{i + 1})\, \cup L(x_1, x_n)$

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

$\Phi(X_n) \Leftrightarrow \left(\,\overline{X}_n\cap B_r(p) = \varnothing \right )$

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

$P(p, q, S) \Leftrightarrow \left(\exists(a_1, a_2, \cdots, a_n), a_1 = p, a_n = q, a_i\in\mathbb{R}^2 \right )\left( \bigcup_{i = 1}^{n - 1}L(a_i,a_{i + 1})\: \bigcap S = \varnothing \right )$

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:

$\Theta (X_n) \Leftrightarrow \left(\exists b\in\mathbb{R}_+ \right ) \left(\forall a \in \mathbb{R}^2 \right) \left(P(a, p, \overline{X}_n ) \Rightarrow d_E(a, p) < b\right )$

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

$D_E(X_n) = \sum_{i = 1}^{n - 1}d_E(x_i, x_{i + 1}) + d_E(x_1, x_n).$

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:

$\textup{Min}\,X_n = X_n : \left(\forall A_n = (a_1, a_2, \cdots, a_n)\in \mathbb{R}^{2n}\right)\left(\Phi(A_n)\wedge\Theta(A_n) \Rightarrow D_E(A_n) \ge D_E(X_n) \right )$

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

$\lim_{n \to \infty}D_E(\textup{Min}\,X_n)$

as the circumference of the circle $C_r(p)$.

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.

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

Thanks for the write-up.

##### Share on other sites
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.

##### Share on other sites
Yeah, I don't know what I'm doing.

As I said,

Quote:

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]

##### Share on other sites
Quote:
Original post by JoeCooper
Quote:

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.