B A Pi Question: Why do we use the awkward approximation 22/7 ?

[Agent Smith]

And some of them always stand inside the curve being approximated.

Yes, but as the NumberOfSides(Polygon) increases, all the points are on the curve/circle. This is not true for the staircase. There are 2 sets of points with a staircase, giving us 2 lines, one connects the top edges of the stairs and the other connects the bottom corners of the stairs. Both sets can't be on the same line/curve, producing the error $\pi = 4$.

 

[PeterDonis]

as the NumberOfSides(Polygon) increases, all the points are on the curve/circle.

Only in the limit. At each finite step, the corners of the polygon are not on the circle.

There are 2 sets of points with a staircase, giving us 2 lines, one connects the top edges of the stairs and the other connects the bottom corners of the stairs. Both sets can't be on the same line/curve

For each finite step, this is true. But that doesn't mean it has to also be true in the limit; there is no general rule about limits that entails it.

The correct procedure is to first ask if the limit of the stairsteps is well-defined at all. If it isn't, which is what I argued for in post #99, then it is meaningless to assert any properties of the limit, since there is no well-defined limit for any such properties to apply to.

 

[Agent Smith]

See my post #99 just now. If the limit of the stairsteps is not well-defined, then the limit of the perimeter is not well defined either.

Got it!

 

[Agent Smith]

Only in the limit. At each finite step, the corners of the polygon are not on the circle.


For each finite step, this is true. But that doesn't mean it has to also be true in the limit; there is no general rule about limits that entails it.

The correct procedure is to first ask if the limit of the stairsteps is well-defined at all. If it isn't, which is what I argued for in post #99, then it is meaningless to assert any properties of the limit, since there is no well-defined limit for any such properties to apply to.

I should've been clearer. The inscribed polygon's vertices are on the curve being approximated. Archimedes used 2 polygons (circumscribing one and the inscribed one). I have to do the math but my hunch is the inscribed polygon's perimeter more closely approximates the circle's circumference, the outer polygon having the same issue as the staircase approximation. As the number of sides of the polygon increases, the inscribed polygon basically becomes the circle.

 

[PeterDonis]

the outer polygon having the same issue as the staircase approximation

No, it doesn't. I made the difference clear in post #99.

As the number of sides of the polygon increases, the inscribed polygon basically becomes the circle.

This is also true of the circumscribed polygon, as I showed in post #99.

 

[Agent Smith]

No, it doesn't. I made the difference clear in post #99.


This is also true of the circumscribed polygon, as I showed in post #99.

I have yet to do the math. To my knowledge, $\frac{22}{7}$ is the upper limit of the rational approximation of $\pi$ (the circumscribing polygon) correct to $2$ decimal places. I have no idea what the lower limit is and how correct it is (the inscribed polygon).

I believe for the inscribed polygon, we're looking at the vertices and for the circumscribing polygon we're looking at the tangential points.

 

[A.T.]

Insofar as Archimedes' and Zu Chongzhi's $\pi$ approximation is concerned, we're computing the perimeter of the circumscribing and inscribing regular polygons.

So which regular polygon has the level of "circularity" that corresponds to the $\pi$ approximation $22/7$?

 

[Agent Smith]

So which regular polygon has the level of "circularity" that corresponds to the $\pi$ approximation $22/7$?

The one on the outside, yes? My hunch is the inside polygon's perimeter is more accurate.

 

[A.T.]

The one on the outside, yes? My hunch is the inside polygon's perimeter is more accurate.

It's not about inside or outside. How many sides does the regular polygon have, that corresponds to the $\pi$ approximation $22/7$?

 

[Agent Smith]

It's not about inside or outside. How many sides does the regular polygon have, that corresponds to the $\pi$ approximation $22/7$?

Well, I don't know. Archimedes used a 96-gon

 

[jbriggs444]

Well, what's the explanation for the error then?

I gave the explanation: The limit of the perimeters is not equal to the perimeter of the limit.

$\pi \ne 4, \pi = 3.14159...$. It can only mean that the curve we're assuming is an approximation of the actual curve (the circle, etc.) isn't what we assume/think it is.
We're overmeasuring or overcounting.

No. It means that the limit of the perimeters is not equal to the perimeter of the limit.

We could investigate where the extra $0.8584073464102067615373566167205...$ is coming from. I'm sure that would be easy for you, being a science person. Can you take a look into that.

I already know. The error is in your expectation that the two figures should be equal.

 

[A.T.]

Well, I don't know. Archimedes used a 96-gon

The point is that not every possible approximation of $\pi$ can be represented by a regular polygon. And when you don't restrict it to regular ones, then you have many differently looking polygons for the same approximation of $\pi$. Thus the idea that a given approximation of $\pi$ corresponds to a specific amount of "circularity" is flawed.

 

[jbriggs444]

I'm not sure this is actually true. For a polygon that is inscribed in the circle or circumscribed around the circle, the angle between the sides approaches 180 degrees as the number of sides increases without bound. In other words, the polygon approaches being a smooth curve, with no angles at all (each "angle" of 180 degrees is just a tangent line to the circle at the "angle" point).

Before we can decide what is "actually true", we should have a definition in hand. What does it mean for a sequence of shapes to "approach a shape in the limit"?

I have a definition in mind.

We are working in a flat two-dimensional metric space. A "shape" is simply a collection of points in the space. I will not try to impose additional requirements such as connectedness or smoothness for a "shape". Such properties are unimportant to the definition that I am trying to phrase.

Suppose that we have an infinite sequence of shapes, $S(i)$. We want to take the limit of this sequence.

The limit, if it exists, is the set of points $p$ such that

For every radius epsilon > 0
There is a minimum index $n$ such that
For every $m > n$
point $p$ is within radius epsilon of some point on shape $S(m)$

If no points satisfy this criterion then we say that the limit does not exist.

I claim that under this definition, the limit of a sequence of circumscribed stairstep shapes with decreasing step size about a circle of fixed radius is the circle.

I claim that under this definition, the limit of a sequence of circumscribed regular polygons with increasing side count about a circle of fixed radius is the circle.

I claim that under this definition, the limit of a sequence of inscribed regular polygons with increasing side count within a circle of fixed radius is the circle.

There is some speculation to the effect that normalized Pythagorean triples are dense on the unit circle. If so, one could form a sequence of sets, each containing finitely many rational coordinate pairs such that the sequence would converge in the limit to a fully populated unit circle.

 

[jbriggs444]

We are working in a flat two-dimensional metric space.

I have an improved definition. It is essentially identical to the previous one. But it applies to arbitrary topological spaces, even those for which no metric is provided.

A "shape" is still a set of points in the space.
We are still trying to define the limit of a sequence $S(i)$ of shapes.

The limit is the set of points $p$ such that

For every open ball $B$ containing $p$
there is a minimum index $n$ such that
for every $m > n$
$S(m) \cap B$ is non-empty

The difference between this definition and the previous one is that the previous one used a ball of radius $r$ centered on $p$. This one relaxes that and simply uses an open ball containing $p$.

[Be gentle. I've never taken a formal course in topology]

 

[A.T.]

One way of compactly stating the issue is that the limit of the perimeters is not equal to the perimeter of the limit. There is no principle of mathematics by which they should be equal. The two ideas do not "commute".

See also this video:

 

[jack action]

Well, what's the explanation for the error then? $\pi \ne 4, \pi = 3.14159...$. It can only mean that the curve we're assuming is an approximation of the actual curve (the circle, etc.) isn't what we assume/think it is. We're overmeasuring or overcounting. We could investigate where the extra $0.8584073464102067615373566167205...$ is coming from. I'm sure that would be easy for you, being a science person. Can you take a look into that.

Edit: Disregard this comment per post #133.

If one really wants to find $\pi$ with the "step" method, one must add the hypothenuse length of each step.

From my account, in the following image:

With the square in the top-right corner of the image, the approximate perimeter of the circle is:
$$4 \times \sqrt{\left(\frac{d}{2}\right)^2 + \left(\frac{d}{2}\right)^2}= 2.828d$$
The approximate perimeter of the circle in the middle-left position would be:
$$8 \times \sqrt{\left(\frac{d}{2}\sin 45°\right)^2 + \left(\frac{d}{2}(1-\cos 45°)\right)^2}= 3.061d$$
The approximate perimeter of the circle in the middle-right position would be:
$$8 \times \left( \sqrt{\left(\frac{d}{2}\sin 22.5°\right)^2 + \left(\frac{d}{2}(1-\cos 22.5°)\right)^2} + \sqrt{\left(\frac{d}{2}(\sin 45° - \sin 22.5°)\right)^2 + \left(\frac{d}{2}((1-\cos 45°) - (1-\cos 22.5°))\right)^2} \right)= 3.1214d$$
By using this method, the error drops really fast, and I'm sure we can show that the limit will be $\pi d$ as the number of steps goes to infinity.

 

[Vanadium 50]

What is this thread about? We're over 100 messages in and I have no idea. It's certainly not about the title, which is either incorrect or so vague as to be meaningless.

 

[PeterDonis]

circumscribed stairstep shapes

As I understand the "stairstep" construction, the shapes are not circumscribed after the initial square; there are points on the "stairstep" that are inside the circle, and only the sides that are the remnants of the original sides of the square are tangent to the circle. But I may be misunderstanding the construction.

 

[PeterDonis]

the approximate perimeter of the circle

I don't know what you mean. The circle's perimeter is $\pi d$ always. The perimeter of the square is $4$, and so is the perimeter of every "stairstep" construction derived from it. What "approximate perimeter of the circle" are you talking about?

 

[jbriggs444]

As I understand the "stairstep" construction, the shapes are not circumscribed after the initial square; there are points on the "stairstep" that are inside the circle, and only the sides that are the remnants of the original sides of the square are tangent to the circle. But I may be misunderstanding the construction.

As I understand the construction, we begin with a square within which a circle is circumscribed.

Then we cut out corners so that each pair of two orthogonal sides (e.g. over and down) is replaced by two new pairs (e.g. over and down and then over and down again). The new shape still circumscribes the circle. It does not extend into the interior anywhere. We continue cutting out corners, doubling the number of orthogonal sides at each step, ad infinitum.

The intermediate stairstep shapes all retain the property that every second vertex is positioned on the circle that is circumscribed. At least if we count the 4 points of tangency at the top, bottom and the two sides as vertices. They also retain the property that no part of the stairstep path extends into the interior of the circle.

 

[PeterDonis]

The intermediate stairstep shapes all retain the property that every second vertex is positioned on the circle that is circumscribed.

Ah, I see, that's what I was missing.

This construction still has the issue I described before, that it does not approach a smooth curve in the limit. Your topological definitions of the limit do not require smoothness, but I'm not sure if those definitions are sufficient for the limit of the perimeter to be well-defined.

I do agree that there is no general rule that the limit of the perimeter must be the same as the perimeter of the limit.

 

[jbriggs444]

Ah, I see, that's what I was missing.

This construction still has the issue I described before, that it does not approach a smooth curve in the limit. Your topological definitions of the limit do not require smoothness, but I'm not sure if those definitions are sufficient for the limit to have a well-defined perimeter.

It does approach a curve in the limit. The curve that is approached is a circle. A circle is smooth.

The fact that the "smoothness" of the sequence of curves does not converge upon "smooth" in the limit is irrelevant. The limit of the smoothness is not necessarily equal to the smoothness of the limit. Those two concepts do not commute.

 

[PeterDonis]

It does approach a curve in the limit. The curve that is approaches is a circle.

Does it? Perhaps that's where my question should have been focused. I know it seems intuitively like it does, but intuitions cannot always be trusted.

 

[jbriggs444]

Does it? Perhaps that's where my question should have been focused. I know it seems intuitively like it does, but intuitions cannot always be trusted.

Use the definition.

 

[PeterDonis]

Use the definition.

What definition?

 

[jbriggs444]

What definition?

The one I posted. Either the preliminary one in #113 or the relaxed one in #114.

 

[jack action]

I don't know what you mean. The circle's perimeter is $\pi d$ always. The perimeter of the square is $4$, and so is the perimeter of every "stairstep" construction derived from it. What "approximate perimeter of the circle" are you talking about?

Edit: Disregard this comment per post #133.

The "approximate perimeter of the circle" is the perimeter of the irregular polygon composed by the sum of the hypotenuses of each "step", i.e. $\sqrt{\text{run}^2 + \text{rise}^2}$, lying outside the circle: a 4-side polygon with the square (top-right corner), an 8-side polygon with the middle-left image, and a 16-side polygon with the middle-right image. The more sides you have, the better the approximation.

 

[PeterDonis]

The "approximate perimeter of the circle" is the perimeter of the irregular polygon composed by the sum of the hypotenuses of each "step"

Which has nothing whatever to do with the argument that was referred to that uses the stair step construction. So it is irrelevant to this thread.

 

[jack action]

Which has nothing whatever to do with the argument that was referred to that uses the stair step construction. So it is irrelevant to this thread.

The question I was answering was:

Well, what's the explanation for the error then? $\pi \ne 4, \pi = 3.14159...$. [...] We could investigate where the extra $0.8584073464102067615373566167205...$ is coming from.

And the answer is that it's the difference between the sum of the sides adjacent to the right angles and the sum of the hypotenuses formed by those sides.

 

[PeterDonis]

it's the difference between the sum of the sides adjacent to the right angles and the sum of the hypotenuses formed by those sides

But according to the stairstep construction, both of these things approach the same limit. So how can they give different answers? You do not address this, which is the actual question at issue, at all.

 

[jbriggs444]

The question I was answering was:

And the answer is that it's the difference between the sum of the sides adjacent to the right angles and the sum of the hypotenuses formed by those sides.

The figure in question is $4-\pi \approx 0.8584073464102067615373566167205$.

That is not the difference in perimeter between any particular circumscribed stairstep shape with $2^{n+1}$ sides and the perimeter of the corresponding inscribed $(2^n)$-gon. The one perimeter is always 4. The other perimeter is always some under-estimate for $\pi$.

It is the difference between the limit of the sequence of perimeters of the stairstep shapes and the limit of the sequence of perimeters of the corresponding inscribed polygons. The one limit is 4. The other limit is $\pi$.

 

[Mark44]

As I understand the construction, we begin with a square within which a circle is circumscribed.

A circle inside a square and tangent to the four sides of the square is said to be inscribed. A circle outside the square for which the corners of the square touch the circle is said to be circumscribed.

BTW we've ventured quite a way from the original subject of the thread, of why we use the "awkward" approximation to $\pi$ of 22/7.

 

[jack action]

But according to the stairstep construction, both of these things approach the same limit.

Sorry, my bad. I should have done one more step in post #116 to find out it was going over $\pi$. Disregard my comment.

 

[jbriggs444]

BTW we've ventured quite a way from the original subject of the thread, of why we use the "awkward" approximation to $\pi$ of 22/7.

In an effort to bring us back somewhat on track, let us explore the question of what regular n-gons have a perimeter to "diameter" ratio closest to 22/7. It turns out that the answer is between 90 and 91.

We begin by noting that $\frac{22}{7}$ is greater than $\pi$. So we should be looking at circumscribed polygons.

The formula for the perimeter of such a polygon with radius $r$ (measured from centroid to the middle of an edge) is:$$2nr \tan \frac{\pi}{n}$$We can tabulate this for a number of sides going from 3 on up. After writing a bit of code:

[...]
Number of sides: 4, radius: 0.5, side_length: 1, perimeter: 4 target = 3.14285714285714
[...]
Number of sides: 90, radius: 0.5, side_length: 0.0349207694917477, perimeter: 3.1428692542573 target = 3.14285714285714
Number of sides: 91, radius: 0.5, side_length: 0.0345367179994631, perimeter: 3.14284133795114 target = 3.14285714285714

If we flip tan to sine then virtually the same code can look at inscribed polygons where the radius is measured to the vertices.

[...]
Number of sides: 6, radius: 0.5, side_length: 0.5, perimeter: 3 target = 3.14
[...]
Number of sides: 56, radius: 0.5, side_length: 0.0560704472371918, perimeter: 3.13994504528274 target = 3.14
Number of sides: 57, radius: 0.5, side_length: 0.0550877603558654, perimeter: 3.14000234028433 target = 3.14

Here is the "circumscribed" version of the code.

#!perl

use strict;

sub tan { sin($_[0]) / cos($_[0]) };

my $i;
my $radius = 0.5;
my $pi = 3.1415926535897932384626;
my $target = 22/7;
my $perimeter = 999;

for ( $i = 3; $perimeter > $target; $i++ ) {
    my $half_angle = $pi / $i;
    my $side_length = 2 * $radius * tan($half_angle);
    $perimeter = $i * $side_length;
    print STDOUT "Number of sides: $i, radius: $radius, side_length: $side_length, perimeter: $perimeter target = $target\n"
};

 

[Agent Smith]

I don't get it. If the jagged staircase does converge on a smooth curve, $\pi = 4$ shouldn't happen. We would be measuring/computing the same thing. How can $1$ thing have $2$ different lengths.

In the video link provided by @A.T. we see that integration has a jagged staircase element to it (more and more of thinner and thinner rectangles, sum their areas and we get the integral). The only difference here is we're not measuring the length of the curve, but the area. Gracias A.T.

@jack action , merci for the explanation. There's an extra $4 - \pi = 0.8584073464102067615373566167205$ when "computing" $\pi$ using the staircase. This mysterious extra length needs to be explained.

$1$ quadrant's (quarter circle) arc length = $2 \pi r / 4 = 2 \pi \times 0.5 / 4 = 0.78539816339744830961566084581988$.

The staircase has for $1$ quadrant (quarter circle) a length of $0.5 + 0.5 = 1$ and
The difference between the staircase and the quadrant arc of the circle = $1 - 0.78539816339744830961566084581988 = 0.21460183660255169038433915418012$

We can see that $4 \times 0.21460183660255169038433915418012 = 0.8584073464102067615373566167205$

Perhaps we might need to go TO INFINITY AND BEYOND to make the staircase argument work!

 

[Agent Smith]

@Vanadium 50 , it was a simple high school curriculum question on $\pi$ and it (d)evolved into the staircase paradox.

 

[Vanadium 50]

Now it's a "paradox"? When I was learning about it, it was just "making a mistake".

 

[PeterDonis]

If the jagged staircase does converge on a smooth curve, $\pi = 4$ shouldn't happen.

You have already been told that this argument is not valid. Limits in general do not have to work that way.

 

[A.T.]

In the video link provided by @A.T. we see that integration has a jagged staircase element to it (more and more of thinner and thinner rectangles, sum their areas and we get the integral). The only difference here is we're not measuring the length of the curve, but the area. Gracias A.T.

As the video explains: You have to show that the error goes to zero. This is the case for the area, but not for the perimeter.

 

[jack action]

@jack action , merci for the explanation. There's an extra $4 - \pi = 0.8584073464102067615373566167205$ when "computing" $\pi$ using the staircase. This mysterious extra length needs to be explained.

It was a mistake on my part. See post #133.

 

[jbriggs444]

If the jagged staircase does converge on a smooth curve, $\pi = 4$ shouldn't happen.

"Should" is a word that often makes me cringe.

In my career as an IT troubleshooter, the word was most often used by users when complaining that the behavior that they were experiencing did not match the behavior that they expected.

I could ask the users why they thought that their stuff "should" do such and such. Rarely were they able to say. Eventually, I would usually give up. End users usually have no clue about how their programs work, what demands they make on the network and what level of performance can be expected. I would have to reverse engineer the application and figure out for myself how it worked, how it could be expected to behave and what changes could be feasibly made to address the perceived issues.

To me, "should" labels an expectation that has no underlying logic.

If you can carefully explain why the limit approached by the perimeters of a sequence of ever finer stairstep shapes should match the perimiter of the limiting shape that is approached then we would have something to explain.

Just saying that the limit of the perimeters "should" match the perimeter of the limiting shape is not sufficient.

We would be measuring/computing the same thing. How can $1$ thing have $2$ different lengths.

Nope. We are not finding two lengths for the same thing.

We are comparing the limit of a sequence of lengths of jagged stairstep shapes with the length of the smooth limiting shape.

In the video link provided by @A.T. we see that integration has a jagged staircase element to it (more and more of thinner and thinner rectangles, sum their areas and we get the integral). The only difference here is we're not measuring the length of the curve, but the area. Gracias A.T.

An integral is not a sum of areas. It is a limit approached by a set of sums of areas.

https://en.wikipedia.org/wiki/Riemann_integral

Perhaps we might need to go TO INFINITY AND BEYOND to make the staircase argument work!

A course in real analysis would be helpful.

 

[Agent Smith]

@A.T. That's right, the error has to become $0$ and that's where we should start. We first compute the error before we build the staircase. Say the perimeter of the circle = C. The perimeter of the square = $P_S = 4$. The error = $P_S- C$. We then begin our staircase argument, but the perimeter(staircased square) remains constant at $4$ i.e. $P_S - C = k = 0.85840734641...$ (a constant). This construction, if we could call it that, doesn't work; if it did $\left(P_S - C\right) \to 0$

@jack action I don't think it's a mistake. In one sum we're summing the base and height of the right triangles in a staircase and in the other we're taking only the hypotenuse (the curve). The triangle inequality law states that the sum of $2$ sides of a triangle > the length of the other side.

All the construction seems to be doing is dividing AB and BC into smaller and smaller parts and then later summing them all back up to AB and BC. $n \times \frac{\text{AB}}{n} = \text{AB}$ and $n \times \frac{\text{BC}}{n} = \text{BC}$.

If the staircase is an approximation of the circle's circumference, we're counting the wrong thing. Maybe we should be counting/adding the hypotenuses of the triangle ABC, instead of adding AB + BC. Nescio.

 

[Agent Smith]

To me, "should" labels an expectation that has no underlying logic.

Sorry for the imprecise language, This question's Affix is B: Basic. Would appreciate if the discussion could be kept as simple (as possible). I should've said:
If the staircase is a (good) approximation of the arc then, as the construction is carried on to infinity then, error $(\text{Arc length} - \text{Stair case length}) \to 0$

We are comparing the limit of a sequence of lengths of jagged stairstep shapes with the length of the smooth limiting shape.

Si.

An integral is not a sum of areas. It is a limit approached by a set of sums of areas.

Thank you for the clarification. I erroneously believed my description of an integral, based on ever thinner rectangles meant the same thing as "a limit approached by a set of sums of areas".

 

[PeterDonis]

If the staircase is an approximation of the circle's circumference, we're counting the wrong thing.

I'm not sure what you mean. The fact that the perimeter of the staircase is $4$ for all $n$ is a simple consequence of the construction.

If you count the lengths of the hypotenuses instead, you're not using the staircase construction, you're doing something else. That something else might well have a different limiting behavior of its perimeter, but that's irrelevant to the limiting behavior of the staircase construction itself.

The simple answer is that the limit as $n \to \infty$ of the perimeter of the staircase is $4$, even though the "limiting curve" of the staircase (at least by a definition given earlier in this thread) is the circle. Welcome to the actual theory of limits, where your intuitions about how they "should" work are not always correct.

 

[A.T.]

If the staircase is a (good) approximation of the arc then, as the construction is carried on to infinity then, error $(\text{Arc length} - \text{Stair case length}) \to 0$

No, because it's only a good approximation in terms of area enclosed by the arc, not in terms of arc length. There is no general meaning of "good approximation of an arc". You have to be specific about which quantitative property of the arc is well approximated.

 

[Agent Smith]

 

[A.T.]

View attachment 350207

Scaling the whole thing doesn't change the ratio of perimeter to d, which is still 4.

 

[jbriggs444]

If the staircase is an approximation of the circle's circumference, we're counting the wrong thing.

Yes. If your goal is to compute the circumference of a circle, taking the limit of a sequence of 4's is the wrong way to go about it.

Edit: One more thing that I will point out. It is a subtlety about the staircase construction that you may not have considered. You included this series of drawings.

Here the rectangles that are carved away from the staircase corners appear to be squares. That is a sub-optimal choice.

If you carve away squares then the corner pieces that are carved away will be smaller and smaller fractions of this particular step. In the limit, this particular right-center-top step will lose about $\frac{1}{n}$ of its scale with each iteration.

So the convergence properties of this particular version of the stairstep construction are not great. Convergence is still assured. The sum of a harmonic series is infinite. We will still succeed in reducing every step size to a limit of zero. But the number of iterations required will be exponential.

We have a free choice in exactly what rectangle to carve out of each stairstep with each iteration. Where should we place the corner that lies on the circular arc?

It is possible to make that choice pathologically so that the stairstep shape does not converge to a circle. With purposely contrived choices, one can keep some of the steps from ever shrinking beyond a certain point.

One way to assure good convergence is to make a choice of chopped-out-corner-rectangle that splits the perimeter of each step exactly in half.

If one is attempting to use the definition up-thread and rigorously prove that the stairstep construction converges to a limiting shape that is a circle then convergence of the step size is a crucial detail and a calculable rate of convergence is helpful.

e.g. "At step $n$, no point on the constructed stairstep shape is more than distance $\frac{1}{2^n}$ from the enclosed circular shape"

 

[Agent Smith]

@jbriggs444, it's getting too complex for the likes of me, mon ami.

As you rightly pointed out, the diagram I drew fails to do justice to the actual events on the arc-staircase. We do have a iterative process (fold corners in on the curve/arc), but length-wise there's variability that I can't handle at the moment with my limited knowledge of math.

Let's look at the original staircase paradox:

As the story ends, $\sqrt 2 = 2$

I was trying to look at it from a vectors POV and isn't it true that: $\overrightarrow {AB} + \overrightarrow {BC} = \overrightarrow {AC}$. We break down the vector $\overrightarrow {AC}$ into it's $2$ components, the horizontal $\overrightarrow{BC}$ and the vertical $\overrightarrow {AB}$. However $|\overrightarrow {AC} | \ne |\overrightarrow {AB}| + |\overrightarrow {BC}|$. Doesn't the staircase paradox violate the triangle inequality theorem?

Shouldn't we also be able to work backwards? Start with the diagonal and construct a staircase that ultimately becomes a square for that diagonal? The same for the arc/curve. Go from curve to staircase. How would you argue that I wonder?

 

[jbriggs444]

Let's look at the original staircase paradox:
[...]
I was trying to look at it from a vectors POV and isn't it true that: $\overrightarrow {AB} + \overrightarrow {BC} = \overrightarrow {AC}$.

Yes, that is a correct statement.

We break down the vector $\overrightarrow {AC}$ into it's $2$ components, the horizontal $\overrightarrow{BC}$ and the vertical $\overrightarrow {AB}$. However $|\overrightarrow {AC} | \ne |\overrightarrow {AB}| + |\overrightarrow {BC}|$. Doesn't the staircase paradox violate the triangle inequality theorem?

What does the triangle inequality actually say?$$|\vec{AC}| \le |\vec{AB}| + |\vec{AC}|$$That requires neither equality nor inequality. The last time I checked,$$2 \le 2$$and $$\sqrt{2} \le 2$$In a general sense, the triangle inequality applies to what we can consider as a measure of "distance". To what mathematicians would call a "metric".

If one applies the stairstep method to evaluate the "length" of an arbitrary curve, one arrives at what is often called the "taxicab metric".

The taxicab metric obeys the triangle inequality. But it does not match the Euclidean metric which also obeys the triangle inequality.

Shouldn't we also be able to work backwards? Start with the diagonal and construct a staircase that ultimately becomes a square for that diagonal? The same for the arc/curve. Go from curve to staircase. How would you argue that I wonder?

Certainly, one can invert the sequence of intermediate shapes so that the successive stairstep approximations are each more coarse-grained than the last.
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The limit is still there. At the fine-grained end. But thinking of this as a "process" runs into a serious snag: What is the first step? The answer is that there is none. Which makes it pretty hard to regard the reversed thing as a process.

That is also the snag in the forward process. There is always a next step. But never a last step.

If one carefully examines the formal definition of a limit, one sees that time plays no role. There is no process. No need for a last step. There is (or is not) a result that fulfills a condition. A condition phrased with epsilons, deltas, for alls and there exists.

Yes, as an intuitive notion there is a process and a limit is the thing approached by the process. However, that notion does not necessarily carry over into the formal definition.