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

[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.
'
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.

 

[Agent Smith]

@jbriggs444 , just wondering ... we can justify the staircase using vectors (breaking up a vector into vertical and horizontal components). However the magnitudes don't track each other (as we already discussed) i.e. the lengths(vectors) fail to tally.

If you look at the post which got this party started, the folding-in-of-the-corner is described as a process ... to be contd. to infinity. Perhaps we can be more specific and call it a computation (it looks as though the corner is reflected across a line, the diagonal), to repeated unto infinity.

 

[jbriggs444]

@jbriggs444 , just wondering ... we can justify the staircase using vectors (breaking up a vector into vertical and horizontal components). However the magnitudes don't track each other (as we already discussed) i.e. the lengths(vectors) fail to tally.

If you look at the post which got this party started, the folding-in-of-the-corner is described as a process ... to be contd. to infinity. Perhaps we can be more specific and call it a computation (it looks as though the corner is reflected across a line, the diagonal), to repeated unto infinity.

"Repeated unto infinity". There is a rub. What does that mean? Where is the last step that gets us "to infinity".

The formal definition of limits gives us a way to carefully state a "result" for a computation that does not terminate.

"Reflected across a line, the diagonal" is a bit ambiguous. The diagonal of what? Presumably the diagonal of a chosen rectangle, one corner of which is a corner on the current staircase and the opposite corner of which is on the circle. The diagonal across which we are reflecting would then run between the other two corners. Actually, that cannot be right. That reflection would intrude into the interior of the circle.

So you must be chopping out squares, not rectangles.

Regardless, there is no guarantee anywhere here that the sequence of perimeters for the shapes arrived at during the process matches the perimeter of the shape that is converged toward in the limit. On the contrary. It will not and does not.


Edit: I want to go off on a tangent here and relate my personal experience with the notion of infinity. Perhaps it will resonate with you.

I was in my sophomore year in college. Doing well with a number of courses under my belt. Calculus. Differential equations. I navigated through everything that involved integrals, derivitives and limits with my own private notion of infinitesimals. Numbers that were small enough so that their square was zero. If I had a coherent notion of "infinity", it was as a sort of process that iterated forever.

I was under no illusions that my notions were rigorous and correct. But they got the job done. I was careful not to voice them in class.

Then I signed up for a 400 series course called "Advanced Calculus".

It was not about calculus. We started with the Peano Axioms for the natural numbers. The natural numbers are, of course, an infinite set.

For two or three weeks, I struggled to reconcile the natural numbers under these axioms with my private notion of infinity as some sort of result of a generalized process.

It finally clicked. I grasped the notion of the natural numbers as a completed whole that satisfies the set of axioms given by Peano. No process anywhere. Just a set and some rules that it obeys.

In the remainder of the course, we explored the notion of "equivalence relations" and defined the signed integers in terms of the set of equivalence classes of ordered pairs of natural numbers under one equivalence relation. Then the rational numbers as a set of equivalence classes of ordered pairs of signed integers under another equivalence relation. Then the real numbers as the set of Dedekind cuts of the rationals. [Cauchy sequences are more mainstream, but our course used Dedekind cuts].

It was the most fun I've ever had in a mathematics course.

 

[Agent Smith]

@jbriggs444 then may I, am I permitted to, say that we need a more complex definition of reflection, one that can be used for such kinds of "transformations". May be not ... we already have enough on our plate, oui?

 

[A.T.]

Let's look at the original staircase paradox:

If you find it counter intuitive, that the stair case construction keeps the perimeter constant, while the area changes, consider the opposite case:

Start with a square and stretch it into a longer and longer rectangle, while keeping the area constant. Here it's the perimeter that changes, up to infinity, despite a constant finite area. And it's not that weird, is it?

 

[jbriggs444]

@jbriggs444 then may I, am I permitted to, say that we need a more complex definition of reflection, one that can be used for such kinds of "transformations". May be not ... we already have enough on our plate, oui?

I do not think that the steps of the transformation process are important.

We could think of chopping off a right triangle (that is not necessarily isoceles) on the diagonal. Then rotating that triangle by 180 degrees and chopping that shape out of the staircase where the diagonal side was.

But, as I said, the exact description of the transformation step is not important. What is important is what it means for the newly modified stairstep shape.

1. The perimeter is left unchanged.
2. The maximum distance of the stairstep shape from the enclosed circle has decreased.

I've already suggested a way in which we can force the decrease in maximum distance to be a factor of 2 at every iteration.

 

[Agent Smith]

@A.T. that's a really good analogy. Asante sana. What's happening in the staircase paradox is that the area is changing (decreasing) but the perimeter is constant. What you're saying is that the area can remain constant while we can change the perimeter. Does this mean the $2$ measurements (perimeter & area) of an object are independent of each other? In other words, a "procedure" that makes areas change can't be used to draw conclusions about the perimeter and vice versa; to the extent that's true, the staircase argument fails. This is interesting to say the least.

What is important is what it means for the newly modified stairstep shape.

1. The perimeter is left unchanged.
2. The maximum distance of the stairstep shape from the enclosed circle has decreased.

Wondering what that translates to, in terms of perimeter and area of the enclosed circle.

 

[Agent Smith]

I lost so much land and I thought the silver lining was reduced fencing costs.

 

[A.T.]

In other words, a "procedure" that makes areas change can't be used to draw conclusions about the perimeter and vice versa; to the extent that's true, the staircase argument fails.

Yes, exactly. For some arbitrary iterative procedure, the area and perimeter can have completely different behaviors.

 

[jbriggs444]

Wondering what that translates to, in terms of perimeter and area of the enclosed circle.

The sequence of areas enclosed by the stairstep shapes converges to the area enclosed by the circle.
The sequence of perimeters of the stairstep shapes do not converge to the perimeter of the circle.

One way of thinking about the perimeter discrepancy is that the stairstep shapes are not "smooth". In some sense they get progressively "rougher". Unfortunately, rigorous definitions of "smoothness" are somewhat technical. One finds phrases like "continuously differentiable".

Look at the definition of arc length here.

 

[Agent Smith]

What do you guys make of the following?

$A = \pi {r_1}^2 \implies r_1 = \sqrt {\frac{A}{\pi}}$
$nA = \pi {r_2}^2 \implies r_2 = \sqrt {\frac{nA}{\pi}} = \sqrt n \times \sqrt{\frac{A}{\pi}} \implies r_2 = \sqrt n r_1$

Because the perimeter/circumference of a circle is a function of its radius, we can say that change(area of circle) is always accompanied by change(circumference of circle) and vice versa.

 

[Frabjous]

What do you guys make of the following?

$A = \pi {r_1}^2 \implies r_1 = \sqrt {\frac{A}{\pi}}$
$nA = \pi {r_2}^2 \implies r_2 = \sqrt {\frac{nA}{\pi}} = \sqrt n \times \sqrt{\frac{A}{\pi}} \implies r_2 = \sqrt n r_1$

Because the perimeter/circumference of a circle is a function of its radius, we can say that change(area of circle) is always accompanied by change(circumference of circle) and vice versa.

You can directly solve for circumference as a function of area.

 

[jbriggs444]

What do you guys make of the following?

Without context, it seems pointless.

$A = \pi {r_1}^2 \implies r_1 = \sqrt {\frac{A}{\pi}}$

This sounds like a way to impute a "radius" for an irregular figure. It is based on the radius the figure would have if we smoothed it out into a circle while retaining the same enclosed surface area.

A similar concept of "areal radius" is used to express the "radius" of things like black holes that have no radius in the usual sense. @PeterDonis could expound at length on this. He has relevant expertise.

$nA = \pi {r_2}^2 \implies r_2 = \sqrt {\frac{nA}{\pi}} = \sqrt n \times \sqrt{\frac{A}{\pi}} \implies r_2 = \sqrt n r_1$

So this is a way of imputing a "radius" for a scattering of $n$ identical shapes?

Because the perimeter/circumference of a circle is a function of its radius, we can say that change(area of circle) is always accompanied by change(circumference of circle) and vice versa.

For a circle [in Euclidean geometry], its circumference, radius, diameter and enclosed area are all related. If you know any one, you can calculate all of the others. That should be obvious.

What is the point?

 

[PeterDonis]

the perimeter/circumference of a circle is a function of its radius

Yes, but that doesn't mean it's true of any plane figure whatever. The formulas you gave are only valid for a circle.

 

[Agent Smith]

What is the point?

We can't craft a staircase paradox with a circle, because unlike a square, its perimeter does change with its area.

 

[Vanadium 50]

We can't craft a staircase paradox with a circle, because unlike a square, its perimeter does change with its area

What? Of course a square's perimeter changes with area.

As far as "paraxoes", doing a calculation incorrectly is not a paradox.

 

[Agent Smith]

As far as "paraxoes", doing a calculation incorrectly is not a paradox.

Si.

 

[jbriggs444]

We can't craft a staircase paradox with a circle, because unlike a square, its perimeter does change with its area.

You do not seem to understand the staircase "paradox". It has nothing to do with area. The definition of convergence that we are using is based on points and their distance from curves. No mention of area anywhere. Nor is there even a paradox to be found. Just a broken intuition.

Retrain your intuition, please!

It would be correct to say that a sequence of circles that converges (in the sense given up-thread) to a given circle has a sequence of perimeters that converge to the perimeter of that same circle. For instance, a sequence of circles, each of radius $r+\frac{1}{i}$ would converge to a circle of radius $r$. The sequence of perimeters: $(2r + \frac{2}{i})\pi$ would converge to $2 \pi r$.

This is rather uninteresting.

 

[Agent Smith]

@jbriggs444 regarding the $\pi = 4$ paradox/broken intuition, the areas of the circle and the construction based on folding in the corners of the square do converge, but the perimeters don't. In what way is our intuition off the mark? Are we seeing the areas converging and misinferring the perimeters are too?

 

[jbriggs444]

@jbriggs444 regarding the $\pi = 4$ paradox/broken intuition, the areas of the circle and the construction based on folding in the corners of the square do converge, but the perimeters don't. In what way is our intuition off the mark? Are we seeing the areas converging and misinferring the perimeters are too?

The sequence of perimeters does converge. To 4.

Your intuitive expectation that the sequence "should" converge to $\pi$ is off the mark.

Again, the sense in which the sequence of stairstep shapes converges to the circle shape has nothing to do with area. It has to do with closeness of points on the stairstep shapes to points on the limiting curve.

 

[Vanadium 50]

Is there anything to this thread beyond "it converges to 4" "but I want it to converge to π!"

As they say, two and two continue to make four despite the whine of the amateur for three and the cry of the critic for five."

 

[Agent Smith]

Is there anything to this thread beyond "it converges to 4" "but I want it to converge to π!"

As they say, two and two continue to make four despite the whine of the amateur for three and the cry of the critic for five."

Good to hear the voice of sanity ... once in a while. @jbriggs444 is more advanced along his mathematical journey than I am and hence this agonizingly prolonged, but inevitable death of this sutra.

 

[PeterDonis]

It seems like the subject has been discussed enough. Thread closed.