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

[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.