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

[Steve4Physics]

I'm told hexagonal wheels work okay given exactly the right profile of 'washboard' surface.

And, if I may butt-in, in some less mathematically advanced parts of the world, they have only gotten this far...

(from https://www.startupselfie.net/wp-content/uploads/2023/04/Square-Wheeled-Bicycle-The-Q.jpg)

 

[berkeman]

in some less mathematically advanced parts of the world, they have only gotten this far...

But you can see they have eliminated the brakes, since they are not needed. Quite a weight-saving feature!

 

[jbriggs444]

Nescio, measure the perimeter and divide it by the "width".

How does one measure the width of a square circle? Is it $2r$ or $2 \sqrt{2} r$?

The closer the ratio approaches $\pi$ the more circular it is. Archimedes & Zu Chongzhi used polygons to compute $\pi$. A circle is an infinite-sided polygon?

There is no such thing as a polygon with infinitely many sides.

One might use a limiting process to define the shape converged to by a sequence of polygons with increasing numbers of sides. But the limit converged upon by a sequence of polygons is not necessarily a polygon.

 

[PeterDonis]

measure the perimeter and divide it by the "width"

Do you mean width or diameter? As I've already pointed out in response to @jbriggs444 (who gave a link with a definition of "width"), they're not the same.

 

[mathwonk]

interestingly (to me), there is no explicit requirement in the following wikipedia article on polygons that the number of sides be finite, (although it appears to be taken for granted), and in fact I could find nothing stated in the article that is not true for a (closed) figure with an infinite number of straight sides. Note e.g. that they do not seem to say that every edge has two vertices as endpoints, nor that every end point of an edge is also the common endpoint of another edge. they do use the language of consecutive edges, but do not seem to require that every edge have two (or even one) adjacent edges.
https://en.wikipedia.org/wiki/Polygon

 

[Agent Smith]

Do you mean width or diameter? As I've already pointed out in response to @jbriggs444 (who gave a link with a definition of "width"), they're not the same.

The width of a circle is its diameter???

 

[jbriggs444]

The width of a circle is its diameter???

That tells us the width of a figure that is a circle.

What about a figure that is not a circle? Since that is the sort of figure we are discussing.

 

[Agent Smith]

That tells us the width of a figure that is a circle.

What about a figure that is not a circle? Since that is the sort of figure we are discussing.

Nescio, what were Archimedes and Zu Chongzhi doing, approximating a cricle's circumference with polygons?

limiting process

Cogito, that's it!

$\displaystyle \lim_{\text{sides} \to \infty} \text{Regular Polygon} = \text{Circle}$
How does that square with $\displaystyle \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

 

[jbriggs444]

Nescio, what were Archimedes and Zu Chongzhi doing, approximating a cricle's circumference with polygons?

That is not responsive to a question about defining the "width" of a figure that is not a circle.

$\displaystyle \lim_{\text{sides} \to \infty} \text{Regular Polygon} = \text{Circle}$
How does that square with $\displaystyle \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

Where is your definition for the limit of a sequence of shapes? You can't use a definition without stating it or referencing it first.

If you had both limit definitions in hand, you might then be able to draw parallels between them.

But regardless, that leaves us no closer to defining the "width" of a figure that is not a circle. I have a definition in mind. But I want to see you put forth a bit of effort here and state yours.

 

[A.T.]

Nescio, measure the perimeter and divide it by the "width". The closer the ratio approaches $\pi$ the more circular it is. Archimedes & Zu Chongzhi used polygons to compute $\pi$. A circle is an infinite-sided polygon?

Polygons do not have a unique width. How do you quantify the resemblance of a hexagon to a circle?

 

[Nik_2213]

IIRC, before 'series' derivations, the polygon route to Pi was based on 'regular' shape. Akin to the way, perhaps, that a slightly wonky wheel would wear down to 'even'...

FWIW, wasn't there a delightful 'Golden Age' SciFi tale, where an alien culture's religious taboo on full circles --So wheels, pulleys etc etc-- was up-ended thus ?
Visitor introduced three-arced Reuleaux triangle rollers, to the utter consternation of clergy...
Wicked 'malicious compliance'...

 

[PeterDonis]

How do you quantify the resemblance of a hexagon to a circle?

It depends on whether the hexagon is inscribed in the circle or circumscribed around the circle. At least, that's how it worked in Archimedes' method for putting bounds on $\pi$ by considering the perimeter of inscribed and circumscribed polygons with increasing numbers of sides. The "diameter" in all cases was the diameter of the circle; but the relationship of that to the perimeter of the polygon was different for the inscribed vs. circumscribed polygon.

 

[A.T.]

It depends on whether the hexagon is inscribed in the circle or circumscribed around the circle.

That's just two of infinitely many possibilities for the width of a hexagon.

 

[PeterDonis]

That's just two of infinitely many possibilities for the width of a hexagon.

Yes, but those two are the ones that are most relevant to comparing it to a circle.

 

[A.T.]

Yes, but those two are the ones that are most relevant to comparing it to a circle.

Why? Why not the mean of the two or something else in between them?

 

[PeterDonis]

Why?

Because they are the two natural relationships between a circle and a regular polygon--inscribing and circumscribing.

 

[A.T.]

Because they are the two natural relationships between a circle and a regular polygon--inscribing and circumscribing.

I can come up with such "natural relationships" all day: Circle cuts regular polygon sides in 3 equal parts feels natural to me.

But even if, for the sake of the argument, we stick to inscribing or circumscribing, it's still not clear which of the two should be used to quantify the resemblance of a polygon to a circle.

 

[PeterDonis]

even if, for the sake of the argument, we stick to inscribing or circumscribing, it's still not clear which of the two should be used to quantify the resemblance of a polygon to a circle.

That's right; there is no unique way of doing it. We can appeal to the "naturalness" argument I gave (which, as you note, is not rigorous) to reduce the number of choices to two, but no further.

 

[WWGD]

How does one measure the width of a square circle? Is it $2r$ or $2 \sqrt{2} r$?

There is no such thing as a polygon with infinitely many sides.

One might use a limiting process to define the shape converged to by a sequence of polygons with increasing numbers of sides. But the limit converged upon by a sequence of polygons is not necessarily a polygon.

Burago, Burago , Ivanov , in its book " Metric Geometry" describes sequences of Metric Spaces (Including Polygonal ones) and convergence properties. I lost track of my copy, unfortunately.

 

[Agent Smith]

Polygons do not have a unique width. How do you quantify the resemblance of a hexagon to a circle?

Insofar as Archimedes' and Zu Chongzhi's $\pi$ approximation is concerned, we're computing the perimeter of the circumscribing and inscribing regular polygons. The greater the number of sides, the better the perimeter(polygon) approximates circumference(circle). Also, whatever might be the choice for the width of the polygon, it too approximates the diameter of the circumscribed/inscribed circle. In the end as the number of sides goes to infinity, we have the circle's circumference and its diameter and $c/d = \pi$.

 

[Frabjous]

This is my favorite* geometric calculation of pi.

*favorite does not mean correct

 

[Agent Smith]

This is my favorite* geometric calculation of pi.
View attachment 350043
*favorite does not mean correct

I did a quick google search and it seems this "proof" is erroneous, but I still haven't figured out where the analogy between the staircase and the curve breaks down. The best way to understand the problem with this "proof" is that some of the points on the staircase are not on the curve This issue is not there when we approximate the curve with a polygon (all the points are on the curve), the way Archimedes and Zu Chongzhi did.

 

[Frabjous]

I did a quick google search and it seems this "proof" is erroneous, but I still haven't figured out where the analogy between the staircase and the curve breaks down. The best way to understand the problem with this "proof" is that some of the points on the staircase are not on the curve This issue is not there when we approximate the curve with a polygon (all the points are on the curve), the way Archimedes and Zu Chongzhi did.

Notice that this curve breaks your perimeter/diameter for circle quality construction.

 

[jbriggs444]

I did a quick google search and it seems this "proof" is erroneous, but I still haven't figured out where the analogy between the staircase and the curve breaks down. The best way to understand the problem with this "proof" is that some of the points on the staircase are not on the curve.

Of course the proof is erroneous. We know that going in.

The issue is not that the some of the points on the staircase are not on the curve. Almost none of the points in the circumscribed/inscribed polygons are on the curve either.

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".

 

[Agent Smith]

Notice that this curve breaks your perimeter/diameter for circle quality construction.

Oh, that's right, visually that is, but I did stress that the critical ratio has to approximate $\pi$.

@jbriggs444 , the "perimeter of the limit"?

Almost none of the points in the circumscribed/inscribed polygons are on the curve either.

The points that matter to the approximation are on the curve (the points get closer and closer as the number of sides increase).

In the case of the staircase, some of the points always stand outside of the curve/line being approximated.

 

[jbriggs444]

@jbriggs444 , the "perimeter of the limit"?

Think about a sequence of stair-step approximations to a circle with more and more sides that are each tinier and tinier. That sequence of stairstep shapes approaches a limiting shape. The limiting shape is the circle, of course.

The "perimeter of the limit" would be the perimeter of the limiting shape, aka the perimeter of the circle, $\pi d$.

Each of the stairstap shapes will have more and more of the tinier and tinier sides. Each shape will have a perimiter. Each will have the same perimeter as it turns out. $4d$ for all of them.

The "limit of the perimeters" would be the limit of the infinite sequence of perimeters. That limit is obviously $4d$

The points that matter to the approximation are on the curve (the points get closer and closer as the number of sides increase).

Please stop using fuzzy language. Phrases like "matter to the approximation" are meaningless. The points on a stairstep also get closer and closer as the number of sides increase.

In the case of the staircase, some of the points always stand outside of the curve/line being approximated.

In the case of a polygon circumscribing a circle, some of the points always stand outside of the circle about which they are circumscribed.

 

[PeterDonis]

In the case of the staircase, some of the points always stand outside of the curve/line being approximated.

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

 

[Agent Smith]

Please stop using fuzzy language. Phrases like "matter to the approximation" are meaningless. The points on a stairstep also get closer and closer as the number of sides increase.

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.

 

[PeterDonis]

That sequence of stairstep shapes approaches a limiting shape. The limiting shape is the circle, of course.

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).

In the case of the stairstep shapes, however, the angle between the sides is constant at 90 degrees; it does not approach being a smooth curve as the number of stairsteps increases without bound. So I do not think the limit of the stairsteps can be a smooth curve. I think the limit might not be well-defined at all.

 

[PeterDonis]

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

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.

 

[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.