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

[PeterDonis]

The points on the figure that I described vary in distance from the centroid. Which counts, I think, as the "center" of an imperfect circle.

If we broaden our scope beyond the particular case that started the subthread (trying to draw a circle with a compass), yes, I would say the figure you describe could qualify as an "imperfect circle"--just not one with constant width.

 

[jbriggs444]

If we broaden our scope beyond the particular case that started the subthread (trying to draw a circle with a compass), yes, I would say the figure you describe could qualify as an "imperfect circle"--just not one with constant width.

Why not constant width?

 

[PeterDonis]

Why not constant width?

Because, as you note, the distance of the points on the curve from the centroid varies.

 

[PeterDonis]

There is a class of figures with a unique diameter. Curves of constant width.

Ah, I see the disconnect: you are using "width" instead of "diameter". They're not the same thing. "Constant diameter" is the property of the exact circle that "imperfect circles" do not share, which is what I was referring to in my earlier post about that. That is not the same property as "constant width".

 

[Nik_2213]

Back to the 22/7 thing:
IIRC, a surveyor's chain of 100 links measured 22 yards. So, like a 'dozen' or 'score', 22 was a familiar number.
Besides, until you can measure and 'machine' accurately enough to need to compensate for temperature, 22/7 is 'close enough'...

Dishonourable mention for those infamous US politicians who, in 1897. nearly enacted Pi as three (integer 3) to ease calculation...
https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node18.html

 

[PeterDonis]

Dishonourable mention for those infamous US politicians who, in 1897. nearly enacted Pi as three (integer 3) to ease calculation...

I would have wanted to ask the legislators how they planned to deal with all the wheels in the state that weren't hexagonal...

 

[ohwilleke]

I'm trying to imagine a case where you need to know π in advance when building something and where 1/25 of a percent isn't good enough.

Machining motorcycle parts.

 

[Agent Smith]

I didn't ask with what. I asked how.

Provide a formula to compute the "resemblance of a given object to a circle".

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?

 

[WWGD]

You can walk through a valley $\cup$, over a hill $\cap$, or on flat land ____ . The details are as so often a bit more complicated.

And that's just for surfaces. Try dimensions 3 or higher.

 

[Nik_2213]

I'm told hexagonal wheels work okay given exactly the right profile of 'washboard' surface.
Vaguely akin to a toothed drive-belt and matching toothed pulley or, yes, a rack & pinion....

 

[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 $c/d = \pi$.