Agent Smith
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With ##\pi##A.T. said:How do you quantify that "resemblance"?
With ##\pi##A.T. said:How do you quantify that "resemblance"?
I didn't ask with what. I asked how.Agent Smith said:With ##\pi##
For an imperfect circle, there is no unique "diameter". So how do you compute the ratio of circumference to diameter?Agent Smith said:for a set of imperfect circles A = {x, y, z}, if x's circumference to diameter ratio is closest to ##\pi## it is more circular than either y or z, oui?
There is a class of figures with a unique diameter. Curves of constant width. The circle is one element of this class.PeterDonis said:For an imperfect circle, there is no unique "diameter". So how do you compute the ratio of circumference to diameter?
Sure, but only an exact circle. The post I responded to was talking about curves that aren't exact circles.jbriggs444 said:The circle is one element of this class.
Such as an n-gon (for odd ##n##) with each side rounded out to be a circular arc about the opposing vertex? Would that be an example of an "inexact circle" with constant width?PeterDonis said:Sure, but only an exact circle. The post I responded to was talking about curves that aren't exact circles.
Please read the subthread between myself and the OP. It makes clear what kind of "inexact circle" we were talking about.jbriggs444 said:Such as an n-gon (for odd ##n##) with each side rounded out to be a circular arc about the opposing vertex? Would that be an example of an "inexact circle" with constant width?
There is no such thing. Nor have I claimed that there is. Again, please read the subthread.jbriggs444 said:an "inexact circle" with constant width?
An "imperfect circle" is:PeterDonis said:There is no such thing. Nor have I claimed that there is. Again, please read the subthread.
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.Agent Smith said:An imperfect circle's points may vary in distance from the center.
You didn't go back through the whole subthread. (To be fair, the OP didn't link to previous posts in the subthread in the post you quoted.) See post #45:jbriggs444 said:An "imperfect circle" is
Agent Smith said:We won't ever be able to draw a perfect circle, ok. But for a set of imperfect circles A = {x, y, z}, if x's circumference to diameter ratio is closest to ##\pi## it is more circular than either y or z, oui?PeterDonis said:Which you will never be able to do perfectly. The compass point you're drawing with has a finite width, the compass itself has a finite amount of "play" in its spacing, and you have a finite error in your ability to keep the "center" point of the compass exactly in the same place. So no, nothing you draw this way will be an exact circle. Whether it is a good enough approximation to a circle will depend on what you are going to use it for.
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 said: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.
Why not constant width?PeterDonis said: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.
Because, as you note, the distance of the points on the curve from the centroid varies.jbriggs444 said:Why not 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".jbriggs444 said:There is a class of figures with a unique diameter. Curves of constant width.
I would have wanted to ask the legislators how they planned to deal with all the wheels in the state that weren't hexagonal...Nik_2213 said:Dishonourable mention for those infamous US politicians who, in 1897. nearly enacted Pi as three (integer 3) to ease calculation...
Machining motorcycle parts.Vanadium 50 said: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.
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?A.T. said:I didn't ask with what. I asked how.
Provide a formula to compute the "resemblance of a given object to a circle".
And that's just for surfaces. Try dimensions 3 or higher.fresh_42 said: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, if I may butt-in, in some less mathematically advanced parts of the world, they have only gotten this far...Nik_2213 said:I'm told hexagonal wheels work okay given exactly the right profile of 'washboard' surface.
But you can see they have eliminated the brakes, since they are not needed. Quite a weight-saving feature!Steve4Physics said:in some less mathematically advanced parts of the world, they have only gotten this far...
How does one measure the width of a square circle? Is it ##2r## or ##2 \sqrt{2} r##?Agent Smith said:Nescio, measure the perimeter and divide it by the "width".
There is no such thing as a polygon with infinitely many sides.Agent Smith said: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?
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.Agent Smith said:measure the perimeter and divide it by the "width"
The width of a circle is its diameter???PeterDonis said: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.
That tells us the width of a figure that is a circle.Agent Smith said:The width of a circle is its diameter???![]()
Nescio, what were Archimedes and Zu Chongzhi doing, approximating a cricle's circumference with polygons?jbriggs444 said: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.
Cogito, that's it!jbriggs444 said:limiting process
That is not responsive to a question about defining the "width" of a figure that is not a circle.Agent Smith said:Nescio, what were Archimedes and Zu Chongzhi doing, approximating a cricle's circumference with polygons?![]()
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.Agent Smith said:##\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##
Polygons do not have a unique width. How do you quantify the resemblance of a hexagon to a circle?Agent Smith said: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?
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. said:How do you quantify the resemblance of a hexagon to a circle?
That's just two of infinitely many possibilities for the width of a hexagon.PeterDonis said:It depends on whether the hexagon is inscribed in the circle or circumscribed around the circle.
Yes, but those two are the ones that are most relevant to comparing it to a circle.A.T. said:That's just two of infinitely many possibilities for the width of a hexagon.
Why? Why not the mean of the two or something else in between them?PeterDonis said:Yes, but those two are the ones that are most relevant to comparing it to a circle.
Because they are the two natural relationships between a circle and a regular polygon--inscribing and circumscribing.A.T. said:Why?
I can come up with such "natural relationships" all day: Circle cuts regular polygon sides in 3 equal parts feels natural to me.PeterDonis said:Because they are the two natural relationships between a circle and a regular polygon--inscribing and circumscribing.
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.A.T. said: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.
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.jbriggs444 said: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.
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##.A.T. said:Polygons do not have a unique width. How do you quantify the resemblance of a hexagon to a circle?
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 said:This is my favorite* geometric calculation of pi.
View attachment 350043
*favorite does not mean correct
Notice that this curve breaks your perimeter/diameter for circle quality construction.Agent Smith said: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.
Of course the proof is erroneous. We know that going in.Agent Smith said: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.
Oh, that's right, visually that is, but I did stress that the critical ratio has to approximate ##\pi##.Frabjous said:Notice that this curve breaks your perimeter/diameter for circle quality construction.
The points that matter to the approximation are on the curve (the points get closer and closer as the number of sides increase).jbriggs444 said:Almost none of the points in the circumscribed/inscribed polygons are on the curve either.
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.Agent Smith said:@jbriggs444 , the "perimeter of the limit"?
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.Agent Smith said: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 a polygon circumscribing a circle, some of the points always stand outside of the circle about which they are circumscribed.Agent Smith said: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 said:In the case of the staircase, some of the points always stand outside of the curve/line being approximated.
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.jbriggs444 said: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.
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).jbriggs444 said:That sequence of stairstep shapes approaches a limiting shape. The limiting shape is the circle, of course.
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 said:Well, what's the explanation for the error then?