It suddenly occurred to me while looking elsewhere that while we can be precise about the area of a square, we can’t be precise about the area of a circle. The reason, of course, is that we can not be precise about Pi. (At least I don’t think we can – doesn’t it just run on in a random string ad nauseum?)

This seems to say something significant about the very nature of a circle and it brought me back to a quarter of a century ago when I was playing with – and writing about – the computer language, Logo. AT that time I wrote a simple, recursive routine that drew a triangle, then a square, then a pentagon, hexagon, etc. Eventually the figure on the screen had so many sides any practical person would call it a circle. And that was a revelation to me because I then understood that a circle was simply another polygon, but with a heck of a lot of sides.

Now, 25 years later, I see how Pi fits into that concept, for the circle never becomes perfect and so pi – the ratio between the radius and the circumference – can never be defined.

I think?

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sophismata, on November 14, 2008 at 11:58 am said:“Can never be defined”?!?

Or is it that the method you chose only approximates asymptotically?

James, on November 14, 2008 at 12:20 pm said:Two thoughts:

1) We’ve calculated the value of pi out to enough digits that it would exceed the precision of our measuring instruments for most everyday applications and most, if not all, extraordinary applications. Comparing it to the square, if you wanted a given area of the square, you would have to measure out a certain size for each side. While the calculation would be precise, your ability to measure precisely would be a separate limitation. I believe this limitation would be much greater than than the lack of precision in any calculation involving pi.

2) Ignoring the question of the precision of measuring instruments, you probably could devise a way to be precise about the measurement of the area of a circle. Here’s one off the top of my head: determine the formula for the relationship between some mass of oil and the area that oil covers as it spreads out over water. Assuming you have an oil that spreads out to a known molecular thickness (let’s say 1 molecule thick) you could know how much area a given mass of oil would cover, and the mass to area ratio would not involve pi. The area would naturally form a circle, and there you would have your precise circle of area. However, your precision in area would cost you precision in knowing the diameter of that circle.

So, you could have one or the other but not both.

Still, your real limitation would be because of the first point — measurement limitations. Except perhaps in astronomical-scale calculations.

Maggie, on November 14, 2008 at 1:10 pm said:My feelings ran along a similar line to James’, although I didn’t think of oil 1 molecule thick, LOL, (I did think of calculating the amount of paint necessary to paint a circle) but rather that the precision to which we’ve calculated pi is absurdly precise compared to any precision we actually need for working with circles. I believe, I’m happy to have somebody correct me.

But, for example, precision in a simple C program using just doubles, not even long doubles, is 10^-37 to 10^37, with 10^-37 being a factor of a billionth off from the mass of an electron.

I think irrational numbers are somewhat disturbing. They are infinite and uncountable, most real numbers are irrational. Just take pi and change one digit and you have a whole new irrational number.

Take a 45-45-90 triangle and the ratio of the sides to the hypotenuse also involves an irrational number – 2^1/2. So there is something else we can’t define precisely.

But then, the rational numbers aren’t always that great, either. Take 2. I have two apples. That’s countable, but they’re not the same, so what does it really mean? ๐

I’m going running now so that’s a time to think about your actual point, which was relating pi to the idea that a circle is a polygon with infinite sides. I don’t know that I’ll think anything intelligent, but I promise to think about it.

Greg Stone, on November 14, 2008 at 1:24 pm said:I agree completely that the lack of precision is meaningless from a practical stand point – but doesn’t it say something about the fundamental nature of the circle vs. the square? I’m not sure what – and when you start talking irrational numbers, well you get in way beyond my depth. This is where I really start to regret my lack of understanding of math beyond high school algebra and trig. But what you said about two apples really intrigues me.

Greg Stone, on November 14, 2008 at 1:57 pm said:James wrote: “Assuming you have an oil that spreads out to a known molecular thickness (letโs say 1 molecule thick) you could know how much area a given mass of oil would cover, and the mass to area ratio would not involve pi. The area would naturally form a circle, and there you would have your precise circle of area. However, your precision in area would cost you precision in knowing the diameter of that circle.”

James , you’ve lost me here. Why would the area covered by oil naturally form a circle? I know a sphere is an ideal shape in natre, but my practical experience tells me that when I spill oil the shape it takes is quite irregular? And I don’t understand why this would impact our knowing the diameter with precision? I’m not challenging these ideas – just trying to understand them?

But in the final analysis I guess it seems to me that the perfection of the circle is an illusion. All circles are simply many-sided polygons and so at some level of examination have a bad case of the jaggies, And I am assuming – and I will grant this is a big assumption – that having a case of the jaggies relates to the imprecision of pi. Now that’s a real leap – but it is that thought -and whether or not this sets the circle aside as a different in a meaningful way,that got me musing down this path. ๐

Maggie, on November 14, 2008 at 2:33 pm said:Well isn’t a circle an abstract concept? If we’re talking about something that is in the shape of a circle, doesn’t it eventually have the jaggies, even if we have to go atomic? But an actual circle — does that have the jaggies? I don’t think so. I confess my mind went completely elsewhere when I was running. That happens sometimes. Sometimes I can focus on a problem, and sometimes it goes off in its own direction. But do triangles bother you? Because there’s that 45-45-90, which has a hypotenuse that is 2^1/2 times the size of the other sides, an irrational number.

Greg Stone, on November 14, 2008 at 2:51 pm said:Maggie wrote: “Well isnโt a circle an abstract concept?”

Ehhh . . . is that what I’m saying? That the circle exists only in the mind? Maybe so. I’m not sure what difference this makes. Help me out. And I guess my question, then, is a square any different?

Just getting back to this business of the area and the simple math I can handle, we can calculate the area of a square with apparent precision. We can’t do so with a circle – at least not by the normal formula that uses pi. It’s not that I’m interested in calculating the area – but I wonder if there’s a relationship between pi being an irrational number and what I see as the inherent imprecision of a circle. And frankly, if I get the answer I’m not sure what it does for me. The revelation a quarter century ago that the circle was a many-sided polygon was real eureka moment – but in the end it didn’t seem to change my life. It just sure seemed important at the time. ๐

Hey – this is another interface and the comment type here is huge!

sophismata, on November 14, 2008 at 8:24 pm said:” All circles are simply many-sided polygons ”

That is not true. ๐

As you pointed out earlier, a many-sided polygon *seems* to look like a circle, but it isnt. It’s just an approximation. The more sides, the more it looks like a circle; but it still isn’t a circle.

You are defining an object X via an approximation method, so your definition will have to be limited. But a circle is a well-defined, non-abstract figure: the set of points that are at a certain distance from another defined point.

James, on November 14, 2008 at 8:27 pm said:Greg wrote:” Why would the area covered by oil naturally form a circle?”

The forces of cohesion between the oil molecules should tend to form a circle in the same way a bubble forms a sphere. Assuming a frictionless water surface. ๐

Even though it’s an imaginary example, it could be made to work practically, at least for small circles. But then it’s the relationship between the diameter and the area that depends on pi; know one and you need pi to find the other.

You could say that it’s not a true circle because, as Maggie points out, circles are imaginary.

So, you could, of course, just

sayyou know the area of a circle. And then, by definition, you know it. And now you have to calculate the diameter.But I’m just being picky about your comment re: area of circles. You were really talking about the relationship, and it is the relationship that is interesting. I was just trying to point out that it’s not the area itself that is special.

Maggie’s triangle example is another excellent example; again, it’s the relationship. You could make a square with a diagonal of known length, but then the sides of the square would be irrational because (again) it’s the relationship between the sides and the diagonal, not anything inherent in the diagonal itself.

Greg Stone, on November 14, 2008 at 8:44 pm said:Sophismata wrote:”You are defining an object X via an approximation method, so your definition will have to be limited. But a circle is a well-defined, non-abstract figure: the set of points that are at a certain distance from another defined point.”

Aha! Thanks. I did not know that definition, but it makes sense. You’re right, my many-sided polygon is an approximation of a circle. A point, I assume, has no dimensions. So a circle is really all the points that are a given distance from another, single point? Being a point rather than an infinitely small – but of some dimension-line, it avoids having the jaggies.

Rohedi, on March 24, 2009 at 8:44 am said:“It suddenly occurred to me while looking elsewhere that while we can be precise about the area of a square, we canโt be precise about the area of a circle.”

Oh this statement similar to the Heisenberg uncertainty in Physics, that when we can be precise about the time travel of particle, we can’t be precise about it’s position. Nice posting.

Rohedi, on March 24, 2009 at 4:48 pm said:I don’t understand why about 2500 years after Archimedes finding of the exact pi number is still regarded difficult, and mathematicians agree 3/14 is celebrated as pi day because they still assume the pi as mysterious number. Even the pi day coincedentally with the birthday of Einstein that we know he was the person at 20st century. I think the celebration of pi day has under estimated Einstein why he didn’t complele his reputable with analytic exact formula for the pi number. Next, If the cause of irrational number for pi is the square root of 2 is correct, this is no problem because our student can calculate with calandra method, even they can teach them numerical method when the exact pi number contains cubic root or higher order of root. I would like to say that this world needs a simple analytic exact formula of pi not pie cake.

Rohedi, on May 2, 2009 at 6:20 pm said:Well, now Rohedi invites you all to look a simple exact formula of Pi number at http://eqworld.ipmnet.ru/forum/viewtopic.php?f=3&t=148. Oh yeah, if you would know Pi(Phi) that is the expression of Pi formula as function Phi Golden Ratio please click Rohedi’s address.