PDA

View Full Version : Is Pi r squared away?

virtually yours
11-23-2001, 11:12 AM
Hi,
How accurate really is the pi formula does it stand up to great distance travel (as in space/great orbits) or do some corrections have to be made?
Are there any other theories to find out radius,etc. of a circle,if so what are they?
Virtually yours

TwistofFate
11-23-2001, 11:17 AM
In Pi calculations, the more digits for Pi you use, the more accurate your answer.

I'm sure there are some crackpot theories out there, and possibly some accurate ones too, but they probably use a constant that performs the same function as Pi.

Interesting question, but I'm not sure if there is.

Gartog
11-23-2001, 11:28 AM
I belive the 'pi formula' is accurate to a perfect degree, if you use a infinatly value of pi.

I used to know the proof for this, but don't at the moment and I don't have a copy with me. I'm sure someone will provide one if asked however.

Orbits in space are never exactly circular as they are affected by many gravitational sources. A circle is an abstract concept that can not be created IRL although it can be approximated to a high degree of accuracy.

Patty O'Furniture
11-23-2001, 11:44 AM
Originally posted by virtually yours:

Is Pi r Squared...

Pi r not square, pi r round. Cake r square.

11-23-2001, 12:05 PM
Originally posted by virtually yours
the pi formula
I can't, and frankly don't see how anybody else can, answer your question without knowing what "pi formula" you're talking about. The only thing you mention in the OP is the radius of a circle, but that formula does not involve pi (divide the diameter by 2). Are asking about the formula for determining the area of a circle (hence the "pi*r2" reference)? Or are you just asking how accurately we can compute pi for use in solving equations?

:confused:

BTW, Attrayant, this habit of yours of beating me to the punch is seriously getting on my nerves. ;)

ultrafilter
11-23-2001, 12:05 PM
Originally posted by Gartog
I used to know the proof for this, but don't at the moment and I don't have a copy with me. I'm sure someone will provide one if asked however.

Just for simplicity, let's assume that all of our circles are centered at the origin. If they aren't, we'll go over to a coordinate system where the origin is at the center of our circle.

So we've got a point on a circle of radius r. What are its (x, y) coordinates? Well, since we're using the standard distance formula here, we know that x2 + y2 = r2. We'll solve for y. This gets us that y = sqrt(r2 - x2), or y = -sqrt(r2 - x2). If you plot these equations, you'll find that the first gives you the semicircle above the x-axis, and the second gives you the semicircle below the x-axis. These are disjoint shapes; therefore, the area of our circle is the sum of the areas enclosed by each semicircle and the x-axis.

So the area of our circle is the integral over [-r, r] of sqrt(r2 - x2) minus the integral over [-r, r] of -sqrt(r2 - x2). This works out to be twice the integral of sqrt(r2 - x2) on [-r, r]. You do your trigonometric substitution, you integrate, and you get that the area of the circle is pr2.

Of course, the OP is probably asking about the error in using a finitely-long approximation to p. Well, in that case, you're calculating (p - e)r2 for some small positive real number e. The error is er2, which is proportional to r2. As you might expect, the closer your approximation is, the smaller the error--however, there is no one value of e which will get you a reasonable approximation for very large radii.

lucwarm
11-23-2001, 12:10 PM
The "pi formula" is exact within Euclidean Geometry - It follows from the postulates, axioms, etc. of that system.

So perhaps a more general way to ask the question is . . . how good of a model of the Real World is Euclidean Geometry?

By all accounts, it's pretty good -- but, I doubt that it's perfect.

sailor
11-23-2001, 12:16 PM
The answer is that the area of a circle is equal *exactly* to PI times the radius squared. The fact that you may have practical limitations in measuring the radius, or that you are using a limited number of decimals for PI does not change that. If you establish units such that the radius equals sqrt(1/pi) then the area of the circle is 1 .

PhiloVance
11-23-2001, 12:19 PM
At first I thought this was Superman's new call:

Pi R squared, aaawwwwaaaaayyyy. :D

erislover
11-23-2001, 12:30 PM
There are all sorts of series which converge on pi as [n][/i] approaches infinity. Numerical analysis can tell you how far away you are from the "real" number with your finite approximation. Rest assured that we can calculate pi to a much greater degree of accuracy than is necessary for space travel.

Actually, pi isn't the problem, because that number can be said to be a constant. The real problem is how accurately we can measure the properties (accelleration properties, mass, etc) of our spaceship!

bibliophage
11-23-2001, 12:53 PM
Originally posted by lucwarm
The "pi formula" is exact within Euclidean Geometry - It follows from the postulates, axioms, etc. of that system.

So perhaps a more general way to ask the question is . . . how good of a model of the Real World is Euclidean Geometry?

By all accounts, it's pretty good -- but, I doubt that it's perfect. Let me expand on this. In perfectly flat spacetime, Euclidean geometry is a fact, so that C=2pr. In our region of spacetime, it is merely an exceptionally close approximation, so that C»2pr (the difference being too small to measure). But Euclidean geometry is not a good approximation everywhere. In the vicinity of a black hole, C<2pr.

DPWhite
11-23-2001, 12:55 PM
Orbits are not circular. IIRC Kepler's laws correctly, they are ellipses, which are only circles in special instances. (An ellipse can be drawn by putting two pins for each locus (loci plural) and connecting them with a string, then drawing with a pencil as far away as the sting lets you around the loci. They sweep out equal areas in equal times when they are in orbits: this means that they go faster when closer to the object they are orbiting. IIRC the earth has only a very small eccentricity of about 5% so the orbit is almost a circle. Maybe someone else can share with us the stuff about the progression of the eccentricity of the orbit of Mercury, which is so close to the space time distortion of the sun that relativity has an effect on it. Ellipses are fun!

11-23-2001, 02:00 PM
Given the level to which this discussion has been elevated, it goes without saying that I stand corrected.

Scott Dickerson
11-23-2001, 03:21 PM
"How accurate really is the pi formula does it stand up to great distance travel (as in space/great orbits) or do some corrections have to be made? Are there any other theories to find out radius,etc. of a circle,if so what are they?"

The various replies, interesting and well-informed though they might be, don't seem to me to be addressing the OP's question. It's a sophisticated one.

Note first that he is specifically interested in Pi as it appears in formulae involving the circle.

In this regard, Pi can be defined as the ratio between the circumference of a circle and its diameter. Treating this as a division problem, it is a quotient: Pi is "how many times the diameter's length goes into the circle's length."

The deep question is whether this number can be determined by some process that does not require empirical measurement at some stage or other. To make the contrast clear: the perimeter of a square is precisely four times the length of one side, because part of the definition of the word "square" is that it have four sides of equal length. By contrast, the measured circumference of a circle plays no role in the definition of "circle"--it's defined in terms of equality of radii. All attempts to derive the relation between some radius and some part of the circumference (such as the use of inscribed and circumscribed regular polygons) requires some prior derivation of e.g. the ratio of the hypoteneuse of a right triangle to a side. And again, what that ratio is numerically is an empirical fact to be discovered, not a matter of "true by definition."

So I think the general question is this. To what extent can we justify our confidence that conclusions based upon empirical findings under one set of circumstances (on the surface of a certain planet, over a certain set of lengths) can be extrapolated to a universal law? Precisely how do we know, for sure and certain, that the Pi ratio has the same numeric value in Andromeda as it does in Albany?

We don't.

ultrafilter
11-23-2001, 03:32 PM
Originally posted by Scott Dickerson
The deep question is whether this number can be determined by some process that does not require empirical measurement at some stage or other.

Yep, and it's easy, too. The sum from over n from 1 to infinity of n-2 is equal to p2/6. So the square root of 6 times this quantity is p, and that's independent of any measurement you might want to take. There are other, simpler expressions for p, but that's the first one I could remember.

FWIW, p is defined as the ratio between the circumference and diameter of a circle in Euclidean space. This is guaranteed to be constant. However, there are parts of the universe in which the Euclidean approximation is not a good one, but we can still make computations there based on the models we have, and we can even test those models by comparing our computations to measurements of some physical quantities.

g8rguy
11-23-2001, 09:39 PM
Originally posted by DPWhite
Orbits are not circular. IIRC Kepler's laws correctly, they are ellipses, which are only circles in special instances. (An ellipse can be drawn by putting two pins for each locus (loci plural) and connecting them with a string, then drawing with a pencil as far away as the sting lets you around the loci. They sweep out equal areas in equal times when they are in orbits: this means that they go faster when closer to the object they are orbiting. IIRC the earth has only a very small eccentricity of about 5% so the orbit is almost a circle. Maybe someone else can share with us the stuff about the progression of the eccentricity of the orbit of Mercury, which is so close to the space time distortion of the sun that relativity has an effect on it. Ellipses are fun!

Yes, ellipses are grand, but we just replace pr2 by pab, and away we go! Yay p!

More fun facts to share...

Circular orbits also of course sweep out equal areas in equal times; this is just conservation of angular momentum, and holds for any orbit around a radial potential, whether that's gravitational or some other form.

Way back when, people noticed that the perihelion of Mercury precessed around the sun (in other words, the spot where Mercury was closest to the sun in its orbit changed each time it went around). Most of this can be explained by the influence of the other planets (especially Jupiter), but there was a remaining small amount (I don't remember exactly how much, but in the tens of arcseconds per century, which is pretty tiny) that couldn't be explained. Einstein eventually solved this problem by invoking GR, and pretty much nailed the answer; this was one of the first signs that GR is right. And now you know (if you didn't already).

Boyo Jim
11-23-2001, 09:54 PM
I think what you're asking if there is some way to "measure" a theoretical astronomically large circle without using pi.

A perfect circle is a human construct, and is defined in terms of pi. So a perfect measurement of it necessarily involves pi. If a two dimensional area or circumference CAN be measured more accurately another way, then it is not a circle. Measuring the area of a circle can be done in other ways, for instance dividing up the area into many small squares and adding up the areas of the squares, but this involves approximations around the edge of the circle, and is inherently less accurate.

virtually yours
11-24-2001, 01:36 AM
Originally posted by yojimboguy
I think what you're asking if there is some way to "measure" a theoretical astronomically large circle without using pi.

A perfect circle is a human construct, and is defined in terms of pi. So a perfect measurement of it necessarily involves pi. If a two dimensional area or circumference CAN be measured more accurately another way, then it is not a circle. Measuring the area of a circle can be done in other ways, for instance dividing up the area into many small squares and adding up the areas of the squares, but this involves approximations around the edge of the circle, and is inherently less accurate.
Yes but to my understanding that is what pi does also

Boyo Jim
11-24-2001, 07:24 AM
Measuring the area of a circle can be done in other ways, for instance dividing up the area into many small squares and adding up the areas of the squares, but this involves approximations around the edge of the circle, and is inherently less accurate. [/B]
Yes but to my understanding that is what pi does also [/B][/QUOTE]

No, pi (the symbol)is exact, though the number 3.1416... is an approximation. So it is correct to say the circumference of a circle is EXACTLY 2 x pi x radius, but
2 x 3.1416 x radius is not exact.

But remember that no one has ever seen a perfect circle. Even a drawing done with a protractor is an approximation because of the width of the pencil (among other things). If you and I measure a drawn circle with a tape measure, we may get slightly different results. You might measure the line along the interior length, I might do it by the exterior length. Even the width of the line might vary as the pencil lead wears down or ink flow varies. In geometry, the width of the curve constituting the circle is Zero.

In fact, virtually EVERY measurement taken in the real world is an approximation, because the object is measured only to the limits of our measuring devices. There is some unquantifiable amount more or less than our smallest measuring increment that we must approximate or ignore. So it's fair to same that the same measuring issues exist with squares, triangles or any other geometric shape you care to name. By picking a shape, you must accept an abstract definition of that shape is in order to discuss it.

So if you draw a six inch line on a piece of paper, is it really six inches? You and everyone else can put it under a microscope to measure it more accurately, or you can write "6 inches" as a label, so that number will be accepted and used by others and they don't have to spend all day doing their own measuring. Pi serves the same function. It will give you exactitude for a theorectical circle, and a high degree of accuracy for a real-world circle, but the real-world circle only approximates being a circle by mathematical definition.

The other way to look at it is: if you see something in the real world that LOOKS like a circle, you find out how circular it really is by comparing your measurements of it to a theoretical circle. The closer your circle's area and circumference measurements are to what the formulas using pi say they should be, the more "circular" your circle is.

DSeid
11-24-2001, 10:39 AM
The interested reader is referred to "A History of Pi" by Petr Beckmann.

As he points out, the digits of Pi beyond the first several are of no practical scientific or engineering value, 10 decimal places would be enough to calculate the circumference of the earth to within a fraction of an inch, if it was a perfect sphere. 17 is enough to prevent rounding off errors in rigorous computer applications.

Much of science is inferential. We observe and make tentative generalizations based on repeated observations. Math is mainly deductive. We accept a few postulates and given those postulates we see what follows. (I state "mainly" in reference to the Monte Carlo method.) Pi is always the correct constant if Euclidian postulates hold. Accept a different set of posulates and other deductions follow. What postulates best describe our universe and at what level of analyses? That's not math, it's physics and cosmology.

ultrafilter
11-24-2001, 11:46 AM
Originally posted by DSeid
Math is mainly deductive. We accept a few postulates and given those postulates we see what follows. (I state "mainly" in reference to the Monte Carlo method.)

I seriously doubt that anything other than deductive reasoning has been accepted as a method for theorem proving. If you're talking about applications, then yes, the Monte Carlo methods are very important for finding approximate solutions. Care to elaborate?

DSeid
11-24-2001, 10:23 PM
I am not a mathematician; just a pediatrician who tries to reduce his personal ignorance load when he can. As such, I am dependent on my sources.

"The History of Pi" is the source for that statement, and note that I said "math" not "theorum proving".

In that book, the Monte Carlo method is characterized as being used to solve a wide range of problems,
...suppose we wish to find a complicated multiple integral ... let the computer "shoot" n-tuplets of random numbers. These represent a coordinate in (n-dimensional) space and the coordinate either lies in the volume determined by the integral ("hit") or it does not ("miss"). Then we let the computer shoot at the target, say, half a million times. The number of hits is then proportional to the n-tuple interval.

Beckmann also states that many theorums were used on the basis of experimental evidence long before being proved. He uses Heaviside and "operational calculus", later explained by an integral transform and the delta function really being a distribution.

But not for proof. And that's as good of an elaboration as I am capable of.

ultrafilter
11-25-2001, 12:38 PM
Originally posted by DSeid
"The History of Pi" is the source for that statement, and note that I said "math" not "theorum proving".

The two terms are more or less interchangeable. Even solving a simple equation is proving a theorem.

However, for applications, less rigor may be acceptable at times.

In that book, the Monte Carlo method is characterized as being used to solve a wide range of problems,
...suppose we wish to find a complicated multiple integral ... let the computer "shoot" n-tuplets of random numbers. These represent a coordinate in (n-dimensional) space and the coordinate either lies in the volume determined by the integral ("hit") or it does not ("miss"). Then we let the computer shoot at the target, say, half a million times. The number of hits is then proportional to the n-tuple interval.

Beckmann also states that many theorums were used on the basis of experimental evidence long before being proved. He uses Heaviside and "operational calculus", later explained by an integral transform and the delta function really being a distribution.

OK, this does sound more like my understanding.

Cyberhwk
11-26-2001, 01:36 AM
Actually someone corrected Cecil here (http://www.straightdope.com/classics/a3_357.html) by saying...
"Knowing pi to 39 decimal places would nearly suffice for computing the circumference of a circle enclosing the known universe with an error no greater than the nucleus of a hydrogen atom"

:eek::eek::eek::eek::eek:

virtually yours
11-29-2001, 10:09 PM
"Knowing pi to 39 decimal places would nearly suffice for computing the circumference of a circle enclosing the known universe with an error no greater than the nucleus of a hydrogen atom"

Hey that's more like it!! that really answers my question.
Ok ie the cosine law ,does that have the same limtitations of pi or is that more "exact" without a bunch of decimal points?I'm asking cause I trying to work something out.
I mean specifically the way that the sine/cosines are determined.as opposed to the way pi is used

ultrafilter
11-29-2001, 10:20 PM
Sines and cosines can be determined very accurately through use of their Taylor series. These series converge fairly quickly, and they are what people use to calculate the trigonometric functions.

RM Mentock
11-30-2001, 06:34 AM
Originally posted by Cyberhwk
Actually someone corrected Cecil here (http://www.straightdope.com/classics/a3_357.html) by saying...

Cecil corrected? Absurd.

I don't know why Cecil hasn't corrected that "correction." Dr. Neil Basescu made a silly error of his own. Notice that he says "a reasonable value for the radius of the universe is 2 x 10^34 angstroms. That's just 20 billion years (the time since the big bang) times the speed of light (the upper limit on the rate of expansion)."

Here's the math:

2 x 10^10 year times 3.2 x 10^7 sec/year times 3 x 10^8 meter/sec times 10^10 angstroms/meter = 1.92 x 10^36 angstroms, which is a hundred times the value that Basescu calculated.

See this thread for the original discussion of that issue. (http://boards.straightdope.com/sdmb/showthread.php?threadid=328) With reasonable assumptions, Cecil was right in the first place.

virtually yours

I'm not sure this is relevant to your inquiry now, but it seems relevant to the OP. Two hundred years ago, Gauss actually led a survey to study whether the sum of the angles of large triangles add up to 180 degrees. He triangulated some mountain tops. Had they not added up to 180, of course, then pi r squared would have been off also. He didn't find any discrepancy. And was criticized for a waste of time and money.

DSeid
11-30-2001, 10:17 AM
RM,

Of course if that study had been done accurately enough, with mountains seperated far enough, the angles would not have added up to 180 degrees. The surface of the Earth is not a Euclidian plane.

RM Mentock
11-30-2001, 01:32 PM
Originally posted by DSeid
Of course if that study had been done accurately enough, with mountains separated far enough, the angles would not have added up to 180 degrees. The surface of the Earth is not a Euclidian plane.

Sorry, couldn't resist. Gauss was no fool, he knew that. But he was using line-of-sight measurements, which would not be influenced by the curvature of the earth's surface. When you think about it that way, Gauss was probably the canonical no-fool.