Pi

[Guadalupe]

Based on my scientific research, I discovered an interesting fact about the method used in finding Pi.

History shows that geometry was the method used in finding Pi. This method is called, “Method of Exhaustion”. Geometry was and is today, the only method used to measure the inside of the line that defines the circumference of a perfect circle, giving us 3.14159265… as Pi.

The word circumference means, the outside surface of a circle or sphere and not the inside surface of a circle or sphere.

The word diameter means, a straight line passing through the center of a circle or sphere and meeting the circumference or outside surface at each end. It doesn’t mean, a straight line passing through the center of a circle or sphere and meeting in the inside surface at each end.

Based on this wealth of information and deductive reasoning I was able to see and understand why the use of geometry as the method for finding Pi was and is indeed in error.

It is my contention that with today’s modern technology, we’re able to draw and precisely measure the circumference or outside surface of a perfect circle to find a new Pi. A Pi that is finite.



Copyright 2008 by Guadalupe Guerra

[Moontanman]

Originally Posted by Guadalupe

Based on my scientific research, I discovered an interesting fact about the method used in finding Pi.

History shows that geometry was the method used in finding Pi. This method is called, “Method of Exhaustion”. Geometry was and is today, the only method used to measure the inside of the line that defines the circumference of a perfect circle, giving us 3.14159265… as Pi.

The word circumference means, the outside surface of a circle or sphere and not the inside surface of a circle or sphere.

The word diameter means, a straight line passing through the center of a circle or sphere and meeting the circumference or outside surface at each end. It doesn’t mean, a straight line passing through the center of a circle or sphere and meeting in the inside surface at each end.

Based on this wealth of information and deductive reasoning I was able to see and understand why the use of geometry as the method for finding Pi was and is indeed in error.

It is my contention that with today’s modern technology, we’re able to draw and precisely measure the circumference or outside surface of a perfect circle to find a new Pi. A Pi that is finite.

Well, since mathematically a line has only one dimension and that is length the inside and outside of a circle should be the same measurement, Correct?

[freeztar]

Originally Posted by Guadalupe

It is my contention that with today’s modern technology, we’re able to draw and precisely measure the circumference or outside surface of a perfect circle to find a new Pi. A Pi that is finite.

Ok, what have you got?

[modest]

Originally Posted by Guadalupe

Based on my scientific research, I discovered an interesting fact about the method used in finding Pi.

History shows that geometry was the method used in finding Pi. This method is called, “Method of Exhaustion”. Geometry was and is today, the only method used to measure the inside of the line that defines the circumference of a perfect circle, giving us 3.14159265… as Pi.

You are confusing one method of finding pi with the value itself. In terms of geometry, you are confusing the circumfrance of the two polygons with the circumfrance of the circle. The polygons are a bound for pi, and both the inside and the outside polygons bound it.

There's a good source explaining here.

Archimedies used a 96 sided polygon. The polygon outside the circle measures 223/71 and the inside polygon is 22/7. This means pi cannot be greater than 223/71 nor smaller than 22/7. This does not mean pi itself has 2 different values.

The rest of your post extrapolates from this mistake.

~modest

[KALSTER]

Irrespective of Modest’s correct explanation, Moontanman has given all that is necessary to appreciate the error of Guadalupe’s assertions. Whether by using the limit of the circumference of a polygon that tends towards having infinite sides on either the inside or the outside, the resultant value for pi could be worked out to any arbitrary precision in both cases with values that come nearer to each other the more precisely they are worked out. (Or so I think. Using limits might actually work it out to exact precision, I can't remember this stuff)

[C1ay]

Originally Posted by Guadalupe

History shows that geometry was the method used in finding Pi. This method is called, “Method of Exhaustion”. Geometry was and is today, the only method used to measure the inside of the line that defines the circumference of a perfect circle, giving us 3.14159265… as Pi.

The word circumference means, the outside surface of a circle or sphere and not the inside surface of a circle or sphere.

A circle is simply a set of points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. It has no inside and outside circumference, only a single circumference.

[Guadalupe]

We cannot use the diameter in the equation when measuring the inside of a circle or sphere. Otherwise, the definition of the word, “diameter” would be meaningless.

Only by measuring the circumference of a perfect circle can we use the diameter in the equation to find Pi.

I understand that somethime change is not always easy but it’s important to find the right equation in solving a problem.

[C1ay]

Originally Posted by Guadalupe

We cannot use the diameter in the equation when measuring the inside of a circle or sphere. Otherwise, the definition of the word, “diameter” would be meaningless.

You're missing the point. The set of points that make up a circle does not have an inside or an outside diameter, it has only A diameter that runs through the center of those points. PI, by definition, is the ratio of the circumference of a circle to its diameter and any given circle has only one circumference and only one diameter.

[CraigD]

MoonTanMan and Modest have done a good job, I think, of explaining the misunderstanding of misunderstanding of geometric methods of estimating Pi such as Archimedes’s regular 96-sided bounding and bound polygons that I think leads Guadalupe to conclude that the present day accepted value of Pi is wrong. In short, the claim

Originally Posted by Guadalupe

History shows that geometry was the method used in finding Pi. This method is called, “Method of Exhaustion”.

is correct.

Although very early (eg: 1900 BC) published values of Pi lacked explanation of the methods used to find them, I know of no evidence that any Pi-approximating techniques other than ones similar to Archimedes’s were known from about 200 BC through about 1400 AD. Variations Liu Hui's algorithm (ca: 263 AD) appears to have been the best and latest used of these techniques.

However, the claim

Originally Posted by Guadalupe

Geometry was and is today, the only method used to measure the inside of the line that defines the circumference of a perfect circle, giving us 3.14159265… as Pi.

is incorrect.

From the mid 1600s (from around 1400 in India, but communication between Indian and European mathematicians resulted in many well-known results of the former being independently rediscovered centuries later by the latter), the best and most used methods of estimating Pi involved arithmetic series. The earliest of these is, I think:

, known as the Leibniz formula, but undisputedly know 300 years earlier by Mādhava of Sangamagrama.

Since then, many other Pi-approximating arithmetic series have been discovered, many much more computationally efficient (that is, they require fewer arithmetic operation to give more precise approximation of Pi) than early ones. Rabinovitz and Wagon’s 1995 spigot algorithm is a well known example.

An important idea to be gleaned from this history is that Pi is not only of geometric significance, but has a sort of deep relationship with numbers of any kind. Whenever one “plays with arithmetic”, one discovers infinite series with values equal to or integer multiples of Pi or powers of Pi, and other useful irrational constants, such as e.

PS: As a matter or terminology, the phrase A Pi that is finite doesn’t make sense. Pi is finite: it’s greater than 3 and less than 4. A number being irrational is not the same as it being infinite.

[TheBigDog]

Guadalupe! Long time no post. I have missed you and your quest for Pi. This is indeed a new angle (pun intended), but the fact remains that we understand Pi pretty damn well. We have gone so far as to show that only so many digits of precision are needed (think we settled on 14) for even the most demanding accuracy. I think Craig showed how moving from 13 to 14 digits of Pi in measuring the diameter of the Universe would only improve the accuracy of the measure by less than ten meters. I would need to do some searching to find the post. So adjusting Pi to an even more accurate (and precisely expressible) number would be novel, but serve no practical purpose to science.

Good seeing you again.

Bill