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Thread: pi 3.14...

  1. #1 pi 3.14... 
    Forum Sophomore numb3rs's Avatar
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    calculate pi as far back as you can go
    3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
    58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
    82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
    48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
    44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
    45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
    72458 70066 06315 58817 48815 209
    u can use a calculater but no online cheating. :x


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  3. #2 Re: pi 3.14... 
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    Quote Originally Posted by numb3rs
    calculate pi as far back as you can go
    3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
    58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
    82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
    48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
    44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
    45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
    72458 70066 06315 58817 48815 209
    u can use a calculater but no online cheating. :x
    that's a full PI? I thought it was infinite.


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  4. #3  
    Forum Professor serpicojr's Avatar
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    The decimal expansion of π does go on forever without repeating.
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  5. #4  
    Forum Junior DivideByZero's Avatar
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    if a decimals 0.abcdef... goes of forever will it be rational or irrational?
    (assuming a, b, c, d, etc are completly random numbers from 0 to 9)
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  6. #5  
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    Quote Originally Posted by DivideByZero
    if a decimals 0.abcdef... goes of forever will it be rational or irrational?
    (assuming a, b, c, d, etc are completly random numbers from 0 to 9)
    If the numbers are completely random, it will be considered irrational.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  7. #6  
    Forum Professor serpicojr's Avatar
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    To be a little more precise, if the digits eventually repeat, then the number is rational. By eventually repeat, I mean something along the lines of:

    1/6 = 0.1666...

    where the 6 repeats forever. More generally, I'm talking about expansions like:

    0.a<sub>1</sub>...a<sub>n</sub>b<sub>1</sub>...b<sub>m</sub>b<sub>1</sub>...b<sub>m</sub>b<sub>1</sub>...b<sub>m</sub>...

    where the string b<sub>1</sub>...b<sub>m</sub> repeats forever. This includes the case when the decimal expansion terminates, as you can think of a terminating expansion as ending with an infinite number of 0's. You can also show that any terminating decimal expansion is equal to a decimal expansion with 9's repeating forever.

    If the above does not happen, then you have an irrational number. So we can construct irrational numbers using this fact, for example:

    0.101001000100001000001000000100000001...

    where I keep sticking one extra 0 between successive 1's.

    (This whole discussion is independent of the base B we're using, except replace 9 by B-1 in the statement about terminating decimal expansions and, of course, 1/6 is not equal to that expansion when B ≠ 10.)
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  8. #7 Re: pi 3.14... 
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    Quote Originally Posted by numb3rs
    calculate pi as far back as you can go
    3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
    58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
    82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
    48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
    44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
    45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
    72458 70066 06315 58817 48815 209
    u can use a calculater but no online cheating. :x
    One thing that bothers me, is what they are doing with a number, and calling it pi. Years ago in my school they took pi very seriously, in a good way.

    Pi was the lifetime work of a man called Archimedes. He was stated to have created a very large marble wheel, and then a very long marble path for it to roll upon. He spent years polishing and honing his wheel and path.

    When he was done, he had a ratio that was just larger then 22/7. He wanted perfection, however at the time without calculators, he could not imagine anyone being interested in his ratio, if it was a complex fraction. So he decided after years and years of working on the problem to call pi 22/7.

    But today I hear individuals claim that pi is 3.1459......, the truth is that pi is actually larger then 22/7, in real life.

    By actual building and testing of a precision wheel, I found many phenomena that are overlooked, by people claiming to be experts, or claiming to be the authority on pi.

    This all started because my family worked for an Aero Space contractor, that also knew pi was larger then 3.14159, but felt that at the time when we were just getting over a war with Vietnam, that this point would not be exciting enough to start a debate.

    Years later on the Internet I mentioned it, and someone dared me to make a wheel and measure it. Even though I had pointed out, that the methods they had used to measure a wheel were obviously flawed.

    They had used a tape measure, but a band contracts and stretches as it is bent around a round object. So that the actual point the band measures, is at the diameter of the wheel being measured, plus one thickness of the metal tape used to measure it. Or half a thickness on either side.

    I do this kind of work so it is common knowledge to me.
    What I found by making the wheel and rolling it, was truly incredible.

    I had just made a very nice pass with the cobalt cutter, and decided to take the wheel off the lathe and do a test run, on my polished table. So I gave it a good wipe with a clean towel, and did the same for the table.

    I figured with the possible tiny debris, and oil from the lathe and equipment, that if anything the wheel would roll a little bit further, because of the increase in diameter that the particles would cause.

    But to the average fellow the wheel looked like it belonged in an operating room.

    I give the wheel a roll, and to my surprise, it rolls to a ratio of 3.14159, this made me weak in the knees. Because everything I had ever gotten from the Grumman guys was right on.

    But I am a good sport and good friend. So I figured I would for the sake of science, clean up the wheel, and table with xylene. And then roll it for a real scientific measurement that I would post with an apology to the guys. I actually thought that the wheel would show once cleaned a number less then 3.1459 if it was already at 3.1459.

    So I roll the wheel and I admit I was not real happy about doing it. But I roll it, and it rolls long. I mean much longer then before. I can see the first mark on the table and the wheel just rolled past that mark. So my first thought is wow, wishful mis-roll. So I roll it again, it rolls right on the new longer roll mark. I do it many times. And then I get very confused. And take a break.

    I for the life of me cannot figure out what took place. If anything in my mind at the time, the wheel should have rolled longer with the dirt on it. How could it have rolled longer with the dirt off of it. It couldn't, right?

    Well a couple hours later, after I drew some cadd drawings of a wheel and a bump, and then rotated objects on the screen again and again. It hit me like a ton of bricks. A dirty wheel simulates a dirty bumpy road. And everyone knows that a dirty bumpy road is not the shortest distance between two lines.

    So with a 70-75 T-8 aluminum wheel rolled on the same material, flat rolling surface. I was able to conclude that pi is larger then 22/7 and closer to 3.14308 by actual testing.

    The idea of the computer people using a polygons sides, combined cumulative length, to determine the length, of a circles circumference, is probably the least mathematical thing I have ever seen. It should be the new definition for comparing apples and oranges.

    But then the original computer people decided to replace the division key with the fraction symbol. And most know they have two different meanings in mathematics.


    Sincerely,


    William McCormick
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    Forum Professor serpicojr's Avatar
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    I'm not sure where you got your anecdote about Archimedes, William, but Wikipedia offers another story: Archimedes showed that π < 22/7 by circumscribing a regular 96-gon about a circle. I'd love to continue this discussion with you, and I'd really love to convince you that your estimate of π is close but wrong.
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    Forum Masters Degree bit4bit's Avatar
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    How do we know they didn't just start making them up after the first 5 digits or so?
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    Quote Originally Posted by bit4bit
    How do we know they didn't just start making them up after the first 5 digits or so?
    Because if you have a good enough computer, you can calculate many of them yourself?
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  13. #12  
    Forum Masters Degree bit4bit's Avatar
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    I know, I was joking. It's not like anyone can recite the first million digits of pi off the top of their head.
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    Quote Originally Posted by serpicojr
    I'm not sure where you got your anecdote about Archimedes, William, but Wikipedia offers another story: Archimedes showed that π < 22/7 by circumscribing a regular 96-gon about a circle. I'd love to continue this discussion with you, and I'd really love to convince you that your estimate of π is close but wrong.
    I had gone to Freeport public schools. At the time they were the math capital of the world, and I was a mathematician and honors student.

    I was taught that Archimedes wanted a perfect ratio of a wheels diameter to circumference.

    I was taught that he created a rather large wheel and rolled it on a rather large solid marble walkway.

    There was even debate as to what size wheel he created to do the experiment. It was stated that the wheel was as tall as he was. At the time Greece used a different form of measurement, then we do today. If indeed at the time we were correct, Archimedes stood almost seven feet tall. And so did his wheel.

    I have heard today that they claim that the cumulative measurements of the sides of an inscribed polygon, and the cumulative measurements of the sides of a circumscribed polygon, somehow average out to pi.

    And although most college students will never even have a chance to see the huge difference that having the correct ratio of pi can do for you in life. It is a rather small difference, only a couple thousandths difference in whatever unit you are measuring in.

    Most machinery and equipment compensates for this discrepancy by adding yet another corrective factor. This is unnecessary if you have the correct ratio of pi.

    Rules like "And a little bit more" really do not cut it in the real world. When someone from college comes in and demands that the machinery is calibrated to the 3.1459 standard, it gets a bit comical. If not embarrassing.

    As I mentioned pi is actually closer to 22/7 by actual measurement, and in day to day life use, it comes out to be 3.14308.

    Sincerely,


    William McCormick
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  15. #14  
    Forum Professor serpicojr's Avatar
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    Quote Originally Posted by William McCormick
    I had gone to Freeport public schools. At the time they were the math capital of the world, and I was a mathematician and honors student.
    I guarantee you that Freeport (and I'm not sure which one you're talking about, because there are quite a few, and none is distinguished above any other) was never the math capital of the world. But since we're talking math credentials, let me give you mine: I'm a math PhD student at a leading university on the East coast, so I'm surrounded by a lot of good math and could argue I'm smack dab in the middle of the math capital of the world.

    I was taught that Archimedes wanted a perfect ratio of a wheels diameter to circumference.

    I was taught that he created a rather large wheel and rolled it on a rather large solid marble walkway.

    There was even debate as to what size wheel he created to do the experiment. It was stated that the wheel was as tall as he was. At the time Greece used a different form of measurement, then we do today. If indeed at the time we were correct, Archimedes stood almost seven feet tall. And so did his wheel.
    He may have done this, but he also realized that this is not the mathematical way to go about finding π. As I stated above, he used pure geometric arguments to show that π < 22/7.

    I have heard today that they claim that the cumulative measurements of the sides of an inscribed polygon, and the cumulative measurements of the sides of a circumscribed polygon, somehow average out to pi.
    Close. It's the limit, not the average, of perimeters that can be taken to find π. And it's not that surprising. The argument is pretty simple--inscribed polygons have perimeter shorter than the circumference, circumscribed polygons have perimeter greater than the circumference, and as you let the number of sides increase to infinity, the perimeters or circumscribed and inscribed polygons approach the same number, π. In particular, this shows, as I stated above, that π < 22/7. If you don't understand this argument, I'd be happy to go into it deeper.

    And although most college students will never even have a chance to see the huge difference that having the correct ratio of pi can do for you in life. It is a rather small difference, only a couple thousandths difference in whatever unit you are measuring in.
    I don't know, the math curriculum at my university stresses error, and I'm going to assume that the science and engineering departments do, too. So I'm not sure where you're pulling this from.

    Most machinery and equipment compensates for this discrepancy by adding yet another corrective factor. This is unnecessary if you have the correct ratio of pi.
    You're the self-proclaimed machinist, so I'll have to believe you here. However, I'm going to assume that some sort of measurement bias is occurring--the way you measure radius or circumference (or both) must be slightly inaccurate. I believe the error probably creeps in when you calculate the length by rolling the circle and measuring on the flat surface.

    Rules like "And a little bit more" really do not cut it in the real world.
    Very true. I can calculate π to arbitrary precision. Can you do that by rolling circles around?

    When someone from college comes in and demands that the machinery is calibrated to the 3.1459 standard, it gets a bit comical. If not embarrassing.
    Yes, I am sure he goes back to school and has a good laugh with his colleagues at your expense. Either that or he goes to the wig shop after he pulls all his hair out.

    As I mentioned pi is actually closer to 22/7 by actual measurement, and in day to day life use, it comes out to be 3.14308.
    No. You're not right. Perhaps for your applications, your assumptions are slightly off and this approximation is actually better than the real value of π, but this is not the value of π. In real life applications, say solving differential equations for engineering calculations, you'd run into a whole slew of problems if you used your value for π.
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    As a side note, in my time we used to calculate out 22/7 on paper to see if at some point the pattern of numbers would change. That is how this game started. There were no calculators, other then mechanical ones that would only go out a couple places.

    Although the pattern looked pretty good, someone had speculated that at some point it would change. I just got a repeating number, but I never went out that far. Someone speculated that out a great number of places that the pattern might change.

    I don't know if that was just to keep our minds off of, blowing up the school, or if they really believed that. But that was how all this started.

    Then in the fifth grade they bought the honor students a one million dollar main frame computer. That we started to work on. It did not have a division key that I recall, but it would not perform in line problems like it was supposed to be able to do.

    Being that the whole premise of computers and the grants, was that the computer was going to advance in line mathematical problem solving, it seemed like the computer company missed the boat.

    As early as 1928 they used the in line division symbol in electrical and telephone training manuals.

    The division symbol means to take everything on the left and divide it by everything on the right. That is not what a fraction symbol does. The computer effectively destroyed mathematics. But a lot of money was on the line.

    The in line division key in conjunction with the proven fastest order of math, would allow the input of in line math formulas with almost no parenthesis in real applications of math to build real things. It was a tested and proven superior order.

    The order went addition, subtraction, multiplication and then division. That meant that the division symbol and the multiplication symbol could be placed anywhere in the formula and they would take everything on the left and multiply or divide by everything on the right. Because addition and subtraction had already been done.

    Years ago there were no, negative exponents. And there would not be today, if there was a division key on the keyboard. We used to divide by an exponent to get or show a fraction of a whole.

    This allowed for the very different fraction symbol, to have its very own unique roll in the formula. Without difficult to enter parenthesis. In real life you enter fractions into a formula all the time. Fractions simplify entry.

    http://www.Rockwelder.com/Electricity/Capacitor.pdf



    Sincerely,


    William McCormick
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    Very true. I can calculate π to arbitrary precision. Can you do that by rolling circles around?
    No, I can't.

    I have to use the exacting ratio of pi which by wheel roll was shown to be 3.14308.

    What I believe is being missed is the size of the atom. And the number of sides the infinite count of atoms causes in the dense material, and creates around the wheel. Infinite by reason of, no one individual can count them in one lifetime.

    A polygon created by the computer is just not going to give you the actual number of sides a real object has.


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    William McCormick
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    Close. It's the limit, not the average, of perimeters that can be taken to find π. And it's not that surprising. The argument is pretty simple--inscribed polygons have perimeter shorter than the circumference, circumscribed polygons have perimeter greater than the circumference, and as you let the number of sides increase to infinity, the perimeters or circumscribed and inscribed polygons approach the same number, π. In particular, this shows, as I stated above, that π < 22/7. If you don't understand this argument, I'd be happy to go into it deeper.
    To be honest your assumption is that a polygon inscribed and a polygon circumscribed, is somehow related to a circle with a truly infinite number of atoms making up the closest thing to a perfect circle there is.

    That is probably the least mathematical comparison that I have ever come across.

    Everything there is a guess about the reality. With no actual mathematical evidence.

    If they were so sure that the little bumps on an infinitely sided and calculated circumscribed polygon, would be exact, then why don't they do that? And, I already know why, and I am not being rude. I was trying to alert you to illogical thinking on the part of colleges.

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    William McCormick
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    I guarantee you that Freeport (and I'm not sure which one you're talking about, because there are quite a few, and none is distinguished above any other) was never the math capital of the world. But since we're talking math credentials, let me give you mine: I'm a math PhD student at a leading university on the East coast, so I'm surrounded by a lot of good math and could argue I'm smack dab in the middle of the math capital of the world.
    Let us say you are in the math capital of the world, and your group has come up with the fastest most truly mathematical ways of doing things.

    You would be the most interested in my methods, and very willing to discuss the possibilities with me. You would be pleased if you or I was wrong. You would be helping me look for proof, one way or the other. Or you would find a machinist that will give you his findings and then we would discuss it.

    I have done my own research with polygons and circles. In the cyber world, and in the real world.

    I am saying take a good look at the definitions in geometry, they alone would debunk the inscribed and circumscribed claims to finding the circumference of a circle.

    Someone is just running this fancy calculator on a fancy division problem; that is really not in touch with the reality of the circle and its ratios.

    This animation shows why scientifically and certainly mathematically, you cannot use a polygon to measure a circles circumference.





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    William McCormick
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    He may have done this, but he also realized that this is not the mathematical way to go about finding π. As I stated above, he used pure geometric arguments to show that π < 22/7.
    It may seem like I am in a battle with you. Just the opposite. I love to see your points of view. Because I bet others have the same point of view as well.

    I am sure of many of the things I am saying. In ways that I would not ask you to believe without knowing me better.

    But, math, is the tallying up of "measured" objects. Math is the creation of ratios and formulas to manipulate real measured objects. You have to have a real wheel that you can roll on a real table and come up with a real ratio. If only to test your mathematical theory.



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    William McCormick
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    Forum Professor serpicojr's Avatar
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    It's forum policy to post just one response at a time or, if you make a response and later decide to post another one, to edit the old response to include your new ideas. It's difficult to have to respond to each post in kind.

    Now... I think I see where the difficulty in our discussion is coming from. You have a very physical, mechanical view of math. My own is very axiomatic and ideal.<sup>1</sup> To you, calculating π is building a circle and measuring it. To me, calculating π is defining a circle and using logic to deduce it. Both are valuable perspectives to have in math. Without the former, math would have no use, and you and I certainly understand this. Without the latter, we would not make any progress in math, and it seems that you do not know this.

    You scoff at the polygon argument for "calculating" π. This is a shame, because the very man you lauded as a great mathematician during your first post, Archimedes, is the originator of this idea. This is not a computer calculation. It is a logical argument thousands of years old. You suggest that the definitions of geometry refute this argument; on the contrary, this argument uses the axioms and constructs of geometry and nothing more (well, there's algebra and arithmetic in there, but those are just part of math).

    To understand the difference between our thinking, consider that you and your classmates diligently performed long division ad infinitum to discover whether 22/7 always repeats or not. Individuals like myself prove that it repeats forever, that it's equal to 3.142857142857... with the 142857 repeating forever, by, say, positing that the pattern is correct and then noting:

    x = 3.142857142857...
    x-3 = 0.142857142857...
    1000000(x-3) = 142857.142857...
    1000000(x-3) - 142857 = 0.142857... = x-3

    And so:

    999999(x-3) = 142857
    x-3 = 142857/999999 = 1/7 (check!)
    x = 3+1/7 = 22/7

    See the power of my kind of mathematics?

    ----------------------
    <sup>1</sup>You seem to have the idea that I do math via computers, that I prove my results via simulations. This is far from the truth. This is a third perspective yet, different from yours and mine. Don't be fooled, though, the role of computers in math is very important.
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    Quote Originally Posted by serpicojr
    It's forum policy to post just one response at a time or, if you make a response and later decide to post another one, to edit the old response to include your new ideas. It's difficult to have to respond to each post in kind.

    You scoff at the polygon argument for "calculating" π. This is a shame, because the very man you lauded as a great mathematician during your first post, Archimedes, is the originator of this idea. This is not a computer calculation. It is a logical argument thousands of years old. You suggest that the definitions of geometry refute this argument; on the contrary, this argument uses the axioms and constructs of geometry and nothing more (well, there's algebra and arithmetic in there, but those are just part of math).

    To understand the difference between our thinking, consider that you and your classmates diligently performed long division ad infinitum to discover whether 22/7 always repeats or not. Individuals like myself prove that it repeats forever, that it's equal to 3.142857142857... with the 142857 repeating forever, by, say, positing that the pattern is correct and then noting:

    x = 3.142857142857...
    x-3 = 0.142857142857...
    1000000(x-3) = 142857.142857...
    1000000(x-3) - 142857 = 0.142857... = x-3

    And so:

    999999(x-3) = 142857
    x-3 = 142857/999999 = 1/7 (check!)
    x = 3+1/7 = 22/7

    See the power of my kind of mathematics?

    ----------------------
    <sup>1</sup>You seem to have the idea that I do math via computers, that I prove my results via simulations. This is far from the truth. This is a third perspective yet, different from yours and mine. Don't be fooled, though, the role of computers in math is very important.
    I thought I would have a better chance responding to the different responses, separated by separate quotes in one post to me, by separate posts. I apologize for any inconvenience.



    I guess it depends on when you were born, or where. In days of old in the high tech parts of America, we completed math, science and history pretty much, and were moving on to higher goals with these basics. We had the capability to go to other solar systems, without farfetched time travel in the sixties.

    At that time the government of the United Stated decided to end progress in these fields. There was no conspiracy, however, the government openly stated that due to inner city violence, and other hostile nations. That mostly, hostile minorities would receive a form of counterintelligence education.

    That of course opened the door to counterintelligence being used on all American students. I was in a classroom with a Universal professor, who had a tear in his eye. Told us that by next year he will not be able to teach the Universe as it is. And in fact as of next year the phony neutron particle will be taught as a real particle, by Federal teaching standards.

    This blatant disregard for science lead to Brookhaven Labs receiving a grant for the impossible dream. They claimed that they were going to create a perfect vacuum and isolate particles smaller then an electron. At the time we knew you could not even isolate an electron.

    We thought that maybe this was a funny kind of extortion or rebellion against Washington DC. However now it appears they are for real. Or are at least willing to give false education to many young Americans.

    There is no such thing as a perfect vacuum for anyone that does not know that. A vacuum or near vacuum is created by allowing the pressure that is still present in an area to force out "Most" of the atoms, into a vessel that has even less atoms, causing a vacuum like condition.

    There is no way to suck them out. And you can only press some of them out. Because it takes some to create pressure to push out the other ones.

    In fact in a perfect vacuum the walls of the container would evaporate for lack of a better term into the vacuum. So the idea that they could remove all the giant atoms to see an electron was silly, but even sillier still was the thought that they could isolate smaller nonexistent particles.

    Deep space is full of gas.

    I guess my point is, I have seen how they have convinced a lot of young people that they are learning precise, higher level math and science. However in reality we already had all the precision that is obtainable or provable on earth. You cannot make things anymore accurately with higher math then we were able to do in the sixties.

    If I am not mistaken you are demonstrating that you are inaccurate to 1/100,000th by displaying an exact ratio of 22/7 using base ten notation.

    (Edited)I meant 1/1,000,000th accuracy.

    I have done that ratio of 22/7 and other ratios that Archimedes had claimed to try. Out many places by hand division. But I just did the 22/7 and it appears that the remainder does in fact just repeat. But if you take the ratio of 286/91 and divide that out you will see that the remainder does change. So without actually doing it, I cannot say whether or not it will repeat forever.

    I apologize my long division is horrible, the remainder does repeat.

    In that link at the top I posted, they are talking about minutes and seconds. A degree is 1/360th of a complete circle, a minute is 1/60th of a degree and a second is 1/60 of a minute. That is actually very high accuracy if you do the math.

    A standard compass actually measures Azimuth, and does not measure bearing. Bearings are read from the east, and if they are shown pointing north, they have been rotated 90 degrees.



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    I guess my point is, I have seen how they have convinced a lot of young people that they are learning precise, higher level math and science. However in reality we already had all the precision that is obtainable or provable on earth. You cannot make things anymore accurately with higher math then we were able to do in the sixties.
    I guarantee you the math I do is directly descended from math that was being done in the 60's, the 40's, the 10's, the 1890's. And I guarantee you that never have we felt, since the Renaissance, that we have exhausted all of the math there is to be done. It's difficult to see this without knowing higher math. Unfortunately, you don't, so you have to defer to my expertise.

    Furthermore, I'm not making any claims that the math we do now is more precise. In fact, my claims about being able to calculate π precisely come from arguments that are thousands of years old--from Archimedes. This is not new math I'm talking about in this discussion. This is good old fashioned math being done by hand.

    I have done that ratio of 22/7 and other ratios that Archimedes had claimed to try. Out many places by hand division. But I just did the 22/7 and it appears that the remainder does in fact just repeat. But if you take the ratio of 286/91 and divide that out you will see that the remainder does change. So without actually doing it, I cannot say whether or not it will repeat forever.
    This is the jump you need to make. To you, calculating the decimal expansion of a fraction is sitting down and doing long division until your hand falls off. This will never work, and thinking for some reason it will is delusional--you're human, you're going to die, and the decimal expansion of 22/7 or 286/91 will never stop, so you'll never finish the calculation.

    But the answer is staring you in the face and you choose to ignore it: the remainder repeats. If the remainder repeats, then the sequence of division must repeat, and so the expansion must repeat. And the remainder must repeat because there are only finitely many remainders to a fixed divisor. This is a proof that all rational numbers have repeating decimal expansions.

    To you, math is seeing the truth with your own eyes, making the truth with your own hands. To me, math is this and noticing patterns and making inferences about the way the world must be in order for those patterns to occur. Your math is lacking in abstract thinking.

    As for your discussion of other sciences, I cannot comment, because my expertise lies mainly in math. I'll have to defer to you that some of the physical ideas bandied about today are silly; if you want to discuss these things, hop on over to the physics forum. I can't really speak about the change in education--I am a product of the educational reforms you lambast--but I can assure you that whatever "new math" you may be afraid of is no different than the math you learned as a child. If it were, my mathematical world would have come crashing down at some point in my life. It hasn't.
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    Quote Originally Posted by serpicojr
    I guess my point is, I have seen how they have convinced a lot of young people that they are learning precise, higher level math and science. However in reality we already had all the precision that is obtainable or provable on earth. You cannot make things anymore accurately with higher math then we were able to do in the sixties.
    I guarantee you the math I do is directly descended from math that was being done in the 60's, the 40's, the 10's, the 1890's. And I guarantee you that never have we felt, since the Renaissance, that we have exhausted all of the math there is to be done. It's difficult to see this without knowing higher math. Unfortunately, you don't, so you have to defer to my expertise.

    Furthermore, I'm not making any claims that the math we do now is more precise. In fact, my claims about being able to calculate π precisely come from arguments that are thousands of years old--from Archimedes. This is not new math I'm talking about in this discussion. This is good old fashioned math being done by hand.

    I have done that ratio of 22/7 and other ratios that Archimedes had claimed to try. Out many places by hand division. But I just did the 22/7 and it appears that the remainder does in fact just repeat. But if you take the ratio of 286/91 and divide that out you will see that the remainder does change. So without actually doing it, I cannot say whether or not it will repeat forever.
    This is the jump you need to make. To you, calculating the decimal expansion of a fraction is sitting down and doing long division until your hand falls off. This will never work, and thinking for some reason it will is delusional--you're human, you're going to die, and the decimal expansion of 22/7 or 286/91 will never stop, so you'll never finish the calculation.

    But the answer is staring you in the face and you choose to ignore it: the remainder repeats. If the remainder repeats, then the sequence of division must repeat, and so the expansion must repeat. And the remainder must repeat because there are only finitely many remainders to a fixed divisor. This is a proof that all rational numbers have repeating decimal expansions.

    To you, math is seeing the truth with your own eyes, making the truth with your own hands. To me, math is this and noticing patterns and making inferences about the way the world must be in order for those patterns to occur. Your math is lacking in abstract thinking.

    As for your discussion of other sciences, I cannot comment, because my expertise lies mainly in math. I'll have to defer to you that some of the physical ideas bandied about today are silly; if you want to discuss these things, hop on over to the physics forum. I can't really speak about the change in education--I am a product of the educational reforms you lambast--but I can assure you that whatever "new math" you may be afraid of is no different than the math you learned as a child. If it were, my mathematical world would have come crashing down at some point in my life. It hasn't.
    That is what I am sure of, if the remainder repeats then so will the pattern in the answer. However the whole continuous pattern is an inaccuracy base ten has in displaying actual ratios.

    In real life, I am saying that the ratio of pi is actually a bit larger then 22/7 so all the places they are going out, based on a theory of polygons inscribed and circumscribed is just crazy. It is purely an exercise in what you are accusing me of. Something that is not needed.

    In my day in class we would work on division problems that appeared to be repeating, and later they would change. But the remainder was changing. Archimedes according to my school had worked on other ratios. And we would divide them out all day long. I hated it, I wanted to be out building things. But you had to go to school. I was supposed to have been an astronaut. Until they cancelled the space program.

    I guess I am saying that I know for sure that pi is different then the number that they are quoting as pi. And I do not see any need to calculate it out that many places. None at all.

    But I do understand what you are saying about repeating patterns.

    One thing I noticed was that you said your math descended from the 40's and 60's. It would also depend on where you got your math. Because different places had different ways of doing math.

    The wheel I used for my actual test has smaller atoms then his stone wheel, and rolled farther then his stone wheel. He just did not have that kind of material around.


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    William, I am sorry that you do not appreciate the polygon argument. It is anything but crazy. If you had a specific part of the argument that you did not agree with, I could show you where your error is. But just dismissing it as crazy is not a valid criticism. Talking about atoms in the context of ideal geometric figures also does not lead to valid criticisms.

    The forbears of my mathematics come from all around the globe. Russia. Germany. France. England. Japan. Poland. India. China. America. Lots of other places. My math is the universally accepted math of the worldwide academic mathematics community.
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    Quote Originally Posted by serpicojr
    My math is the universally accepted math of the worldwide academic mathematics community.
    That is a mathematically incorrect way to show that you posses good math. Popular belief and math have nothing in common. Absolutely nothing. A sign of a weak argument is to use such a statement as you did. If you stand with me you stand with me, Ha-ha.

    Archimedes according to my understanding spent his whole life often in ridicule of others that thought he was insane for perfecting a usable formula for a wheel. It is my understanding that he spent years and years and worked on many stones, before he finally found one big enough that would give him a good proof of pi. He ruined according to my sources many stones and many years of work on them.

    Take a look at the basic geometry of the inscribed and circumscribed polygons. The inscribed polygon is going to form the chord of a circular segment. And those chords will be totaled up for a cumulative length.

    While the circumscribed polygon will create two legs of an obtuse triangle, that will protrude beyond the circumference of the circle. These will be totaled up and then the average of inscribed and circumscribed polygon circumference, will be taken to from the length of the theoretical circumference of the circle. The obtuse triangle formed by the intersection of circle circumference and polygon, its hypotenuse will overlay the circle and be equal to the chord formed by the inscribed polygon.

    I have a really good Cadd program, and I was able to scale what would look like a close match of a polygon and circle circumference up, and while zoomed up you can see that nothing changes really. You are dealing with apples and oranges.

    Look at the ratio formed from a simple inscribed and circumscribed polygon to a more complex inscribed and circumscribed polygon, that in itself is the injection of poor unneeded variables that an important American standard should not have in its foundations.

    By creating polygons with many sides they are just obscuring the ratio between an inscribed and circumscribed polygon. It has nothing, nothing at all to do with the circumference of a circle.

    They are just guessing at where the circumference of the circle will fall within an unrelated structure, made of lines.

    Archimedes also would watch the humidity and only perform his experiments at a certain humidly. Because his stone might expand or contract with humidity. He performed his experiments at the same time of day, or with the sun at the same height. For similar reasons.





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    Hopefully you can see that it is all a guess to compare an arc, with straight lines.

    No one can say what the length of the circle circumference is in percentage to two polygons, one inscribed and one circumscribed, with a known number of sides, mathematically without a real wheel and tests.

    I would be the first to admit that the real wheel will have points that from lines. However they are very, very small. Their actual nature is repulsive force, not actual contact.
    So in fact this causes an actual arc to be formed as it is rolled. That is different then a polygon made of straight lines. This used to be highly studied stuff in my day.

    They are just exercising their computers.

    I believe though that by creating a circumscribed polygon at a given diameter, they could show or prove just how small an atom is. Of course that would destroy all the current theories. Ha-ha.




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    Look at the ratio formed from a simple inscribed and circumscribed polygon to a more complex inscribed and circumscribed polygon, that in itself is the injection of poor unneeded variables that an important American standard should not have in its foundations.
    Sir, you keep on insisting that this is the method used for attaining the circumference of a circle. It is simply wrong. No inscribed circles are used.

    If you draw a polygon inside a circle of known radius and measured the total length of the sides, you’d get the circumference of the polygon. The larger the polygon drawn inside the circle gets, the larger the circumference of the polygon will be. So it is not difficult to see that eventually a polygon with infinite sides would yield the circumference of the circle. Since a polygon with infinite sides cannot be defined, limits are used (maybe serpicojr can provide us with an example?). This would then yield the EXACT circumference of the circle to arbitrary decimals and the precise value of pi to arbitrary decimals can be calculated. Do you have problems with this?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

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    [quote="KALSTER"]
    Sir, you keep on insisting that this is the method used for attaining the circumference of a circle. It is simply wrong. No inscribed circles are used.

    If you draw a polygon inside a circle of known radius and measured the total length of the sides, you’d get the circumference of the polygon. The larger the polygon drawn inside the circle gets, the larger the circumference of the polygon will be. So it is not difficult to see that eventually a polygon with infinite sides would yield the circumference of the circle. Since a polygon with infinite sides cannot be defined, limits are used (maybe serpicojr can provide us with an example?). This would then yield the EXACT circumference of the circle to arbitrary decimals and the precise value of pi to arbitrary decimals can be calculated. Do you have problems with this?
    An inscribed polygon is drawn inside the circle. A circumscribed polygon is drawn outside the circle.

    Yes I do have a problem with it, it is not proof. It was proven that an arc is formed around finely polished objects. Not straight lines, but an actual arc is formed around a circular object.

    By repulsive energy. I am from the school that no two objects ever touch. Not even subatomic particles. I could pass a lie detector test, to support that fact.

    I say do it with the circumscribed polygon if you are so sure of the near infinity, it should yield the same results. But it will not.

    The only way to actually calculate the arc is to roll the circular object, and actually get a measurement and a ratio.

    That is what I did, and I learned some amazing things about the wheel in doing it.

    When I cleaned off the rather clean wheel with xylene, it became almost silent. Except for an almost inaudible super high pitched wine, ultra sonic. It almost sent a shiver up my spine. While it was not totally clean it created a more noticeable metallic harmonic.

    I just thought that the clean wheel rolling longer then the dirty wheel was amazing. When ever you doing anything real, you realize that you don't know anything.

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    An inscribed polygon is drawn inside the circle. A circumscribed polygon is drawn outside the polygon.
    I know. I said inscribed circles.

    Do you really think making a measurement by hand is more precise than working from precise mathematic principles?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

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    I am trying to follow you and maybe you are trying to follow me. But we should agree upon the terms that we are talking about.


    Here is my definition of circumscribe. It is the Geometry definition" draw a figure round another, touching it at points but not cutting it".

    This is my definition of inscribe, "draw a figure round another, so that some or all points of it lie on the boundary of the other".

    I sometimes create a typo that I usually fix. However to me an inscribed polygon is drawn inside the circle. A circumscribed polygon is drawn outside a circle.

    I am suggesting that they are not using the polygon drawn outside the
    circle. And coming up with a wrong answer.

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    Yes, we are saying the same thing but missing each other in the middle. We have a circle. A circumscribed polygon goes on the outside and the inscribed one goes on the inside.

    What I was saying though, was that no part of the polygons in question ever leaves the circle. No circumscribed polygon (ergo, no inscribed circle) is ever considered. It would yield inaccurate results. Perhaps this is where you have the method wrong.
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

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    Quote Originally Posted by KALSTER
    An inscribed polygon is drawn inside the circle. A circumscribed polygon is drawn outside the polygon.
    I know. I said inscribed circles.

    Do you really think making a measurement by hand is more precise than working from precise mathematic principles?

    Without doubt, you have to see a precision wheel roll. When the line on the wheel comes around to one complete roll, it looks like it is in slow motion. You can mark the table exactly with ease, and repeat the experiment again and again. Everyone should have to make a wheel before they leave math class.

    I know why Archimedes spent his life rolling it. Because he was having fun and finding out all kinds of wild phenomena about wheels. With all the things he learned about it, he probably did not even care if anyone believed him. It was his wheel. He owned it. It was one of the few things they could not take away from you back then.

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    Quote Originally Posted by William McCormick
    I am from the school that no two objects ever touch. Not even subatomic particles. I could pass a lie detector test, to support that fact.
    If no two objects ever touch, then you cannot measure the circumference of a circle by rolling it along a flat surface. The path traversed by the circle along the flat surface must be longer than the circumference to account for the distance between the two objects. (But, of course, your physical circle will never be a perfect circle, so you're always going to get some error anyway.)
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    Quote Originally Posted by KALSTER
    Yes, we are saying the same thing but missing each other in the middle. We have a circle. A circumscribed polygon goes on the outside and the inscribed one goes on the inside.

    What I was saying though, was that no part of the polygons in question ever leaves the circle. No circumscribed polygon (ergo, no inscribed circle) is ever considered. It would yield inaccurate results. Perhaps this is where you have the method wrong.
    Ok, I guess my question is are they using a circumscribed polygon drawn around the circle to come up with pi?

    My guess, no. Because I know a lot of individuals that are very good at math and they would banish that notion immediately. Because the number of sides you would need would be infinite, because of the number and orientation of the atoms in the wheel.

    You have to build and roll one. Just don't try to grab the shavings off the cutter with your bare hands, they can wrap around you and cut off fingers.

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    Ok, I guess my question is are they using a circumscribed polygon drawn around the circle to come up with pi?
    No, they use incrementally larger inscribed polygons drawn inside the circle. But they don't actually measure the sides, they do that with mathematics (it provides the necessary precision).

    Because the number of sides you would need would be infinite, because of the number and orientation of the atoms in the wheel.
    And that is exactly where they use limits (the mathematical tool) to deal with the fact that the number of sides tend towards infinity.
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    Quote Originally Posted by serpicojr
    Quote Originally Posted by William McCormick
    I am from the school that no two objects ever touch. Not even subatomic particles. I could pass a lie detector test, to support that fact.
    If no two objects ever touch, then you cannot measure the circumference of a circle by rolling it along a flat surface. The path traversed by the circle along the flat surface must be longer than the circumference to account for the distance between the two objects. (But, of course, your physical circle will never be a perfect circle, so you're always going to get some error anyway.)
    As far as circles go a machined circle will be as close as the world has to offer, at coming close to a perfect circle.

    I don't even have to think about polygons, being even in the race for perfect circle, much less near a perfect circle.

    To bring polygons and pi into the same arena would be like saying, "a tangerine is the most perfect circle we can find to set the pi standard". It would be a lie, just like measuring polygons for pi is a lie.

    Math is secondary to the object being measured. Math is for the guy who measures things in the real world. Math is not for its own purpose.

    That is how ridiculous things like 6^0 power or 6^1 have crept into math. They are ridiculous. 6^2 meant six squared. 6^3 meant six cubed. I admit that the definition of exponent being "The number of times the base number is multiplied by itslef" can be misleading. But as I said it was to show square and cube. Two and three dimensions.

    Like the formula Pi D^2 for area of a sphere. Or Pi D^3/6 for the volume of a sphere. It just makes it easier to remember I guess.

    Calling dividing by zero an error is wrong as well. Dividing by zero causes an infinite loop. You can put nothing into a garbage can all day long, you might look a little odd and someone might call the police, but dividing by zero is infinity.

    As far as no two objects ever touching. In a perfect world I would love to see if there was any way to measure what the actuality is. But matter right now is totally misunderstood in my opinion.

    Yet everyone always pops up with one or two of the old Universal Scientists truths when they need them. It was these truths that isolated the elements.

    Like dense matter being 90 percent space. It is true dense matter is 90 percent space. Yet imagine something being 90 percent space and not letting light through it.

    Just one of those little hidden right in your face truths. The way it was taught to me was that light hits a metal panel that is covering an opening in a totally dark room. Yet no light enters the room. How? I cannot stop water with a piece of metal that is 90 percent space. Yet not a drop of light comes in the room.

    The reason is that everything is electrons. Protons are balls of electrons.

    As ambient high speed radiation that we do not even detect, passes through all things. It is often accelerated by dense matter. From light speed to dark radiation. That is the case with our metal sheet.

    The light hits the panel and the ambient radiation the electrons that almost appear to orbit atoms, is really just being violently slingshot, around the atoms, accelerated. The light is accelerated to dark radiation again. Invisible to humans.


    This is how I was taught we see light with our electrical sensors.

    http://www.Rockwelder.com/Flash/mrbill/mrbill.html

    I know it seems like I jump around a lot, but I believe that you need a well rounded understanding of the whole universe, to understand anything.

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    Quote Originally Posted by KALSTER
    No, they use incrementally larger inscribed polygons drawn inside the circle. But they don't actually measure the sides, they do that with mathematics (it provides the necessary precision).

    Let them try it with a circumscribed polygon drawn outside the circle. They will not get the same results. I know exactly why they only use an inscribed polygon. Because it gives them the answer they like. Not pi.

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    It doesn't matter whether you use circumscribed or inscribed polygons. You get the same result, which is the actual value of π, which is different from the value you calculate.
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    Quote Originally Posted by serpicojr
    It doesn't matter whether you use circumscribed or inscribed polygons. You get the same result, which is the actual value of π, which is different from the value you calculate.
    I would like to see those results. The ones done on a circumscribed polygon drawn around a circle.

    If they do come out to give a ratio of 3.14159 and I doubt they do. You would then have to look into how they are calculating the length of the theoretical polygons side and or the angles they are extrapolating the length of the side from.

    It might take a powerful computer just to calculate the angels sufficiently enough to create the length of the side of the polygon exactly enough.

    But to me if you think that an inscribed polygon can ever equal a circumscribed polygon, you are not a good mathematician. Your theory is flawed either way. And has nothing to do with anything real.

    Get a real wheel and go for it.

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    Quote Originally Posted by serpicojr
    It doesn't matter whether you use circumscribed or inscribed polygons. You get the same result, which is the actual value of π, which is different from the value you calculate.
    I am missing my own point here. Ha-ha.

    And that is a dirty wheel rolled a shorter distance then a clean wheel.

    A dirty wheel with debris on it, rolled a shorter distance. Because it simulates a bumpy road. It is not a true arc.

    That would mean that a polygon even if circumscribed might just roll a shorter distance then an arc.

    I am sure this is what fascinated old Archimedes.


    This is wild stuff.

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    Here is the answer. It pays to find a group of good minds stuck to their convictions. Thank you guys.

    Here is the answer. It is the ratio of angle to polygon side length. As you decrease the side length by increasing the number of sides in your polygon. You are creating a ratio shortening the side at an ever greater ratio to the angle, over that of a lesser sided polygon.

    The near perfect arc I created must be free of this ratio. Therefore it moves forward while minimally changing the angle of the wheel. There for in one turn my wheel goes farther. It levels out the road.


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    First, let's use the convention that all polygons are regular so I don't have to keep repeating myelf. Similarly, the circle is the unit circle (radius 1, circumference 2∏).

    Next we have to notice that, if we inscribe a polygon in a circle, then its perimeter is less than the circumference. If we circumscribe a polygon about a circle, then its perimeter is greater than the circumference.

    Now take an n-gon (polygon with n sides) inscribed in the circle. Cut it up into n isosceles triangles by drawing segments from the vertices of the n-gon to the center of the circle, and then cut each of these in half by drawing the segment from the center of the circle bisecting each side of the n-gon, obtainig 2n right triangles with hypotenuse 1 (each hypotenuse is a radius of the circle). The angle at the center for each right triangle is ∏/n (using radians, and this is independent of what we believe ∏ to be). Thus each side opposite this angle has length sin(∏/n). Thus each side of the n-gon has length 2sin(∏/n), and so the n-gon has perimeter 2nsin(∏/n). Now we use a fundamental fact from trigonometry: if we take the limit of sin(x)/x as x approaches 0, we get 1. In other words, if you drew this function for x > 0, you could extend it continuously to x = 0 by setting it equal to 1. Thus we have (n/∏)sin(∏/n) approaches 1 as n goes to infinity. Multiplying by 2∏, we have that 2sin(∏/n) approaches 2∏ as n goes to infinity.

    Now take an n-gon circumscribed about the circle. Cut it up into 2n right triangles in a manner analogous to the above--if you want me to be more exact, please tell me. In this case, we have that the side adjacent to the angle at the center of the circle has length 1, and so now the side opposite this angle has length tan(∏/n). So the perimeter of the n-gon is 2ntan(∏/n). Noting that tan(∏/n) = sin(∏/n)/cos(∏/n), and since cos(x) goes to 1 as x goes to 0, we have that 2ntan(∏/n) approaches 2∏ as n goes to infinity.

    So let's clear up what we've done. We have the sequence a<sub>n</sub> of perimeters of inscribed polygons, each term being less than but tending to 2∏. We have the sequence b<sub>n</sub> of perimeters of circumscribed polygons, each term being more than but tending to 2∏. You can sit down and drawn a polygon with a large number of sides circumscribed about the circle, as Archimedes did, and you'll find that ∏ < 22/7. It looks like n = 91 is the smallest value that will give you this result.

    Happy birthday.
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    Quote Originally Posted by serpicojr
    First, let's use the convention that all polygons are regular so I don't have to keep repeating myelf. Similarly, the circle is the unit circle (radius 1, circumference 2∏).

    Next we have to notice that, if we inscribe a polygon in a circle, then its perimeter is less than the circumference. If we circumscribe a polygon about a circle, then its perimeter is greater than the circumference.
    No, and part of this is my fault for having the answer but not putting it into a proper mental frame set to communicate to others. I made a wheel and recorded it rolled farther when cleaned then when it was dirty. That means that the bumps of the circumscribed polygon act like a bumpy road. And allow the wheel to rotate much more in degrees, with less forward movement. A phenomena to say the least.

    The circumscribed polygon could be inverted on a flat surface to make this more understandable. You would see the bumpy road.

    If you take a polygon of say ten sides, it will be made up of 10 triangles each with an angle at the vertex of 36 degrees. 360 degrees/10 degrees.
    The sides of this polygon are exactly two inches.

    Now we create a polygon with the same circumscribed diameter, but with twenty sides. The angle is going to be half of the larger sided polygon, so at the vertex of each of the twenty smaller triangles the angle will be 18 degrees. Or 360/20 degrees equaling 18 degrees.

    But the sides of the polygon will not be 1 inch long, or half of the two inch sided polygon. They are going to be far less in length. That means for each degree in wheel rotation the polygon with more sides is going to go a lesser distance. A little over three one hundredths less, per side or 18 degree rotation of the wheel.

    My thought here is that if I did create a near perfect wheel and I am sure it is very round and smooth. It may be free from this ratio and in fact travel a further forward distance with less change in angle then most would be allowed to believe using unrealistic tests, or made up formulas that they believe simulate an arc.

    An arc is not a polygon.

    Sincerely,


    William McCormick
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    Your clean/dirty wheel argument does not hold muster. The dirty wheel is not a convex shape. When you roll a nonconvex shape, the distance it rolls in one revolution is less than its perimeter. This is the argument you're makig. But the circle and all regular polygons are convex shapes. When you roll a convex shape, the distance it rolls in one revolution is equal to its perimeter. The moral of the story? You cannot compare the dirty wheel to the polygon, or rather, you cannot make inferences about the polygon from the dirty wheel. After all, a dirty wheel is not a regular polygon.

    Note that I'm making the same argument against your argument that you're making against mine--you can't make inferences about one shape and use them for a different shape. You're guilty of such by comparing the polygon to the dirty wheel. You're claiming I'm guilty of such by comparing the polygon to the circle. The difference is that I actually have a geometric, logical argument that allows me to compare the circle to the polygon, and I never claim that the two are similar shapes. The point of my argument is not that the circle is a polygon but rather that we can bound the circumference of a circle by the perimeter of a circumscribed polygon. That is the key step. Once we have this, then we can forget about the circle and calculate the perimeters of polygons, and this is what I do in showing that, indeed, the perimeters of circumscribed polygons approach the circumference of the circle.

    Please, have faith in my expertise, try to understand my argument. If you don't understand a part of my argument, don't assume it's wrong; rather, ask me a question and I'll explain it for you.
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    Quote Originally Posted by serpicojr
    Your clean/dirty wheel argument does not hold muster. The dirty wheel is not a convex shape. When you roll a nonconvex shape, the distance it rolls in one revolution is less than its perimeter. This is the argument you're makig. But the circle and all regular polygons are convex shapes. When you roll a convex shape, the distance it rolls in one revolution is equal to its perimeter. The moral of the story? You cannot compare the dirty wheel to the polygon, or rather, you cannot make inferences about the polygon from the dirty wheel. After all, a dirty wheel is not a regular polygon.
    I am claiming that a bump on a wheel is in fact the same thing as the protruding, intersecting point of two sides, of a circumscribed polygon.

    I am stating what seems to be unreal or unbelievable to you. That a circumscribed polygon covers less ground when rolled one revolution, then a nearly perfectly round wheel.

    Because a circle is not a polygon. It just is not. It never will be. I am working on some math to show how, a circumscribed polygon creates the simulation of a wheel going over a bumpy road, even though it is on a flat surface. Then you can just check the math.

    This is why such an intelligent man like Archimedes did spend his whole life rolling a wheel.
    It is unbelievable to actually roll a wheel. It is so exciting that I often forget to put it into a form of communicatoin to share with others. I have shared my finds, but did not explain why they are so amazing. Mostly because it is a couple years old to me.

    I am not sure of the methods used to calculate a polygon, and I have only been guessing that a fully circumscribed polygon would not roll less then a circle that it encompasses. However the more work I do, and the more I have talked to you guys here, the more I suspect that it is totally plausible, that circle will roll farther then a circumscribed polygon around it.

    The corners of a polygon create the effect of a circle rolling over debris, or a bumpy road. Surly you cannot move a wheel up and down, over a bumpy road, and cover the same distance in a single revolution as you could with a totally round wheel.

    Because if that is true we need to put speed pumps everywhere. Ha-ha.

    Sincerely,


    William McCormick
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    I have to prove that a true arc is the shortest distance between point A and point B, on the table, I am rolling my wheel on. Requiring the least amount of revolution per distance covered.

    And that a circumscribed polygon is a longer distance between point A and point B. And requires more revolutions per distance covered.

    If you guys could help me prove this it would be great.

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    William McCormick
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    I will admit that it is not obvious that a circumscribed polygon has a perimeter longer that the circumference of the circle about which its is circumscribed. However, it is true. I cannot help you prove the statement you ask us to help you prove because it is false. Let me come up with an accessible proof of the fact that I claim and then you can buy me a cup of coffee.

    First, I will prove the following: let ABC be a right triangle with right angle at vertex B, so that AC is the hypotenuse. Draw the circle about the point C with length equal to the segment BC, and let P be the intersection of the circle with segment AC. Then the length of the circular arc BP (i.e., the piece contained inside triangle ABC) is less than the length of the segment AB.

    This will imply the polygon statement. If you don't see how, ask me.

    -----------------------------

    EDIT: Oh, wait, nevermind, it's obvious. So, without loss of generality, we may assume |BC| = 1. Thus the circle about C has radius 1, circumference 2∏, and area ∏. Let T be the measure of angle BCP. Then area of the circular sector spanned by this angle is T/2. The area of the right triangle is:

    |AB|*|BC|/2 = |AB|/2

    Clearly the area of the triangle is bigger than the area of the sector. So...

    |AB|/2 ≥ T/2

    |AB| ≥ T

    Finally, the length of the circular arc BP is T. This is what I claimed to be true.
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    Quote Originally Posted by William McCormick
    Surly you cannot move a wheel up and down, over a bumpy road, and cover the same distance in a single revolution as you could with a totally round wheel.
    Over a bumpy road, the actual distance travelled by the wheel would have to include the contours of the bumps of the road – if the bumps are significantly large in relation to the wheel. This is not the same as the horizontal distance travelled by the wheel, i.e. the horizontal translation of the centre of the wheel (which is what it appears to me you’re trying to get at).

    And the analogy of a circle rolling over a bumpy surface with a polygon rolling over a flat surface seems rather absurd to me.
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    Another voice of reason!
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    Quote Originally Posted by serpicojr
    Another voice of reason!
    You do understand that bumps on the wheel will act like the intersecting sides of a circumscribed polygon?

    If not I will draw a diagram.

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    William McCormick
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    Quote Originally Posted by serpicojr
    I will admit that it is not obvious that a circumscribed polygon has a perimeter longer that the circumference of the circle about which its is circumscribed. However, it is true. I cannot help you prove the statement you ask us to help you prove because it is false. Let me come up with an accessible proof of the fact that I claim and then you can buy me a cup of coffee.

    First, I will prove the following: let ABC be a right triangle with right angle at vertex B, so that AC is the hypotenuse. Draw the circle about the point C with length equal to the segment BC, and let P be the intersection of the circle with segment AC. Then the length of the circular arc BP (i.e., the piece contained inside triangle ABC) is less than the length of the segment AB.

    This will imply the polygon statement. If you don't see how, ask me.

    -----------------------------

    EDIT: Oh, wait, nevermind, it's obvious. So, without loss of generality, we may assume |BC| = 1. Thus the circle about C has radius 1, circumference 2∏, and area ∏. Let T be the measure of angle BCP. Then area of the circular sector spanned by this angle is T/2. The area of the right triangle is:

    |AB|*|BC|/2 = |AB|/2

    Clearly the area of the triangle is bigger than the area of the sector. So...

    |AB|/2 ≥ T/2

    |AB| ≥ T

    Finally, the length of the circular arc BP is T. This is what I claimed to be true.
    I don't fully understand why you would want to do something like that to make a responsible comparison of a circle to a polygon. You could get the same results by drawing a circle in a square. That would be irresponsible.

    In fact the whole silly theory of using a computer to create a ratio to compare two different geometric figures totally unrelated, is based on that ratio of square to circle changing.

    Save yourself a lot of heartaches get a real wheel. Or hold onto the dark ages to the end.

    Sincerely,


    William McCormick
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    Here is another analogy that is good for both of us.

    If you tied a piece of string around the earth, and then raised it a half inch above the earth, the string would only need to be pi longer to make that happen.

    So if a polygon is a way to measure a circles circumference, certainly the polygon being a few billionths of an inch above the circle is not going to change much.

    But my thinking is that the quickest way around a circle is a bunch of straight lines. Not an ARC. I would look into this before you have to throw away all the rules of math.



    Sincerely,


    William McCormick
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    Quote Originally Posted by William McCormick
    I don't fully understand why you would want to do something like that to make a responsible comparison of a circle to a polygon.
    I'm sorry, I assumed you'd be able to see how my statement proves the fact that a regular polygon circumscribed about a circle has perimeter longer than the circumference. It's really quite simple. Get yourself a paper and pencil. Circumscribe a regular polygon around the circle. Draw segments from the center of the circle to each vertex of the polygon. Now you have n isosceles triangles, n being the number of sides of the polygon. Cut these in half by drawing segments from the center of the circle to the points of tangency between the circle and the polygon. This creates 2n identical right triangles. Note that each of these right triangles satisfies the description given in my statement above. Now the perimeter of the polygon is the sum of the lengths of the triangle segments considered in my statement. The circumference is the sum of the lengths of the circular arcs considered in my statement. This is all very careful and very responsible.

    You could get the same results by drawing a circle in a square. That would be irresponsible.
    Um, I don't think so. If you can show me how, then I'll eat your pineapple upside down cake.

    In fact the whole silly theory of using a computer to create a ratio to compare two different geometric figures totally unrelated, is based on that ratio of square to circle changing.
    William, if you read a single thing that I write with any care and decency, let it be this one paragraph. Although I have suggested the use of computer or calculator as a means of calculating things once or twice, I have never suggested that any of my arguments should be done on a computer or depend upon computers. Everything I'm talking about is paper and pencil, maybe ruler and compass if you don't trust your skills of logic and abstraction. The only silly thing here is your practice of setting up straw men, writing my name on them, and slapping them around with your cane.

    Save yourself a lot of heartaches get a real wheel. Or hold onto the dark ages to the end.
    How can you dismiss my argument for using computers (which it doesn't, but I'll bear with you for a second) in one breath and then claim that I live in the Dark Ages with the next? At least have some consistency in your criticism. Of course, if you had been paying attention to what I've been saying, you'd know my argument is older than computers and the Dark Ages--it dates back to the Greeks, to that wonderful mathematician of old, Archimedes. What a guy! Did you know he calculated the area of a circle by using polygons, too?

    If you tied a piece of string around the earth, and then raised it a half inch above the earth, the string would only need to be pi longer to make that happen.
    I love this fact. It seems counterintuitive on the surface, and it shows the power of thinking abstractly.

    So if a polygon is a way to measure a circles circumference, certainly the polygon being a few billionths of an inch above the circle is not going to change much.
    Exactly my point! You're beginning to see the light, William! The polygon is a good APPROXIMATION to the circle, and it gets better as you let the number of sides increase!

    But my thinking is that the quickest way around a circle is a bunch of straight lines. Not an ARC.
    Uh... why? The only argument that you have given to this is the clean/dirty wheel argument, and this has already been dismissed by myself and JaneBennet. You can't just say, "William McCormick thinks this and therefore it's true." (Only Fermat, Weil, and Langlands can do that.) And, of course, it isn't true because I proved the opposite. Now would be the perfect time for you go back and put some effort into understanding my proof. I guarantee you it's pretty simple and very correct.

    I would look into this before you have to throw away all the rules of math.
    William, I double dog dare you to find a single instance where I have thrown away any rule of math. I guarantee you that I haven't.

    Please, read over my proof.
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    Q:What do you get when you take the sun and divide its circumference by its diameter?


    A: Pi in the sky!!

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    You are standing by the theory, that a polygon side can be compared to an arc, for setting standards. It is so comical and abstract that it makes no sense at all. My bottom line.

    You can compare the ratios of each to themselves and to other things but you cannot use one interchangeably with another.

    As you increase the number of sides in your polygon eventually the vertex of the triangle that creates the pie like slices from the center of a polygon to the sides intersections, will be 0.00000000000..... degrees. I am stating for the record at some point the arc that does not change at all, at some point overtakes the polygon side in length. Because there is no more measurable difference in the angle.

    You get basically an arc and a straight line, contained between two parallel lines. At that point the arc is easily recognized as the longer object. Because the arc never changed the polygon did.

    It is like our planet when we stand on it we do not readily see the curvature. Yet it exits super evidently from a distance. It becomes a pea sized object that has a pronounced arc.

    Compare one side of a circumscribed square to the circle it surrounds. Look at that ratio. That is what I am saying occurs at a near perfect circle or arc.

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    William McCormick
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    William, you're hopeless.
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    Are old people really wizer?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

    "Gullibility kills" - Carl Sagan
    "All people know the same truth. Our lives consist of how we chose to distort it." - Harry Block
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    Quote Originally Posted by Selene
    Q:What do you get when you take the sun and divide its circumference by its diameter?


    A: Pi in the sky!!

    Yay, let’s have some comic relief in this thread!

    A man sends his son to school to learn maths. When his son comes home from school, he says to him, “Well, sonny boy, what did ya learn at school today?”

    “Dad,” says his son, “we learned how to find the area of a circle: pi r squared.”

    “Bah, nonsense!” exclaims his father. “Everybody know that pi r round!”
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    Quote Originally Posted by JaneBennet
    Quote Originally Posted by Selene
    Q:What do you get when you take the sun and divide its circumference by its diameter?


    A: Pi in the sky!!

    Yay, let’s have some comic relief in this thread!

    A man sends his son to school to learn maths. When his son comes home from school, he says to him, “Well, sonny boy, what did ya learn at school today?”

    “Dad,” says his son, “we learned how to find the area of a circle: pi r squared.”

    “Bah, nonsense!” exclaims his father. “Everybody know that pi r round!”
    Wow the old guy is a fool, he doesn't even know they make square pies.

    Sincerely,


    William McCormick
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    Quote Originally Posted by JaneBennet
    Quote Originally Posted by Selene
    Q:What do you get when you take the sun and divide its circumference by its diameter?


    A: Pi in the sky!!

    Yay, let’s have some comic relief in this thread!

    A man sends his son to school to learn maths. When his son comes home from school, he says to him, “Well, sonny boy, what did ya learn at school today?”

    “Dad,” says his son, “we learned how to find the area of a circle: pi r squared.”

    “Bah, nonsense!” exclaims his father. “Everybody know that pi r round!”
    Ha ha ha...i like it!

    I was getting worried then.

    I thought everyone had come down with a severe dose of Intellectualitis.

    I was just preparing me hypodermic!
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    Quote Originally Posted by serpicojr
    William, you're hopeless.
    Did you get what I meant. About two parallel lines with a straight line between them and an arc between them. At that point it is obvious that the arc is longer.

    That is all I was saying, to take a look at one side of a circumscribed square around a circle. You can see that between to parallel lines that the arc is longer. I was not being facetious either. I was really offering you another look at it from the other end of the spectrum.

    An arc is not the side of a polygon, so at some point a many sided polygon will have a perimeter shorter then the arc of the circle it contains. This is why Archimedes spent his life on this. It is wild to see a precision wheel roll. You can see something very strange, in the way it moves. It is fascinating.


    Sincerely,


    William McCormick
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    But William, I am not describing an arc and a segment between two parallel lines. I agree, in that scenario, the arc is longer. But that is not the comparison I claim to make. My lines are not parallel--they intersect at the center of the circle--so the arc ends up being shorter than the segment. Just because the lines look parallel when the angle becomes miniscule doesn't mean they are.

    Now why is it comical and abstract when I compare a line segment and an arc but not when you do so? Please be consistent.
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    Quote Originally Posted by serpicojr
    But William, I am not describing an arc and a segment between two parallel lines. I agree, in that scenario, the arc is longer. But that is not the comparison I claim to make. My lines are not parallel--they intersect at the center of the circle--so the arc ends up being shorter than the segment. Just because the lines look parallel when the angle becomes miniscule doesn't mean they are.

    Now why is it comical and abstract when I compare a line segment and an arc but not when you do so? Please be consistent.
    You claim that your comparison of an arc and a line is to show that they are the same or can be at some theoretical place. I know that is not true. It would be like getting a balony sandwich when you were promised a salami sandwitch.

    I am stating that with basic mathematics you can prove that a circle contianed in a polygon will actually have a longer perimeter then the polygon around it. Because the polygon is not infinate in nature. It is finite. The arc by definition is infinite.

    I do understand that you are saying that the two lines that make up the pie sliced triangle created between the ends of one side of the polygon and the center of the polygon, are not parallel in reality.

    However after doubling the sides of the polygon enough times. The actual angle of the two lines will be so nearly parallel, that the difference in distance from the center of the polygon to the polygon side and the circle will become obscured.

    And the longer length of an arc compared to a line, will be predominate.

    Sincerely,


    William McCormick
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    Quote Originally Posted by William McCormick
    I am stating that with basic mathematics you can prove that a circle contianed in a polygon will actually have a longer perimeter then the polygon around it.
    William, you seem not to understand. I already used basic mathematics to prove that a circle contained in a polygon has a circumference shorter than the perimeter of the polygon. The proof is in this thread. You claim you can prove the opposite. You fail to supply proof. In fact, I know you can't, because it's false, and I've already proved such. Read my proof. If you don't understand it, ask me questions. Don't dismiss it because you can't understand it or because you don't want to believe it's true. It's all very simple, basic, true math. You can understand it if you allow for the possibility that it is true (which it is) and if you put some effort into understanding it (which you have refused to do up until now).

    William.

    Read my proof.

    Read my proof.

    Read my proof.

    William.

    Read it.
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    Quote Originally Posted by serpicojr
    William, I double dog dare you to find a single instance where I have thrown away any rule of math. I guarantee you that I haven't.
    You claim that an arc is the shortest route around a circle. I claim that short straight lines making up a circumscribed polygon around the circle, is the shortest way around a circle.

    You have to understand what the ARC is. It is infinite.

    To go the furthest in one revolution a circle will go farther then a nearly perfect high number of sided, polygon appearing to be a circle.

    Sincerely,


    William McCormick
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    Not only did I make such a claim, I proved it. Your claim is unsubstantiated; in fact, it is false, because my claim is true. Again...

    Read my proof.

    If you don't understand it...

    Ask me.
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    Quote Originally Posted by serpicojr
    William, you seem not to understand. I already used basic mathematics to prove that a circle contained in a polygon has a circumference shorter than the perimeter of the polygon. The proof is in this thread.
    Did you use the oldest and most used ratio? The one that America adopted and used for most of Americas history?

    That would be 22/7, claimed by Archimedes to be off by about 1/100,000th part. If you did not, then you are just making claims that when 3.14159 is plugged into a formula that a kindergartner can come up with the right answer. And it will show by double check that 3.14159 was in fact the ratio used in the formula.

    And sure if you keep plugging 3.14159 into the formula instead of at least 22/7 then you are going to get wrong answers.

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    William McCormick
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    Buddy, I didn't use anything about the specific value of ∏. All I used is that, if we take a unit circle (radius 1), then there is a number ∏ such that 2∏ is the circumference of the circle (clear), and we have ∏ is also the area of the circle (which you can prove by, say, calculus). Then I used simple formulas for areas of triangles and circular sectors and for the length of a circular arc.

    Please, do not resort to name-calling. You don't know how silly it makes you look.
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    http://j-walkblog.com/index.php?/web...ving_big_rocks

    We need a break, this is an awesome movie, even if you have seen it.

    This is math and science.


    When it loads the HTML page, I think you have to hit the bottom play button, located at the bottom of the controls for the movie.

    If you hit the play button on the screen I believe that it loads another occurrence of the movie.

    I used to watch hour long specials on Stonehenge, as all kinds of highly decorated sorts, contemplated everything from aliens to Buck Rogers moving the stones with space rays. Sharp as a marble they were, Wally proves it.

    Lazy thats what I say they were.

    Sincerely,


    William McCormick
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  71. #70  
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    Good idea. That's a really impressive video. We can both certainly appreciate what he does--nature gives us a lot of simple but powerful tools to work with, and when we combine them in novel ways, we can accomplish even greater things. The tremendous thing is how simple his ideas are. Once you see them, it's obvious how and why they work, but it takes a sharp guy like him to see the simplicity in the first place.
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    Quote Originally Posted by serpicojr
    Buddy, I didn't use anything about the specific value of ∏. All I used is that, if we take a unit circle (radius 1), then there is a number ∏ such that 2∏ is the circumference of the circle (clear), and we have ∏ is also the area of the circle (which you can prove by, say, calculus). Then I used simple formulas for areas of triangles and circular sectors and for the length of a circular arc.

    Please, do not resort to name-calling. You don't know how silly it makes you look.
    I was not calling you names in anyway shape or form.

    In fact what you did was not a proof of what pi is, but an exercise in using a value you are claiming is pi. An exercise that some exceptional kindergartner could do.
    I don't think he would have thought it up. But he could go through the motions. It was not getting to the bottom of how pi is determined. That is going to take more then just you and me. I was actually saying that you were insulting me.

    I believe you are much smarter then you are putting forward. You just don't want to go to the dirty places to hash it out with me. You give me the standard rhetoric to confuse or put off the issue at hand. I have been going over this since the forth grade. That is a long time ago. Ha-ha.

    I believe that together we could explain how pi gets its value, and what it is. Rather then changing it to something else. So it does not conflict with a polygon.


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    William McCormick
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    Quote Originally Posted by serpicojr
    Good idea. That's a really impressive video. We can both certainly appreciate what he does--nature gives us a lot of simple but powerful tools to work with, and when we combine them in novel ways, we can accomplish even greater things. The tremendous thing is how simple his ideas are. Once you see them, it's obvious how and why they work, but it takes a sharp guy like him to see the simplicity in the first place.
    I am sorry that I am hard on colleges and noble prize winners. But if they don't wake up, I have no doubt what is coming for civilization. Ours is not the first here.

    Sincerely,


    William McCormick
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  74. #73  
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    Okay, no holds barred. If I were to calculate ∏, I would do so analytically, i.e. via calculus. My gameplan would be this:

    1. Think of sine and cosine (and the rest of the trig ratios) as functions of a real variable, with radians being the argument.

    2. Define ∏ as the first positive number so sin(∏) = 0.

    3. Show the relationship between the derivatives of sine and cosine. (This requires a geometric argument very similar to some of my arguments about triangles.)

    4. Show the derivative of arctan(x) is 1/(1+x<sup>2</sup>).

    5. Develop the power series of arctangent.

    6. Deduce a series expression for ∏ using arctan(1) = ∏/4.

    By truncating the series, I will get decent rational approximations. I can use Taylor's inequality to find error bounds for such, so I'll have a quantitative expression for how good my approximations are. How does that sound? Again, the geometry is still there, but it's hidden in step 3.
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    I think that part of the problem is that if you measure a rectangle width and height, and then measure the diagonal across that rectangle, even with just a good ruler or good tape measure.

    You will see that the formulas in cadd are close. But they show the diagonal smaller then what it really is.

    You don't need to be a brain surgeon to see the difference. So before you can compare what I am saying about a wheel circumference we have to get the triangle formulas in order.

    A^+B^=C^ is not true by actual test. It does not really offer an accurate answer. In a short span it is off by a pretty good amount. Luckily in life we do not really rely on it for much in building. But rather for a double check or approximating the quantity. Most things are done with square. And you work around the inconsistencies.

    I guess I am saying that you cannot use your formula because it is flawed.



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    William McCormick
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  76. #75  
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    Quote Originally Posted by William McCormick
    I think that part of the problem is that if you measure a rectangle width and height, and then measure the diagonal across that rectangle, even with just a good ruler or good tape measure.

    You will see that the formulas in cadd are close. But they show the diagonal smaller then what it really is.

    You don't need to be a brain surgeon to see the difference. So before you can compare what I am saying about a wheel circumference we have to get the triangle formulas in order.

    A^+B^=C^ is not true by actual test. It does not really offer an accurate answer. In a short span it is off by a pretty good amount. Luckily in life we do not really rely on it for much in building. But rather for a double check or approximating the quantity. Most things are done with square. And you work around the inconsistencies.

    I guess I am saying that you cannot use your formula because it is flawed.



    Sincerely,


    William McCormick
    William

    It appears that you value empirical, actually measured, ratios and distrust anything achieved analytically.

    Fine.

    Take a sheet of A4 paper and fold down the short side to the long side, giving you a diagonal at 45 degrees. This diagonal is apporximately the length of the long side. Why? Because when A4 paper was devised the ratio of long to short side was chosen as 99 to 70 (297mm x 210mm) in order to replicate as closely as possible the irrational number that is the square root of two. This irrational number itself had its value (1.4142135 to seven decimals) calculated using precisely that a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> that you distrust.

    But that ratio, and Pythagoras, are vindicated not just by A4 paper, but by square tiles. Have you ever arranged a 5 x 12 rectangle? Lay tiles on the diagonal and you will see exactly 13 of them there. I know this because I've tried.
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  77. #76 An Illustrated Proof of the Pythagorean Theorem 
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    Consider the following picture:



    I've taken four congruent right triangles and arranged them to form two squares. One way to calculate the area of the large square is to square its side lengths:

    (a+b)<sup>2</sup> = a<sup>2</sup> + 2ab + b<sup>2</sup>

    Another way is to add up the area of the small square and the areas of the four triangles. The area of the small square is c<sup>2</sup> and the area of each triangle is ab/2, so the total area is:

    c<sup>2</sup> + 4(ab/2) = c<sup>2</sup> + 2ab

    Now since these quantities represent the same area, they're equal:

    a<sup>2</sup> + 2ab + b<sup>2</sup> = c<sup>2</sup> + 2ab

    Subtracting 2ab from both sides, we obtain:

    a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>
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  78. #77  
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    And in case you're still not convinced, here's MC Hammer giving mad props to my picture.
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    Quote Originally Posted by sunshinewarrio

    William

    It appears that you value empirical, actually measured, ratios and distrust anything achieved analytically.

    Fine.

    Take a sheet of A4 paper and fold down the short side to the long side, giving you a diagonal at 45 degrees. This diagonal is apporximately the length of the long side. Why? Because when A4 paper was devised the ratio of long to short side was chosen as 99 to 70 (297mm x 210mm) in order to replicate as closely as possible the irrational number that is the square root of two. This irrational number itself had its value (1.4142135 to seven decimals) calculated using precisely that a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> that you distrust.

    But that ratio, and Pythagoras, are vindicated not just by A4 paper, but by square tiles. Have you ever arranged a 5 x 12 rectangle? Lay tiles on the diagonal and you will see exactly 13 of them there. I know this because I've tried.
    I totally understand what you are saying the square root of 9 is 3. Using A^2+B^2=C^2

    It is beautiful in base ten.

    But if I am working with something important and normally I do not. I would not trust most available calculators or cadd programs, I get strange results. And some of these devices are going out a lot of places.

    Normally what we do is use "x" "y" coordinates, they are precise and do not rely on algorithms and complex formulas. They assure exact placement and easy cross checking between the design platform and the device used for marking or cutting or drilling.

    The thought of adding the root calculation into the mix to calculate the side of a polygon with an almost infinite number of sides, would be reverse engineering to me. It would be dangerous.

    My thought was if you actually found the real length of a wheel circumference, you could then create an exact ratio to diameter of a circle. And then just divide the circumference by the number of polygon sides in your polygon. Then calculate the length of the chord of one of the circular segments.

    But you would have to trust PI. Now we cannot. That is the problem.

    Look at this formula for calculating a polygon area.
    n=number of sides, s=side length, r=radius to center of polygon side. (nsr)/2 = Area.

    This fascinates me. The only problem is finding the length of the polygon side with extreme accuracy.

    If anyone has any explanation of how that works I will take it.

    Sincerely,


    William McCormick
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    It just dawned on me, it is turning all the triangles into squares and then cutting them in half.

    I had the old triangle area formula on my mind. Base times height divided by two. But now I get it.


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    William McCormick
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  81. #80  
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    Quote Originally Posted by William McCormick
    Look at this formula for calculating a polygon area.
    n=number of sides, s=side length, r=radius to center of polygon side. (nsr)/2 = Area.
    You're a blooming genius, William. This inspired the following in me.

    So the area of a regular polygon is nsr/2. Suppose this polygon is circumscribed about a circle. Then the r in your formula is the radius of the circle. Clearly the polygon is bigger than the circle, so since the circle has area ∏r<sup>2</sup>, we have:

    nsr/2 > ∏r<sup>2</sup>

    Dividing both sides by r and multiplying by 2, we have:

    ns > 2∏r

    The quantity on the left is the perimeter of the polygon. The quantity on the right is the circumference of the circle. Thus, using your formula, I have proved that the perimeter of a regular polygon circumscribed about a circle is always greater than the circumference of said circle.

    Do you believe me now?
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    I have been using (1/2) * radius * cirumference to double check my chaos. But I cannot figure out why it gives poor checks.

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    William McCormick
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    Quote Originally Posted by serpicojr
    Quote Originally Posted by William McCormick
    Look at this formula for calculating a polygon area.
    n=number of sides, s=side length, r=radius to center of polygon side. (nsr)/2 = Area.
    You're a blooming genius, William. This inspired the following in me.

    So the area of a regular polygon is nsr/2. Suppose this polygon is circumscribed about a circle. Then the r in your formula is the radius of the circle. Clearly the polygon is bigger than the circle, so since the circle has area ∏r<sup>2</sup>, we have:

    nsr/2 > ∏r<sup>2</sup>

    Dividing both sides by r and multiplying by 2, we have:

    ns > 2∏r

    The quantity on the left is the perimeter of the polygon. The quantity on the right is the circumference of the circle. Thus, using your formula, I have proved that the perimeter of a regular polygon circumscribed about a circle is always greater than the circumference of said circle.

    Do you believe me now?
    I never doubted that part of the argument. In fact many years ago my father made an octagon table for our kitchen. And we were drawing diagrams and stating just that.
    We said that if the polygon was drawn outside the circle then the volume must be greater. We kind of felt like any polygon we could conceive or draw, would also be longer in length.

    However my original teachings in school did not rule out a very high number sided, polygon rolling a shorter distance in one revolution then a true circle. And I kind of kept an open mind to it.

    I have trouble with the length of the sides of the imaginary polygon. I am trying to understand so I can stand with you perhaps, on this. And get rid of Archimedes ratio once and for all if that is the case.

    However as you move into a more complex polygon, I need to know how the lengths of the polygon are being acquired. For this computer simulation.

    I can take a square that is circumscribed around a circle and use Archimedes ratio. And show that area that is left outside a circle but still in the square is what is not in the circle.

    So I just need to understand how the length of the sides of the polygon are determined by the computer simulation. Is it with the intersection of a circle, at certain degrees? Or some other method.

    I am starting to have a really good time here. I am enjoying your points of view.


    Sincerely,


    William McCormick
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    I haven't read all the posts so maybe it's been explained but why is the area area of a regular polygon equivalent with nsr/2?
    373 13231-mbm-13231 373
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    The regular polygon can divided into n congruent triangles with lines joining the centre to the vertices. r is the distance from the centre of the polygon to a midpoint of one of its sides. This is perpendicular to the side, whose length is s; hence the area of each of the congruent triangles is ½rs. There are n triangles so the polygon has area ½nrs.
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  86. #85  
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    Quote Originally Posted by William McCormick
    I have trouble with the length of the sides of the imaginary polygon. I am trying to understand so I can stand with you perhaps, on this. And get rid of Archimedes ratio once and for all if that is the case.
    The point is that we don't need to know what the lengths of the sides are. We know that they have a well-defined length, and since we're assuming regularity, we know all side lengths are the same. This is enough for us to work with. My proof never needs any specific values. I'm able to use geometric arguments to compare area, and then I'm able to use this comparison to compare perimeters and circumferences via algebra.

    If we want to sit down and calculate the exact side lengths of the triangles we're playing with, we can probably use known values of trigonometric functions and trig identities. For example, we have the half angle formula which relates cos(x) to cos(x/2):

    cos(x/2) = ± sqrt((1+cos(x))/2)

    This allows us to calculate, say, all cosines for angles of the form n∏/2<sup>m</sup>, n and m whole numbers, since we know that cos(n∏) = (-1)<sup>n</sup>. And then the Pythagorean theorem states:

    [sin(x)]<sup>2</sup> + [cos(x)]<sup>2</sup> = 1

    (This is the statement for a right triangle with hypotenuse 1.) So using this, we can figure out sine for the same angles. And this is all we need to calculate the perimeter of a polygon with number of sides equal to a power of 2.
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    Quote Originally Posted by serpicojr
    The point is that we don't need to know what the lengths of the sides are.
    Now before I threw Archimedes away, I would need to know how they ascertain the exact length of the side of the polygon that they are totaling the sides of. In a cumulative way that would exaggerate the error normally created by using square root.

    In real life to build things we would intersect rays with a circle, to form a polygon.



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    William McCormick
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    Quote Originally Posted by JaneBennet
    The regular polygon can divided into n congruent triangles with lines joining the centre to the vertices. r is the distance from the centre of the polygon to a midpoint of one of its sides. This is perpendicular to the side, whose length is s; hence the area of each of the congruent triangles is ½rs. There are n triangles so the polygon has area ½nrs.
    I had trouble with how it creates a square. But you put it well.


    It can also use the radius to the center of one side of the polygon, as the height of a parallelogram, and multiply that by the length of one side of the polygon, and then multiply by the number of sides of the polygon, and then just cut that area in half.

    It creates as many parallelograms as there are isosceles triangles, and then just cuts that area in half.

    When you reverse two isosceles triangles you get a parallelogram. The formula for a parallelogram is one side times the height.

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    William McCormick
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    Quote Originally Posted by serpicojr
    The point is that we don't need to know what the lengths of the sides are. We know that they have a well-defined length, and since we're assuming regularity, we know all side lengths are the same. This is enough for us to work with. My proof never needs any specific values. I'm able to use geometric arguments to compare area, and then I'm able to use this comparison to compare perimeters and circumferences via algebra.

    If we want to sit down and calculate the exact side lengths of the triangles we're playing with, we can probably use known values of trigonometric functions and trig identities. For example, we have the half angle formula which relates cos(x) to cos(x/2):

    cos(x/2) = ± sqrt((1+cos(x))/2)

    This allows us to calculate, say, all cosines for angles of the form n∏/2<sup>m</sup>, n and m whole numbers, since we know that cos(n∏) = (-1)<sup>n</sup>. And then the Pythagorean theorem states:

    [sin(x)]<sup>2</sup> + [cos(x)]<sup>2</sup> = 1

    (This is the statement for a right triangle with hypotenuse 1.) So using this, we can figure out sine for the same angles. And this is all we need to calculate the perimeter of a polygon with number of sides equal to a power of 2.
    Many in the trades that use pi, say to just use 3.14 if you are not going to ascertain the actuality of pi. Heck it is much simpler and avoids all the nonsense. You can always make some corrective little additive later.

    You will see that things work well with pi at 3.14 it will just be a ratio of diameter to circumference. It will simplify working with it. Some places want it used to avoid error.

    But when you actually have to make things that require an exact ratio, then you will be back to 22/7 for sure.

    When you make bands of metal that will fit around a round object. You calculate the center line of the band, as it will be around the round object. That becomes your diameter. The diameter you calculate with, will be one thickness of the band larger then the object it wraps around. Then you multiply that by pi and you get a piece that fits around a round. You should use the right pi though.

    And if you are making wheels that measure distance of fresh materials manufactured, it also pays to have the right ratio of pi.

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    William McCormick
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  90. #89  
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    Let me ask you a few questions.

    If you had a big enough circle, the width of the tape measure would be insignificant in measuring the circumference right?

    If you build a square with 1 foot sides, how would you measure its perimeter? What answer would you get?

    If you made a circle with a 6 inch radius, would the thickness of a tape measure make a significant difference in what is measured? Is the circumference of a 6-inch-radius circle less than the perimeter of a 1-foot-per-side square?

    If you build a regular (all angles equal) octagon with 1 foot sides, what would the perimeter be? If you build a circle that barely fits inside the octagon, would you agree that any minute differences between the radius of the circle and the inradius (half the distance between the center of two opposite edges) wouldn't make a significant difference in the final results? What would the circumference of this circle be?

    What if we used a regular 360 sided polygon, with each side 1 foot long? What would the perimeter of such a polygon be? What would the circumference of a circle that just barely fit inside it be?

    How many sides would the polygon have to have to have a smaller perimeter than the circumference of a circle that just barely fits inside of it?
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    Quote Originally Posted by MagiMaster
    Let me ask you a few questions.

    If you had a big enough circle, the width of the tape measure would be insignificant in measuring the circumference right?

    If you build a square with 1 foot sides, how would you measure its perimeter? What answer would you get?

    If you made a circle with a 6 inch radius, would the thickness of a tape measure make a significant difference in what is measured? Is the circumference of a 6-inch-radius circle less than the perimeter of a 1-foot-per-side square?

    If you build a regular (all angles equal) octagon with 1 foot sides, what would the perimeter be? If you build a circle that barely fits inside the octagon, would you agree that any minute differences between the radius of the circle and the inradius (half the distance between the center of two opposite edges) wouldn't make a significant difference in the final results? What would the circumference of this circle be?

    What if we used a regular 360 sided polygon, with each side 1 foot long? What would the perimeter of such a polygon be? What would the circumference of a circle that just barely fit inside it be?

    How many sides would the polygon have to have to have a smaller perimeter than the circumference of a circle that just barely fits inside of it?
    I like your thinking, about the different angles. I have though yet to see anyway to compare a circle that is just an arc. To a line, without pi. I just don't see any connection anywhere.

    I would like to know if there is a precision wheel somewhere that they use for proof of pi?

    As far as measuring the thickness of the band, it is very important, in actual day to day use.

    And if we are going to talk precision, it would probably be out of the question.

    It is funny but in real application if you are using say 1/16" material to make a square Architectural style tube with somewhat sharp radius bent corners, like most standard Architectural box tubing. With an outside measurement of 2" square. The material length only has to be seven and a half inches long. You gain a 1/16" at every side end.

    On round tubular manufacture the wall thickness does in fact play a similar roll. You have to go bigger.

    A metal tape cannot bend around something round and stay the same length on both sides of the tape. It is just an impossibility. So usually the center line of the tape, becomes about the point that is the actual length before it was bent.

    So when I calculate for pipe manufacture, I use the completed center line radius, to make the cut on the flat sheet. I use the larger version of pi.

    Some of the mechanical bending equipment can cause some little differences here and there.

    But we compensate for them. In pipe bending, I use a slightly smaller pi formula. About 2.93, and calculate the center of the pipe when it is finished to get the cut length.

    Sincerely,


    William McCormick
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  92. #91  
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    Sorry, you lost me. However, it sounds like what you're talking about isn't actually changing pi, but changing some correction factor, which in a perfect/ideal/virtual system would be 1.

    So when you measure the circumference, you get something like 2*pi*radius*correction, where correction is based on what and how you're measuring. Does that sound right to you?
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    William, you'll never prove what π is equal to by using a precision wheel. You may come up with estimates, but that's all you can get. Proving what the value of π is requires rigorous mathematical arguments based on Euclidean geometry or calculus. No matter how precise your wheel is, no matter how steady your hand is, no matter how sharp your vision is, some error will always creep in when you try to measure things with a precision wheel.
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    Quote Originally Posted by serpicojr
    William, you'll never prove what π is equal to by using a precision wheel. You may come up with estimates, but that's all you can get. Proving what the value of π is requires rigorous mathematical arguments based on Euclidean geometry or calculus. No matter how precise your wheel is, no matter how steady your hand is, no matter how sharp your vision is, some error will always creep in when you try to measure things with a precision wheel.
    Then your math will always be one step behind me. And in doubt of what my wheel does or doesn't, would or would not roll.

    I am just dying to see the simple formula you use to create the polygon. I mean to go out all those places of possible error, ones foundation must be almost error free.

    It may be I am just interested in seeing how you came up with the polygon measurements. Mostly the side length. I could not do it without a circle using pi as the circumference. Surly you could share that with me.



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    William McCormick
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  95. #94  
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    Quote Originally Posted by William McCormick
    I am just dying to see the simple formula you use to create the polygon.
    I've already given you all the formulas you need, complete with proofs that they show what I claim. Once again... read my proofs.

    Then your math will always be one step behind me.
    No.
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    I can atleast say that for a n-sided circumscribed regular polygon with inradius 1 foot, the side length would be 2*tan(180/n) feet (if you take the tangent in degrees). This should be accurate to the limit of your measuring and manufacturing devices.
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    Quote Originally Posted by MagiMaster
    I can atleast say that for a n-sided circumscribed regular polygon with inradius 1 foot, the side length would be 2*tan(180/n) feet (if you take the tangent in degrees). This should be accurate to the limit of your measuring and manufacturing devices.
    I work building real things and rarely, thank the lord do I need to use sine cosine tangent.

    But I don't see how that works out.


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    William McCormick
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  98. #97 Re: An Illustrated Proof of the Pythagorean Theorem 
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    Quote Originally Posted by serpicojr
    Consider the following picture:



    I've taken four congruent right triangles and arranged them to form two squares. One way to calculate the area of the large square is to square its side lengths:

    (a+b)<sup>2</sup> = a<sup>2</sup> + 2ab + b<sup>2</sup>

    Another way is to add up the area of the small square and the areas of the four triangles. The area of the small square is c<sup>2</sup> and the area of each triangle is ab/2, so the total area is:

    c<sup>2</sup> + 4(ab/2) = c<sup>2</sup> + 2ab

    Now since these quantities represent the same area, they're equal:

    a<sup>2</sup> + 2ab + b<sup>2</sup> = c<sup>2</sup> + 2ab

    Subtracting 2ab from both sides, we obtain:

    a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>

    The three four five triangle is perhaps the best proof. (3x3) + (4x4) = 25 and the square route of 25 is 5. That is what I use to lay out pipe penetrations from a main branch. There is inaccuracy, it is in the measuring devices and the way you can use them. But for things like that it is more then close enough.

    In fact my old friend Ben Jackson who's father was a Russian History professor taught me the 3,4,5 triangle.

    I guess what I am saying is that over the years using the cruddy devices we have to calculate with. So many errors occur while using A^2+B^2=C^2 That the in accuracies are either input related or device related. Often because of the rather large complex fractions you cannot easily see a slight input error.

    So in the end I tend to resort to other methods. In actual day to day use. They are less prone to the weaknesses in our math and equipment.

    We usually site things in. Using a known square object. It is usually the clear winner for accuracy.

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    William McCormick
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  99. #98  
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    Quote Originally Posted by William McCormick
    rarely, thank the lord do I need to use sine cosine tangent.
    I thought you liked math.

    Anyway, the 3-4-5 triangle is not a proof of the Pythagorean theorem; it is an example of a right triangle, and you can show it as an illustration of the theorem. This doesn't tell you anything about the infinitely many other right triangles out there.
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  100. #99  
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    Quote Originally Posted by serpicojr
    Quote Originally Posted by William McCormick
    rarely, thank the lord do I need to use sine cosine tangent.
    I thought you liked math.

    Anyway, the 3-4-5 triangle is not a proof of the Pythagorean theorem; it is an example of a right triangle, and you can show it as an illustration of the theorem. This doesn't tell you anything about the infinitely many other right triangles out there.
    I like math my way. Not the way it is.


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    William McCormick
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  101. #100  
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    So how do you do trigonometry if not with trigonometric functions?
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