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From "A History of Pi" by Petr Beckmann, a mathematician from Czechoslovakia. "In 1936, a tablet was excavated some 200 miles from Babylon. Here one should make the interjection that the Sumerians were first to make one of man's greatest inventions, namely, writing; through written communication, knowledge could be passed from one person to others, and from one generation to the next and future ones. They impressed their cuneiform (wedge-shaped) script on soft clay tablets with a stylus, and the tablets were then hardened in the sun. The mentioned tablet, whose translation was partially published only in 1950, is devoted to various geometrical figures, and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60^2) (the Babylonians used the sexagesimal system, i.e., their base was 60 rather than 10). The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day). The tablet, therefore, gives ... Pi = 25/8 = 3.125."

[The actual ratio is 0.1837762969058794 One side of a hexagon, divided by the circumference of the inscribed circle within the hexagon, mathematically gives you the ratio, of 0.1837762969058794There does not seem to be any ratio, between the full perimeter of the hexagon and the inscribed circle circumference. Maybe their hieroglyphics are not too good? Ha-ha. The ratio of the circumference of a circle, that is inscribed in an octagon, divided by the perimeter of that octagon is 0.9480594157436246That is in the ball park of the Babylonian translation. In actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle. Sincerely, William McCormick

Quote from: William McCormick on 12/08/2012 23:29:14[The actual ratio is 0.1837762969058794 One side of a hexagon, divided by the circumference of the inscribed circle within the hexagon, mathematically gives you the ratio, of 0.1837762969058794There does not seem to be any ratio, between the full perimeter of the hexagon and the inscribed circle circumference. Maybe their hieroglyphics are not too good? Ha-ha. The ratio of the circumference of a circle, that is inscribed in an octagon, divided by the perimeter of that octagon is 0.9480594157436246That is in the ball park of the Babylonian translation. In actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle. Sincerely, William McCormick Reg Hexagon - with side 1 has a perimeter of 6. Circumscribed circle (ie outside around the hexagon touching every vertex AND thus radius of 1) has perimeter of 2pi or 6.2832. Ratio of Hexagon perimeter to Circumscribed Circle = 6/6.2832 = .9549. The Babylonians stated it as 57/60 + 36/3600 which in our notation is 0.96. So firstly, they were a hell of a lot closer than you :-) and secondly, thats not a bad estimation (they would have been closer at 57/60 + 18/3600)

Quote from: imatfaal on 14/08/2012 18:54:40Quote from: William McCormick on 12/08/2012 23:29:14[The actual ratio is 0.1837762969058794 One side of a hexagon, divided by the circumference of the inscribed circle within the hexagon, mathematically gives you the ratio, of 0.1837762969058794There does not seem to be any ratio, between the full perimeter of the hexagon and the inscribed circle circumference. Maybe their hieroglyphics are not too good? Ha-ha. The ratio of the circumference of a circle, that is inscribed in an octagon, divided by the perimeter of that octagon is 0.9480594157436246That is in the ball park of the Babylonian translation. In actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle. Sincerely, William McCormick Reg Hexagon - with side 1 has a perimeter of 6. Circumscribed circle (ie outside around the hexagon touching every vertex AND thus radius of 1) has perimeter of 2pi or 6.2832. Ratio of Hexagon perimeter to Circumscribed Circle = 6/6.2832 = .9549. The Babylonians stated it as 57/60 + 36/3600 which in our notation is 0.96. So firstly, they were a hell of a lot closer than you :-) and secondly, thats not a bad estimation (they would have been closer at 57/60 + 18/3600)That ratio you are describing only works for that one sized, hexagon, with a perimeter totaling six. It serves no purpose. The one ratio I gave you, works on all sized circles and octagons, it is the ratio between the circles circumference, and the octagons perimeter. That could be useful. A circles circumference, that is totally inclosed in a octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it. No matter the size of the circle. You can also use the other ratio I gave you on any circle, and one side of any hexagon. One side of a hexagon, is 0.1837763181265384 percent of the circumference of a circle that is totally inside and yet touching each side of that hexagon. Sincerely, William McCormick

This was created with a very exacting cadd program. "Inclosed" is the same as "enclosed" in America. I am older so I use the inclosed instead of enclosed.

The cadd program is very accurate the numbers in white are mine, and every now and then I will do a copy and paste from a calculator, and grab a fifteen digit number that is a little different at the end, because of the base 9/10 system we use. But the point I am making remains the same. The Cadd program is most accurate. Made for humans. I actually rolled a wheel that I machined and at first it rolled a ratio of 3.14159 and I was actually a little disappointed because others that I know machined a wheel and theirs rolled a longer distance per revolution. Their ratio was more like Archimedes ratio. I cleaned up the wheel with Xylene and rolled it again, I wanted to report the actual length it rolled, thinking it would roll an even shorter length, once cleaned. I found, it rolled a much longer distance clean then if it was dirty. It rolled 3.14308To me what was important was that, a wheel with debris on it rolled a shorter distance in one revolution, then a clean wheel. I rolled it in both conditions many times. And always got the same results. I believe that colleges have been disingenuous over the years. One college experiment claimed that they used a metal tape around a large wheel they made to check the size. The only problem is that the tape expands and contracts so the area near the center line of the metal tape they use. Will actually be the straight laid out length of the tape, when they straighten it to measure it. So if they really got 3.14159 as a ratio of the measurement, the actual size of the diameter of the circle they were measuring would be almost the thickness of the tape measure larger. It was just a personal experiment I did for myself. I learned a lot from it. I machined a 70/75 T8 plate of aluminum into a wheel. And also used the same material to roll the wheel on. Sincerely, William McCormick

Quote from: William McCormick on 25/08/2012 22:31:123.464102" should read 3.464101615"18.849556" should read 18.84955592"20.784612" should read 20.78460969"and the quotient should read 1.102657909 based on the other two values displayed OR1.102657791 based on the more accurate figures quoted aboveIn either event the quoted value of 1.102658285566954 is not correct, and the last half of this very long decimal is totally pointless!In fact a whole lot of the figures quoted -- more than half of them -- are totally pointless.When a scientist quotes a figure like 1.1026, this represents a claim that the value s/he has measured is in the range 1.1025 to 1.1027 (strictly 1.10255 to 10265, but the wider latitude is usually allowed for convenience). Engineers usually like to carry an extra figure or two through the working, but the final result they quote is still in accordance with this convention.So if a chemist claims that the concentration of an impurity is 2 parts per million, that could mean anything between 1.5 and 2.5 parts per million. If s/he says 2.00 parts per million, that is a much stronger claim for a value between 1.99 and 2.01 parts per million.If you are talking about machining, and distances of a few inches, you are optimistic to claim 3 decimal places -- 4 significant figures -- and any more figures than that is either a ludicrous claim or a coded message! In the diagrams above we should be seeing values like 3.000" and 3.464" -- any more figures than this is just plain wrong!

Oh dear, William!Anybody can link pages of a dictionary into a post. It takes a little more skill to read them.From your latest post I gather, although you have not said it very clearly, that you meant that your cadd program was precise, and not particularly exacting.

So, since you haven't answered.How did you measure the distance your wheel rolled (and its diameter) to six digits?And have you realised why your previous statements of the ratios were out by a factor of a hundred.

One side of a hexagon, is 0.1837763181265384 percent of the circumference of a circle that is totally inside and yet touching each side of that hexagon.

This was created with a very exacting cadd program.

"making unreasonable or inconsiderate demands; taxing; arduous."

In actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle.

OK then William, enough with the riddles:(1)QuoteOne side of a hexagon, is 0.1837763181265384 percent of the circumference of a circle that is totally inside and yet touching each side of that hexagon.Your own diagram clearly shows that it is 18.377...%Similarly the next figure in your post should read 94.806...%(incidentally the "..." above are not to be interpreted as part of what should be; they simply indicate that if you wanted to add a whole lot of other figures you would be entitled to, provided that you were using a standard value for π. They represent mathematically exact quantities, not engineering achievements.)(2) QuoteThis was created with a very exacting cadd program. Your own dictionary, whose pages you linked, shows that "exacting" meansQuote"making unreasonable or inconsiderate demands; taxing; arduous." (and gives no indication of any other meaning). It became increasingly clear in your later posts that this was not what you were trying to say about your cadd program.(3) The point about significant figures quoted in a result is far from a trivial one. It could make a vital difference if a chemical assay claimed 2.0 ppm contamination instead of 2 ppm contamination and the actual level proved to be 2.3 ppm. The former claim would be quite wrong and actionable; the latter is correct.If this is applied to your inscribed figure diagrams, then quite apart from using 7 figures in the first place, it is completely nonsensical to quote 16 significant figures when you divide two 7 figure numbers by one another: at best only the first 7 figures of your answer will have any meaning; in your case even the 7th is wrong: digits 8 through 16 are astrological portents!(4)QuoteIn actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle. This brings up a different aspect of experimental error.To claim that a measurement you have made shows that 22/7 is a better estimate for π than the mathematically correct value (I say mathematically correct because π arises in all sorts of strange contexts in mathematics, many of which have nothing to do with geometry, and the well-known value of π -- to a million figures or more -- is certainly not obtained by geometric measurement). That brings into play the exact parameters of your experimental measurement, and makes the detail of how you actually performed the measurement vitally important. Is the method you used for estimating when one "roll" was complete reliable to the necessary accuracy? When you repeated the measurement, was the result of your previous measurement uppermost in your mind? What precautions were made to avoid any slippage in the roll?To measure the accurate value of circumference/diameter within the 1 part in 10,000 that would be necessary to validate your claim would be an exacting task that may well not provide an exact outcome.Finally you ask me for more links in my posts, and especially for links to my own work. My teaching material -- such of it as remains after successors have taken over most of my modules -- is on a restricted site -- the policy of my university. Some of my published work in the peer reviewed literature is accessible on the web, on a pay-to-view basis from several Journal websites. Most of it, though, predates what has been uploaded to the web, or is in scholarly book chapters that are not available on the web. I can send you some of my stuff by private email if you really want it.

Under the synonymous listing for for exacting is "absolute". Here is something from another dictionary about it. I was using it correctly in America. ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

OK then William, enough with the riddles:(3) The point about significant figures quoted in a result is far from a trivial one. It could make a vital difference if a chemical assay claimed 2.0 ppm contamination instead of 2 ppm contamination and the actual level proved to be 2.3 ppm. The former claim would be quite wrong and actionable; the latter is correct.If this is applied to your inscribed figure diagrams, then quite apart from using 7 figures in the first place, it is completely nonsensical to quote 16 significant figures when you divide two 7 figure numbers by one another: at best only the first 7 figures of your answer will have any meaning; in your case even the 7th is wrong: digits 8 through 16 are astrological portents!(4)QuoteIn actual testing I have found that 22/7 is closer to the actual, circle circumference divided by diameter. But 3.14308 were my actual test ratios of a wheel I machined. It was interesting to learn that particles very fine particles on the wheel made it roll a shorter distance. When you completely sanitize the wheel it rolls a longer distance. Just a geometric reality of a shape that is not a perfect circle. This brings up a different aspect of experimental error.To claim that a measurement you have made shows that 22/7 is a better estimate for π than the mathematically correct value (I say mathematically correct because π arises in all sorts of strange contexts in mathematics, many of which have nothing to do with geometry, and the well-known value of π -- to a million figures or more -- is certainly not obtained by geometric measurement). That brings into play the exact parameters of your experimental measurement, and makes the detail of how you actually performed the measurement vitally important. Is the method you used for estimating when one "roll" was complete reliable to the necessary accuracy? When you repeated the measurement, was the result of your previous measurement uppermost in your mind? What precautions were made to avoid any slippage in the roll?To measure the accurate value of circumference/diameter within the 1 part in 10,000 that would be necessary to validate your claim would be an exacting task that may well not provide an exact outcome.Finally you ask me for more links in my posts, and especially for links to my own work. My teaching material -- such of it as remains after successors have taken over most of my modules -- is on a restricted site -- the policy of my university. Some of my published work in the peer reviewed literature is accessible on the web, on a pay-to-view basis from several Journal websites. Most of it, though, predates what has been uploaded to the web, or is in scholarly book chapters that are not available on the web. I can send you some of my stuff by private email if you really want it.

QuoteUnder the synonymous listing for for exacting is "absolute". Here is something from another dictionary about it. I was using it correctly in America. ...sorry, you cannot view external links. To see them, please REGISTER or LOGINSorry, but you were not using the word correctly, even in North America.Your cadd program is not "absolute" in any of the senses of that word.The Merriam-Webster definition 2 refers only to something which requires great precision in its use, and is therefore delicate or dainty. There is no indication anywhere in either of the definitions of the word that it refers to something that produces results with great precision.

Obviously no one here was following the directions well. Even you accused me of putting the perimeter of the hexagon over the circumference, that is what I was supposed to do according the directions given by Boogie.

It has been a cluster barrage of error all around.

QuoteObviously no one here was following the directions well. Even you accused me of putting the perimeter of the hexagon over the circumference, that is what I was supposed to do according the directions given by Boogie.I have no recollection of having done so, and cannot find such a reference anywhere in the thread. Perhaps you could link what you are seeing that I said.QuoteIt has been a cluster barrage of error all around.I am thinking here that you might be misunderstanding the scientific use of "error". It is a technical term, and quite different from the everyday usage as a synonym for "mistake" or "miscalculation". A chemical assay is often quoted as something like "23.15±0.03%". What this means is that when the experiment is reviewed, and all of the uncertainties in the precision of a volume of liquid or an instrument reading are taken into account along with possible inaccuracy in the experimental design, we can say with 95% confidence that the true composition of a material is within 0.03 of 23.15%. 0.03% is referred to as the "experimental error". There is no question of this embracing any misconception or miscalculation. The number of significant figures quoted is a slightly less formal and less precise way of making a statement about the experimental error than the "±" method.

The one ratio I gave you, works on all sized circles and octagons, it is the ratio between the circles circumference, and the octagons perimeter. That could be useful. A circles circumference, that is totally inclosed in a octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it. No matter the size of the circle.

Quote from: damocles on 27/08/2012 03:35:26QuoteObviously no one here was following the directions well. Even you accused me of putting the perimeter of the hexagon over the circumference, that is what I was supposed to do according the directions given by Boogie.I have no recollection of having done so, and cannot find such a reference anywhere in the thread. Perhaps you could link what you are seeing that I said.QuoteIt has been a cluster barrage of error all around.I am thinking here that you might be misunderstanding the scientific use of "error". It is a technical term, and quite different from the everyday usage as a synonym for "mistake" or "miscalculation". A chemical assay is often quoted as something like "23.15±0.03%". What this means is that when the experiment is reviewed, and all of the uncertainties in the precision of a volume of liquid or an instrument reading are taken into account along with possible inaccuracy in the experimental design, we can say with 95% confidence that the true composition of a material is within 0.03 of 23.15%. 0.03% is referred to as the "experimental error". There is no question of this embracing any misconception or miscalculation. The number of significant figures quoted is a slightly less formal and less precise way of making a statement about the experimental error than the "±" method."Your own diagram clearly shows that it is 18.377...%Similarly the next figure in your post should read 94.806...%(incidentally the "..." above are not to be interpreted as part of what should be; they simply indicate that if you wanted to add a whole lot of other figures you would be entitled to, provided that you were using a standard value for π. They represent mathematically exact quantities, not engineering achievements.)"That ratio you mention would indicate the circumference over the perimeter of the hexagon. Boogie had stated perimeter of the hexagon over the circumference, that will give you the 1.102658........ ratio. Sincerely, William McCormick

From William:QuoteThe one ratio I gave you, works on all sized circles and octagons, it is the ratio between the circles circumference, and the octagons perimeter. That could be useful. A circles circumference, that is totally inclosed in a octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it. No matter the size of the circle. This is the "next figure" I was referring to -- nothing to do with Boogie's post. You are erroneously quoting a ratio as a percentage once more. For the record, you were technically quite right about "circumscribed".Quote from: William McCormick on 27/08/2012 04:08:57Quote from: damocles on 27/08/2012 03:35:26QuoteObviously no one here was following the directions well. Even you accused me of putting the perimeter of the hexagon over the circumference, that is what I was supposed to do according the directions given by Boogie.I have no recollection of having done so, and cannot find such a reference anywhere in the thread. Perhaps you could link what you are seeing that I said.QuoteIt has been a cluster barrage of error all around.I am thinking here that you might be misunderstanding the scientific use of "error". It is a technical term, and quite different from the everyday usage as a synonym for "mistake" or "miscalculation". A chemical assay is often quoted as something like "23.15±0.03%". What this means is that when the experiment is reviewed, and all of the uncertainties in the precision of a volume of liquid or an instrument reading are taken into account along with possible inaccuracy in the experimental design, we can say with 95% confidence that the true composition of a material is within 0.03 of 23.15%. 0.03% is referred to as the "experimental error". There is no question of this embracing any misconception or miscalculation. The number of significant figures quoted is a slightly less formal and less precise way of making a statement about the experimental error than the "±" method."Your own diagram clearly shows that it is 18.377...%Similarly the next figure in your post should read 94.806...%(incidentally the "..." above are not to be interpreted as part of what should be; they simply indicate that if you wanted to add a whole lot of other figures you would be entitled to, provided that you were using a standard value for π. They represent mathematically exact quantities, not engineering achievements.)"That ratio you mention would indicate the circumference over the perimeter of the hexagon. Boogie had stated perimeter of the hexagon over the circumference, that will give you the 1.102658........ ratio. Sincerely, William McCormick

Now we have sorted that out, how about answering the other question.How did you measure how far a wheel rolled to six digits?(And how did you measure the diameter to that accuracy and, come to think of it, how did you actually machine it to that accuracy?)