Naked Science Forum
On the Lighter Side => New Theories => Topic started by: William McCormick on 12/08/2012 23:29:14
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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 0.1837762969058794
There 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 0.9480594157436246
That 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
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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 0.1837762969058794
There 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 0.9480594157436246
That 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)
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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 0.1837762969058794
There 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 0.9480594157436246
That 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
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[
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 0.1837762969058794
There 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 0.9480594157436246
That 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
Are you really too dim to realise that, in the same way that the ratio is the same for all octagons, it's the same for all hexagons.
Saying pi "serves no purpose" is pretty dim too.
Finally
"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. "
Nope, it's not.
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When you use terms like dim, to get to the truth, or clear up an misunderstanding, I doubt you are after truth.
I see what is happening here. I have seen this before, it is the language itself, not the understanding of the geometry.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2FcircumcribeScript.jpg&hash=5906696be38ca31e5eeb0c82859a1f6f)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Finscribe.jpg&hash=cd79cbf3f9fac59658e21df97a5cd2e3)
If you are talking about a polygon and a circumscribed circle, the circle is inside the polygon.
If you are talking about a circle and a circumscribed polygon the polygon is inside.
I think the best way to say it, is to explain it in plain terms. Those two words have caused good people to leave forums forever. The people that left actually knew the correct way to say it.
If you wanted to get fancy and or uppity, you might say a polygon in the circumscription of a circle.
It is true that a hexagon in the circumscription of a circle, has a ratio of polygon perimeter to circle circumference of 0.9549297052370732 or a ratio of one polygon side to circle circumference of 0.1591549508728455 This ratio is something that you find in machining hand books
Also true is that one side of a hexagon, is 0.1837763181265384 percent of the circumference of a circle that is totally inside the hexagon, and yet touching each side of that hexagon. I do not know why anyone would dispute that. The perimeter of the same hexagon is 1.10265780962907 percent larger then the circumference of the circle inside the polygon, and touching the center of each side.
A circles circumference, that is totally inclosed in a octagon and touching each side of the octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it.
Here is something I thought you might find interesting.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2FArchimedes.jpg&hash=0c6de0f3c50e60ee88d5b8da5642c376)
Sincerely,
William McCormick
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I don't know why you thought those quotes from the dictionary were helpful. I know what the words mean.
You were wrong to say "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." earlier on, and you are still wrong now.
"I do not know why anyone would dispute that. " Because it's wrong by a factor of about a hundred.
"A circles circumference, that is totally inclosed in a octagon and touching each side of the octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it." Nope, still a hundredfold wrong.
And, since you have a dictionary there, you might want to check the spelling of "enclosed".
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I know from experience, when you are trying to make a point, and doing a lot of calculations you can make a simple arithmetic mistake. I did it a couple times with these simple geometric figures.
But the reality is the reality.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc1.jpg&hash=a987fe582ef5c25ba76bab51af7e8e62)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc2.jpg&hash=1dc325526a709edc470c32f745d8a015)
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.
Sincerely,
William McCormick
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From William
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.
I wonder what your dictionary tells you about the meaning of "exacting"? My American Dictionary (Funk & Wagnalls 1959 Edition) tells me that it means "making unreasonable or inconsiderate demands; taxing; arduous."
Is that what you were really wanting to say about your cadd program? or were you perhaps meaning "(overly) precise"?
And why in the same post are you an advocate of 22/7 for a π approximation (an error of 5 parts in 10,000) while quoting other values to something like 12 figures?
Surely you are aware of the approximation 355/113 for π, much loved of engineers and slide rule jockeys of my generation, because it is easily remembered with the arrangement of odd digits, and useful because it gets the value of π right to a remarkable degree of accuracy: within 1 part in ten million.
By the way, if you really did mean "exacting" in describing your cadd program, please let me know which it is in a private message so that I can avoid that one when I am next buying.
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You like to argue a lot.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Fincloseenclose.jpg&hash=66912ad7827bc12879685e0a90885364)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Fexactdef.jpg&hash=29688c404f1d8312bede39eacaacf4e2)
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 3.14308
To 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
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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.
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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 3.14308
To 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
So, you don't understand experimental error then.
I guess it's a bit like your inability to count digits.
How did you measure the ratio of the diameter to the distance rolled to 6 digits?
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(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc1.jpg&hash=a987fe582ef5c25ba76bab51af7e8e62)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc2.jpg&hash=1dc325526a709edc470c32f745d8a015)
3.464102" 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 OR
1.102657791 based on the more accurate figures quoted above
In 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!
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(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc1.jpg&hash=a987fe582ef5c25ba76bab51af7e8e62)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fhexcirc2.jpg&hash=1dc325526a709edc470c32f745d8a015)
3.464102" 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 OR
1.102657791 based on the more accurate figures quoted above
In 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!
You are just arguing about displaying ratios in our base 9/10 system. The point was that the same kind of ratios exist, between hexagons inscribed or circumscribed.
I took the information that was posted, the way it was written, Boogie just wrote it the way that many today write circumscribed. It is forever beng confused. The way he wrote it technically put the circle inside the polygon. And I could find no ratio near 0.954 or 0.96 like the information posted.
So I looked for some other ratios and found them. Some people claimed those ratios do not exist, and of course they do. So all this arguing is an attempt to belittle a fellow forum member. Not very becoming.
No one is going to measure a solid hexagon, past 1/100,000 places, because the edges of the material will not allow it. So the numbers I posted are ridiculously accurate, for the purpose I posted them.
When you go out past six or seven places, the base 9/10 system we use, distorts the answers a bit, because of the number we plug into the formula. And because I am only using six places of accuracy.
I have plenty of important things to do, I was merely taking some time to show the people that denied a ratio existed, that it existed, I did that. Unless you do not think that ratio exists.
There is a ratio between everything and everything else.
Sincerely,
William McCormick
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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.
I would like to see more links from you. Some of your own personal material posted. I really would.
As far as that cadd program, goes, it makes the drawings for real things, real projects. You actually cannot use more then six places of accuracy for anything you want to build. There is no reason no benefit. You certainly cannot measure it to make sure, so that program is exact for all practical purposes, all real endeavours.
Apparently I have to post, these links or others deny them. I have been accused of the most ridiculous things here. And I have had to waste time while our planets infrastructure collapses playing word games and proving things that should be known by people posting on a science forum. All that extra clutter hides that there is a failure in the underlying system. I am sure of it. You will be before I am done.
Sincerely,
William McCormick
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This is an image from an optiscope, showing about a three by five micron area of a chip. You can see the corner degradation that takes place. The same is true if you look at a razor blade in a comparator, or microscope. You see the distortion in any edge you create.
A micron is about or approximately 3/100,000ths of an inch. You can see the amazing accuracy the chip people can create, however matter itself, limits the amount of downsizing you can do.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Felectricity%2FChips%2Foptiscop.jpg&hash=e06c7eed5016769afce4e4cfe93dd316)
That is why six places is deemed all that is necessary for real life applications.
The chips are actually harvested because they cannot just create a perfect wafer every time. They only take the chips that pass, the rest get thrown away.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Felectricity%2FChips%2Fyield.jpg&hash=4747108d898eacfbf8a6545dd04535c1)
Aim high and take what you can get. These guys use some wild techniques to get that kind of accuracy.
Sincerely,
William McCormick
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An automobile expands and contracts one eighth of an inch, bumper to bumper, in summer and winter temperatures. So going past six place is just a waste of time for real projects.
Sincerely,
William McCormick
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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.
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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.
Never said I measured the roll of my wheel out to six digits. I don't think you can.
Even the numbers I got are just the base 9/10 system we use, trying to express an actual ratio.
My ratios were correct for the hexagon and circle as I described them, as far as I know. You have wasted a lot of communication and have not just stated what is wrong.
Also the speaking in riddles is not helpful or thought of as intelligent. If you have a point, make it. Don't waste communication.
Sincerely,
William McCormick
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OK then William, enough with the riddles:
(1)
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.
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)
This was created with a very exacting cadd program.
Your own dictionary, whose pages you linked, shows that "exacting" means
"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)
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.
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.
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OK then William, enough with the riddles:
(1)
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.
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)
This was created with a very exacting cadd program.
Your own dictionary, whose pages you linked, shows that "exacting" means
"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)
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.
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.
I was trying to save the forum a tremendous amount of embarrassment, I have been through this before. Four guys each have their own opinion of what we are talking about. I put those definitions up there so you could figure it out. I searched for ratios according to the information Boogie had posted, about a circle that was contained in the polygon. Perimeter of the hexagon over the circumference of the circle. As he put it a "hexagon and a circumscribed circle". Technically that means the circle is inside the hexagon.
I have seen this before I have looked on the web and it is actually extremely confusing. The definitions I found and posted are very good.
Those drawings I made in the previous posts are correct to six digits, for a circle completely inclosed in a hexagon and intersecting the center of each hexagon side.
Evidently Boogie meant the hexagon was contained in the space of the circle, he just did not say it that way. So I had been investigated ratios of a circle contained inside of a hexagon. And put the perimeter of the hexagon over the circumference of the circle.
If you do put the hexagon inside of the circle then you do get 0.9549297052370732 by putting the perimeter of the hexagon over the circumference of the circle.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fgeometry%2Fcircumscribed-polygon.jpg&hash=e827dc61e70a630cd5cb2ac95f1e08eb)
I was correctly following Boogie's instructions, I think he just meant, circumscribed hexagon. The way I did the hexagon and circumscribed circle, with the perimeter over the circumference was correct, according to his description. I have been on other forums and we found different rules for circumscribe and inscribe, on the web, but you know how the web changes.
I believe if you inscribe a hexagon plaque with a circle, then the inscription of the circle is inside or within the plaque. If you were to circumscribe a plaque with a circle, then the circle would be outside the plaque.
I don't know why they move everything I say somewhere else?
Sincerely,
William McCormick
-
OK then William, enough with the riddles:
(1)
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.
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)
This was created with a very exacting cadd program.
Your own dictionary, whose pages you linked, shows that "exacting" means
"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)
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.
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.
http://www.merriam-webster.com/dictionary/exacting
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Fexactdef.jpg&hash=29688c404f1d8312bede39eacaacf4e2)
Sincerely,
William McCormick
-
Under the synonymous listing for for exacting is "absolute".
Here is something from another dictionary about it. I was using it correctly in America.
http://www.merriam-webster.com/dictionary/exacting
Sorry, 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.
-
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)
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.
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.
You have to understand, I was merely testing a proposed formula, as the formula was written. I fooled with it and posted what I found. I have no interest in the places after the sixth place. I just cut and pasted to see if we were on the same page.
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. I am not letting myself off the hook either. However my purpose was a pure one. I was not trying to belittle or grade anyone. I was willing to put the proper information up there, get it squared away and move on. I think everyone could have gotten a cool lesson. As it is we have been moved to another place and cut off.
As far as rolling the wheel you would have to see it roll, to understand how smooth it rolls. It is very much like a gear on a gear bar. It is very well stuck to the table. I thought it would roll crooked or spin or slip. But it surprised me. When it rolled it rolled with a very faint super high harmonic.
I thought it would be harder to mark the start and stop of the roll. However the way the start line on the wheel, is either perpendicular to the table surface or not, is very easy to see.
The fact that I got a roll of 3.1415 is probably very significant. Because I had wiped my wheel off, and cleaned it with rags very well. I thought it was very clean. I also wiped down the slab that I was rolling on. When I cleaned both the wheel and the table with xylene for the final test, and it rolled a longer distance I was impressed with the universe. That blew my mind.
I cannot say how many variables are involved in proving what the actual circumference of the wheel is, I thought of a couple myself. But this is not something that is new, expert old machinists knew this. I suspect it has to do with printing rollers, and other things that feed material using a roller.
I do not know exactly what you do, or what kind of papers you have. Do you have any interesting chemical formulas? Rather recently I did some experimenting with sodium acetate, where you over saturate a solution with it. And then put it in the refrigerator. It does not crystallize, rather it stays liquid. When you touch it with a warm finger, it liberates a very decent amount of heat, and solidifies. It will burn flesh if left in contact with it.
I really love links that is why I always try to post some, to bad you cannot post your work.
Sincerely,
William McCormick
-
Under the synonymous listing for for exacting is "absolute".
Here is something from another dictionary about it. I was using it correctly in America.
http://www.merriam-webster.com/dictionary/exacting
Sorry, 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.
If you are married I am going to say a prayer for your wife, God bless her. Ha-ha
Here you can see the same definitions apply to the word exact as well as exacting. Funk and Wagnalls just wrote it the way they did. They did not rule out my usage. I know you can use it that way. I had an English teacher Army trained, he used to jump into the garbage can to get us to remember supersede, is not spelled superseed.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Fexacting.jpg&hash=5d707717e2669f4f91ae4abd657ddb20)
Sincerely,
William McCormick
-
Under the synonymous listing for for exacting is "absolute".
Here is something from another dictionary about it. I was using it correctly in America.
http://www.merriam-webster.com/dictionary/exacting
Sorry, 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.
Here is another where they claim that exacting just means severe.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.rockwelder.com%2Fwords%2Fexacting-severe.jpg&hash=0e81f83b5eb92cbed2f3f8555679e0fd)
Sincerely,
William McCormick
-
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.
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.
It 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.
-
Here is one that kind of says the same thing.
http://www.vocabulary.com/dictionary/exacting
Sincerely,
William McCormick
-
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.
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.
It 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:
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.
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".
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.
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.
It 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
-
"Never said I measured the roll of my wheel out to six digits. "
Oh yes you did.
"I actually rolled a wheel that I machined and at first it rolled a ratio of 3.14159"
-
From William:
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.
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".
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.
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.
It 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
"A circles circumference, that is totally inclosed in a octagon, is 0.9480607501454566 percent of the perimeter of the octagon around it."
I see it now, I put percent after the actual ratio.
The circle that is inside or inclosed in a octagon, has a circumference, that is 94.8060.....% of the length of the perimeter of the hexagon around it. And the ratio is 0.948060......
That was not what had me confused, that is just a typo, I was afraid I messed up the order of the ratio. Ha-ha. Good catch. I wish you would have just said something. In my mind percentage and ratio are equal terms. I am looking at .948060 like .948960/over perimeter of 1, the percent of circumference to perimeter. I just forgot to display it properly. In my mind it instantly translates to a percentage.
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?)
-
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?)
Never said I measured my wheel to six digits of accuracy, that is just the number necessary to display the actual ratio of the roll in the base 9/10 system we use. In other words I took the measurements from the length and diameter of the wheel, and when I divided them, it came out to that number, because of the base 9/10 system we use.
Sincerely,
William McCormick