Here's one for your further amazement... http://www.davidfurlong.co.uk/sekes0.htm
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Circle Division
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One thing that you should keep straight (pardon the pun) is that theory and practice are often different. And they can be very different. I am well versed in basic mathematical principles; math was my minor in college. And I am somewhat proficient at or at lease know a little about how things can be made in the shop or in a metrology lab. There are profound differences.
One example: In Euclidean geometry there is the classic method of constructing a line perpendicular to a given line at a given point and using only a compass and a straight edge. This is rigorously proven to be completely precise and accurate, using the methods of the mathematician. Again, this is within the assumptions of Euclid. I am aware that there are nonEuclidean geometries but will ignore them in this discussion.
Now, all examples of this Euclidean construction are made on a physical piece of paper or on some other object. They are made with a physical straight edge. And they are made with a real, physical compass. And when this method is translated into an actual attempt to construct a "line" that is perpendicular to another "line" there are so many physical limitations that I would have difficulty in listing all of them here. These are just some of the most obvious ones:
The piece of paper or other object is NOT a perfect plane. It will have hills and valleys and other texture features.
The given "line" is not a real line. A real line has a zero width and therefore can not even be seen.
The straight edge is not actually straight. It will also have deviations from that state.
The "points" of the compass are not actual points which would have zero dimensions. They will have some real diameter at the depth that they are pressed into the paper. And even their actual tips will be rounded to some degree.
ETC!
Now, can this Euclidean method be used in the shop or lab? Sure, of course it can. But exactly how accurate it will be depends on a lot of factors, many of which can not be controlled without a lot of trouble.
If I were trying to create an accurate, whatever that means, right angle in the shop or in a lab I would not use lines scribed on paper or other objects. I would not use a straight edge, not even one of the rather expensive ones that are used in shops. And I would not reach for my compass.
One possible method that I would use in a shop or lab in order to make a somewhat accurate 90 degree angle would be to use my lathe to turn a cylindrical square. In making this cylindrical square I would not use any of the Euclidean tools. I would use tools like a micrometer and a dial indicator mounted on an appropriate base and a surface plate, which is the shop or lab version of a plane. And what I would have at the end of this process is not two pseudo lines but rather the flat end and the cylindrical surfaces of the cylinder square which would be at right angles to each other to a high degree of accuracy. Both of these surfaces could actually be seen and they can be used to align other things to that right angle relationship.
My point here is that the best techniques of the mathematician and of the machinist are and must be DIFFERENT. The mathematician is looking at points, lines, planes, straight edges, and compass as theoretical objects with the MIND'S EYE. The mathematician can idealize these things in the mind. And by working with these ideal concepts the mathematician can discover and prove things about an idealized existence. But any physical use of these things will be limited by the degree of accuracy which they can be constructed with.
On the other hand, the machinist is working with real objects and gets to choose those that will make the task (of forming a right angle) as easy as possible. The machinist is not limited by the assumptions of Euclidean geometry, but rather by the real world limits imposed by the tools and materials that are actually available.
This is only one example and in it I choose to refer to only one method that a machinist may choose to employ. There are other examples and other methods. But I believe they illustrate the idea that these are different disciplines and it is no wonder that different paths would be taken. The mathematician, with just a few sentences, completely assumes the perfection of the elements that he/she is working with. A point has only a location, no physical dimensions. A line has only length, no width. A plane is perfectly flat, etc. The machinist works for his/her entire life trying to get ever closer to that perfection that the mathematician does not worry about after just a few minutes of assumption. And the machinists, working in the real world, NEVER actually get there. For one it is a briefly stated starting point and for the other it is an end that can never be actually reached.
Paul A.
SE Texas
And if you look REAL close at an analog signal,
You will find that it has discrete steps.
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Originally posted by nickelcityfab View PostHere's one for your further amazement... http://www.davidfurlong.co.uk/sekes0.htm
"Fourth, Fifth and Sixth dynasties"
â€‹â€‹â€‹â€‹â€‹â€‹â€‹JRLast edited by JRouche; 02152021, 01:43 AM.
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Originally posted by Stargazer View Post
If you were "starting from scratch" (conceptually, not necessarly in real life) how would you relate circle division to making more precise linear measurements?
Conceptually thinking I though I said I dont like circles,. JR
I can tell you why in or on a dinner plate. See, food is big for my people. JR
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Originally posted by Stargazer View Postcircle division to making more precise linear
Two things are wrong.
Circle, Division and Precise. JR
Edit: Real quick! I am not trying to out smart anyone.
Its not my way and getting booted from 9th grade gives me more faith with some of our educators. (Whole story( JR
All I was saying is I like the circle more than the angles. JRLast edited by JRouche; 02152021, 02:40 AM.
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Originally posted by Stargazer View Postwhen you see geometry and mathematical objects come to life in the shop
WoW man! What kind of coffee are you drinking? Its the Kind! JR
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I can remember Dad making foundation layouts using string. He would have a loop of string with three small loops tied into it. The spacing between loops was 3, 4, and 5 units of lengths didn't matter what the units were. Hook one over a nail, stretch one side out and put another nail. Where the third loop came to defined a square corner. You can also use the same method to define a vertical if you have a known horizontal. The water level was good for setting a horizontal over a distance. I used the concept to align my friends pool table.
Then there's the plumb bob. A pointy weight hanging from a string. Great fun to hang it from a tree branch and swing the bob over a carefully laid bed of sand. Not so much fun getting ****t for playing with it. But interesting seeing what patterns develop in the sand. I didn't understand that until only very recently, actually.
Something else I found interesting was how to measure distance without a long tape measure. Find or make a wheel with a known circumference. Put a mark at one point and count the revolutions as you roll (in a very straight line) from one point to wherever you're measuring to. I used my bicycle wheel, and it was X number of revolutions from our house to the first cross road. I remember telling my dad there would be an error because with my weight on the bicycle there would be a flat spot on the tire and the radius would change. Circumference is pie are square, right? He said no, that wouldn't happen. Took me a while to understand that.
Talking about foundations of accuracy I wonder how a modern physicist would think of it. I mean, is there even such a thing as a straight line?I seldom do anything within the scope of logical reason and calculated cost/benefit, etc I'm following my passion
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Originally posted by darryl View PostTalking about foundations of accuracy I wonder how a modern physicist would think of it. I mean, is there even such a thing as a straight line?
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Originally posted by darryl View Post... Divide by 7 is interesting start with divide by 6. Then draw a line through two of those points perpendicular to the diameter line. Where that line crosses the diameter, set your compass from that point to where that line crosses the circle. That's your compass spacing to divide the circle into 7 parts. I'm sure this can go on and on. There is a way to divide by 5, but I don't remember it.
The method you're describing is visualized in this youtube video...
https://www.youtube.com/watch?v=mJfe4T7irE
The line QM, which the author claims is the chordal length for a septagon, has a length given by:
R * [sqrt(3) / 2] = 0.86603 * R
where R is the radius of the circle.
The correct chordal length for a septagon is given by:
R * [2 * sin(180/7)] = 0.86777 * R
================================================== ==================================================
Everyone knows that a regular hexagon can be easily constructed in a circle using only a compass and a straightedge. With the compass set to the radius of the circle, simply walk it around the circle striking off points on the circumference. There will be exactly six points. Connect these points with straight lines and the result will be a regular hexagon.
This raises the question of what other regular polygons can be constructed using only compass and straightedge? Obviously, three and four sides are easy and we know from above that six is possible. What about five or seven?
Karl Friederich Gauss, the German mathematical prodigy and genius, solved the generalized problem. He proved that a regular ngon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of *distinct* Fermat primes (including none).
A Fermat prime is a Fermat number of the form
Fk = 2^(2^k)+1
that is also a prime. The known Fermat primes are:
k Fk
0 3
1 5
2 17
3 257
4 65537
Not all Fermat numbers are prime. The list above includes all the currently known Fermat primes. For example, for n = 5, we have the Fermat number 4,294,967,297 which has the factors 641 and 6,700,417 and so is not prime.
Also, the restriction to *distinct* primes is important. A 9sided polygon cannot be constructed. 9 has prime factors 3*3 and, while 3 is a Fermat prime, ALL the factors of 9 must be DISTINCT Fermat primes for it to be constructible.
So, based on the above, a regular septagon cannot be constructed, but a pentagon can. A procedure for the latter is shown here...
https://www.mathopenref.com/constinpentagon.html
Last edited by mklotz; 02172021, 11:39 AM.Regards, Marv
Home Shop Freeware  Tools for People Who Build Things
http://www.myvirtualnetwork.com/mklotz
Location: LA, CA, USA
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'A true septagon can NOT be constructed with only dividers and straightedge.' Is this true? I'll admit, I never tested the concept, but if this is true then I have been misled. Perhaps a step was left out of the procedure? Perhaps the result was so close that for all intents and purposes is was good enough? I don't know. Maybe I'll pick up some posterboard, a compass and a straightedge and test it out
I seldom do anything within the scope of logical reason and calculated cost/benefit, etc I'm following my passion
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[QUOTE=darryl;n1928764]... Perhaps the result was so close that for all intents and purposes is was good enough? ...[QUOTE]
"Close enough" is not a synonym for "true" in mathematics.
Regards, Marv
Home Shop Freeware  Tools for People Who Build Things
http://www.myvirtualnetwork.com/mklotz
Location: LA, CA, USA
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This post demonstrates the DIFFERENCE between mathematical theory and practical shop methods.
In our shops we stop refining things and call it "GOOD ENOUGH" when we reach a point where the accuracy is good enough for whatever we are making or when we exceed the ability of our shop measuring tools to reliably detect any error.
In mathematical work, there are NO PRACTICAL considerations and they are talking about idealized concepts like points that have no dimensions, lines that have no width and are perfectly straight (whatever that may be), and planes that are perfectly flat. Their compass has ideal points  NO AREA.
In our shops none of those mathematical perfections exist. They can not exist because we are working with actual. physical objects, not theoretical concepts.
So, YES we can take a compass and draw a "circle" on something. We can then step off ANY number of divisions, using that same compass by starting with a good or not so good guess and adjusting that guess as needed. And if we are not demanding an excessive amount of accuracy and do not have a means of measuring the divisions to a greater amount of accuracy than the compass guessandtry method, then we can call it "perfect".
The mathematician does not, CAN not call such a method precise. They KNOW that no matter how small the remaining error is, it will always still be there. The mere fact that neither he/she nor anyone else can actually observe or measure that residual error is of NO CONSEQUENCE to them.
When you were told that you can divide a circle into any number of parts with a compass, that was a completely true statement for the shop environment. So go ahead, divide, mark, make chips, and use what you have made. It will probably work just fine, 99% of the time. Just do not ask a mathematicians to accept it as a mathematical method.
Mathematicians use mathematical methods. And we use practical methods that produce results that are within a small margin of error but are functional for whatever device we are making. And if you have had deep enough training in BOTH of these areas of human pursuit, you would understand the differences and when it is appropriate to use one method or the other. And HOW THOSE DIFFERENCES DO AND DO NOT MATTER.
Originally posted by darryl View Post'A true septagon can NOT be constructed with only dividers and straightedge.' Is this true? I'll admit, I never tested the concept, but if this is true then I have been misled. Perhaps a step was left out of the procedure? Perhaps the result was so close that for all intents and purposes is was good enough? I don't know. Maybe I'll pick up some posterboard, a compass and a straightedge and test it outPaul A.
SE Texas
And if you look REAL close at an analog signal,
You will find that it has discrete steps.
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Paul, you have a wonderful way with words. Very clarifying, thanks. I have always heard that one couldn't divide a circle by 7, but I would have to ask "Under what conditions?" The tools and techniques available to the HSM today, are so refined that error is either immeasurable, or negligible. I am a fan of George Thomas' writings about his "Versatile Dividing Head" and his system takes you down to one hundredth of a degree using extremely basic and simple techniques. Would I be able to measure a hundredth of a degree? probably not. But for my shop and what I do, it is an acceptable compromise. And isn't engineering just a collection of acceptable compromises?
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