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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.
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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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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?
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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 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 nickelcityfab View PostHere's one for your further amazement... http://www.davidfurlong.co.uk/sekes0.htm
"Fourth, Fifth and Sixth dynasties"
JRLast edited by JRouche; 02152021, 01:43 AM.
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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.
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You would love DaVinci's "virtruvian man". How the proportions of the human body are all in the golden ratio, and how they became the foundation of the Imperial system like 8,000 yrs ago. Some of which we still use today. Unfortunately, some of my favorite websites on metrology have "gone dark" especially about relating ancient units of measures and math that they did way back when. I'll have to go looking again.
EDIT: This is the guy I was remembering... Awesome writing, kinda like "Foundations" but it goes back to ancient times. Bummer their website is gone: https://en.wikipedia.org/wiki/Livio_Catullo_StecchiniLast edited by nickelcityfab; 02142021, 05:48 PM.
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Originally posted by nickelcityfab View Post
In the "Foundations" book, they used the master plates to create their master straightedges, which also happened to be master squares  they were square or box in crosssection. Darryl and Rich both mention one of the ancient methods of using just dividers. Either way is perfectly valid. It's amazing how much high school geometry applies to mechanical stuff  using just a dividers and a ruler or other straight edge, you can create whatever you need. Notice that either method still requires the use of a straight edge 
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Originally posted by Stargazer View Post
It masterful book. Thanks again for recommending it. I spent the entire weekend on the circle division section. When I googled the subject, I came across a post from last October on some Internet message board topic on the Whitworth Three Plate method where one poster wrote:
"A similar process can also get you a master square, although there are other ways to do it (such as via circle division)." https://news.ycombinator.com/item?id=24684365
That I can't see to understand how you go from a layout trick (circle division) and obtain a master square. What could he be referring to?Last edited by nickelcityfab; 02142021, 04:49 PM.
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Darryl in Post # 6 nailed it !
Remember fellows, the Protractor is a "New Tool"
In the 1800's and before , every angle was done using circles and chords , or circles and dividers
Since the circle could be divided , and the cord gave an absolute angle . Drawings made in those old days would state the circle diameter and the chord length and dividers were the common tool of the trade Or for example the drawing says " 6 bolts on a 20" circle " and the machinist would mark off the distance with his dividers ....
The chord for a right angle (90) with two equal legs is 1.414141414 ( x the leg)
Rich
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