
Neat picture but I need some explanations. What are the steps I follow with a compass and a straight edge to make the figure you have there?
First build an angle Alpha
Second draw a line perpendicular to bisettrice
Third make a circle with the diameter equal to part of line inside the angle
Forth join upper point of circumference with the bisettrice and the lower with the same
(forgive my english)
Fifth make a round using a compass pointing in upper side of cirle with an aperture above halfh segment,another with the same aperture from point of the bisettrice on the left;join the intersection and you reach 1/4 Alpha.
Sixth you have points on the left of circumference like A B C D E
Seventh join A B D and the center
Eight join A D
Ninth Build a segment parallel to bisettrix (the + simbol...)
Tenth find the half of the segment.
Eleventh join the half with the center,allonge over circumference and you reach 1/3 Alpha...
OK I'll give this a try. Did you come up with this? Have you done the math to prove it?
I can find the center of the small circle but how do I know its radius?
The small circle isn't important
OK I'll try and proceed without it. May I ask why it is in the figure if it is not needed?Originally Posted by clearwar
What I'm trying to do is code you your method in Mathematica. I can then test it numerically if things look good I'll crank through the algebra to prove your method works.
I don't remember why i draw it. I hope you will be able to define an easy formula allows us to reach a third of a dimension. Cause trisecation is useful to divide also a segment too. However i tried to build an exagon in the circle moving parts of the "segment" rapresenting a third of the angle,in autocad,and it
fits perfectly... :?
Here is an animation from what I have coded up in Mathematica. It does not look like the angle has been trisected. The is the angle between the orange lines and the is the angle between the red lines. Be sure to look closely and make sure I have done your reconstruction as you suggest. If the angle was trisected then the ratio would stay at 0.333333.
Without attempting to understand your construction, I can tell you that it is impossible to trisect a general angle with an unmarked straightedge and a compass. This is proved using the methods of field theory, and if often done in classes on Galois theory. Here is a Wiki article explaining the mathematics.Originally Posted by clearwar
http://en.wikipedia.org/wiki/Angle_trisection
Just wanted to edit this. I found I was in error. This only works for bisecting angles, not trisecting angles. In later posts I show a way to trisect and check a trisection. Using just a compass and ruler, and some paper and writing tools.Originally Posted by DrRocket
You just bisect the bisects, till you have twelve bisects.
http://www.Rockwelder.com/Flash/Trisect/Trisect.htm
Then each four bisects create a bisect, one third of the whole angle. That is what I would do in the field. Or I would use cadd to create a straight line with measurements along it that we could duplicate easily in the field. And then just snap to those marks.
Hiram Abiff would do that in his sleep either way.
Sincerely,
William McCormick
Your first approach doesn't work. (Bisecting alone can only get angles of the form . That is some multiple of the original angle, divided by a power of 2.)
Your second approach doesn't use just a compass and straight edge. There are many things cad programs can do that classical constructions can't, but that's entirely beside the point.
Onestly I began from a 180° angle. After i draw red lines.
180° angle is trisected,point Alpha in the center and you'll note it.
I repeat,i built an exagon from 3 points I've reached.
It would work for any angle
No it won't. Please take a look at the link that I provided. It is quite impossible to trisect a general angle using only an unmarked straightedge and a compass.Originally Posted by clearwar
Understand this clearly. It is NOT simply that mathematicians have been unable to find a method to trisect a general angle using a compass and straightedge. They have PROVED that it is IMPOSSIBLE to do so.
Originally Posted by MagiMaster
If you divide an angle into twelve parts. At four parts will be one of the trisection points. We actually do stuff like this, for real life stuff. There is another trick in real life that allows you to not even make all the dividing points just the ones I did in the movie. You can use a compass to make just two more arcs to get the other trisection.
Later I will make another movie just showing what we would do. I was showing in the other movie, the whole story. It may have been confusing.
I made the movie so you could get the idea that it is just like dealing with fractions. Finding the common denominator. No big deal at all to trisect an angle. With just a compass and straight edge.
We often use a surveyors tape as the compass, and a piece of chalk to mark the cement. We even use the surveyor tape pulled taunt as a straight edge as well. Highly accurate.
So actually you need a Surveyors tape, and a piece of chaulk to do it in most cases. Sometimes you need a surveyors tape and some marking stones or rocks, soda cans, whatever is lying around.
There are other ways to do it, that I have used as well. I will try to gather them and demonstrate those too.
Sincerely,
William McCormick
It's true that if you can divide something in to 12 parts, you can divide it into 3. The problem is, you can't divide something into twelve parts with just bisections.
The point of this thread is trisecting an arbitrary (given) angle with only an unmarked straightedge (not a ruler) and a compass. There's any number of ways to do this if you use things besides just a compass and straightedge, but that's not the point.
You are right, my video is in error. Did you catch what I did. If I was actually cutting the metal I would have caught it. But doing it to let someone see it. I did not catch what I did.Originally Posted by MagiMaster
It is hard to trisect an angle. We use dividers and they are not always accurate for trisecting. I guess we normally don't do three way divides.
I will check up on this and get back to you. My error.
Sincerely,
William McCormick
Ok there is an easy way...
1° Build an angle Alpha
2° Find Bisectrix
3° Draw a line perpendicular bisectrix
4° Build a circle in point reached with bisectrix and line draw
5° Build a circle with same aperture on the first and one low the first
6° Draw a line from upper and lower extremity of 3 circles to right or
left to encounter the angle.
7° follow the painting...
This work for any angle smaller of 180°
That won't work because you didn't specify the radius of the circle (or rather, how to construct the radius).
You open the compass as you like...
There's no way, according to your drawing, for an arbitrary circle to work the way you want it to. If I'm mistaken, can you show why?
Edit: Ok, wait. I kind of see what you're saying. I don't think that trisecting the base of an isosceles triangle will trisect it's corner though, but I could be wrong.
Edit again: Yeah, I just plugged some equations into MATLAB and trisecting the base doesn't trisect the angle, although it's fairly close for very small angles.
Yes I can. Consider a circle like an entity or object and axign it a value. Copy this value 2 times over and down the first we obtain an entity formed by 3 equal primitives. Interacting with these 3 primitives with a target (Wide of an Angle) we can perform an abstraction concerning another primitive entity (the random angle).
Not really random in truth,cause it must be >0° and <180°
I don't think so. Base of an isoscele triangle could rappresents its corner apertureEdit: Ok, wait. I kind of see what you're saying. I don't think that trisecting the base of an isosceles triangle will trisect it's corner though, but I could be wrong.
What i have done is rectifing an half of circonference divided in 3 points and moved through bisectrix till the extremes "touch" the borders of the angle
ok,my mistake
I put it in an edit above, but I just checked in MATLAB. Trisecting the base of an isosceles triangle does not trisect its angle. This should be easy enough to see with a very wide, flat triangle.
do it again without using a 180 angle, cause that's the form we CAN trisect. use an arbitrary angle, not 180, and then check what you did.
You can trisect a 90 degree angle and a 45 degree angle with only a compass and an unmarked ruler.Originally Posted by DrRocket
I am working on ways to do the other angles. I can do it now to within a couple thousands of an inch.
This is done with just a compass and ruler, exactly no tolarance.
Sincerely,
William McCormick
Originally Posted by clearwar
You only trisected the straight line. Not the arc that needs to be trisected. I have been doing this stuff for years. The more I think I know and I know a lot, the less I actully know or can remember.
You can do anything though. I am sure of that. I don't always have to do it without a ruler though. But it is nice to be able to do so.
You can trisect a straight line using three random segmants that are slightly larger then the total segment you want to divide. Then put the arbitraty length segment, between the segment you would like to divide on an angle to the line you want to segment.
And just run perpendicular lines down to it.
This has some cool tricks on drawing.
http://www.Rockwelder.com/geometry/G...letriangle.pdf
Sincerely,
William McCormick
The correct mathematical statement is that you cannot trisect an arbitrary angle using only a compass and a straightedge. In particular, you cannot trisect a 60 degree angle.
Certain angles certainly can be trisected. Again, what the theorem is claiming is that some constructable angles cannot be trisected, not that all of them can't. For example, you can trisect 90 degrees obviously, since 30 degrees is a constructable angle. And since you can bisect any angle, you can construct 15 degrees, and therefore trisect 45 degrees. There is nothing new or profound about these observations.
Some of the geometric figures create natual angles. The pentagon creates the angle 36 degrees from one point of the pentagon to the opposite sides ends or intersections.Originally Posted by salsaonline
From there you can create a 36, 18 and 9 degree angle.
I will find a 20 degree angle.
I am trying to figure out how to draw a nonagon, that creates 20 degree angles.
Sincerely,
William McCormick
False.The correct mathematical statement is that you cannot trisect an arbitrary angle using only a compass and a straightedge. In particular, you cannot trisect a 60 degree angle.
False.The correct mathematical statement is that you cannot trisect an arbitrary angle using only a compass and a straightedge. In particular, you cannot trisect a 60 degree angle.
Finding 3 equal segments you can approch angle and realize a trisection.
Point in origin and draw a circle as long as ORIGIN,third point you have
Point in 3rd point you have and make a round as long as 3rd point,4th foint
Join 3rd point with the point you have on the big circle
Allounge a line from points on big circle and 2nd and 3rd versus origin
Make a normal circle wide as ORIGIN,point on wall of angle. Join points.
To those who are shouting "false", just be aware that you are disagreeing with a proven theorem in Galois theory.
When your conclusions run up against over 100 years of mathematical experience, you are best advised to proceed with extreme caution.
Math fell out of favor for a while. There were always the craftsmen that just looked at the castle with puzzlement as to how the castle dwellers could be so far from reality.Originally Posted by salsaonline
When the craftsmen with so little time, so little working capital were doing the things that the castle claimed one day the high taxes they were collecting would allow them to do.
By using a line approximately the size of the section we want to divide. We can certainly create an exact trisection of any angle from all the experiments I have been doing. I have tested this enough to call it plausible. It is actually easier then bisecting a bunch of times.
http://www.Rockwelder.com/Trisecting1/Trisecting1.html
Sincerely,
William McCormick
Just learn some Galois theory or shut up.
Originally Posted by salsaonline
I am saying it is not a question anymore. I can do it. I have done it. Many times.
That 2/3rds intersection is something, a place that only exists where it exists. There is no other place that you can find that 2/3rds mark. No other angle makes it.
Since you can create the trisection on a straight line with an arbitrary length increment. It is highly plausible that you can recreate it exactly, as possible with only a ruler and compass. I have checked other angles and they all show this ratio, exactly. The best pens or pencils, in the world have a deviation of more then what the cadd program is checking to.
Base ten calculations will create larger differences then the 2/3rds, mark. It is a very accurate mark. Many times more accurate then anything we can duplicate in the field.
I told you Hiram Abiff, would do this in his sleep.
Sincerely,
William McCormick
Then would you care to present a careful, stepbystep guide for trisecting an arbitrary angle with only a straightedge and a compass?
Originally Posted by MagiMaster
Sure. Give me a little time to create something. I just got home.
Sincerely,
William McCormick
Don't make a video, just list the steps. It'll be a lot easier to follow if I can take it at my own pace.
This shows that there are a lot of ways to cut up an angle using just a compass and ruler. I would bet there are thousands of ways to accurately cut it up.
Could they be off mathematically to some very small fraction of a fraction of a fraction of a dgregree? Because I do not know, and cannot prove it, I would say sure.
However for anything you could want to build with a compass and ruler, this is more then accurate enough. This is accurate to six places of accuracy. Done with a very accurate program. General Cadd.
http://www.Rockwelder.com/Flash/Tris...isecting3.html
Growing up watching the guys, they used to use other methods that are not as accurate however they are good enough for most building applications.
Sincerely,
William McCormick
Can you just list the steps? I can't follow your video, in part because you made it after the figure was already drawn.
Originally Posted by MagiMaster
http://www.Rockwelder.com/Flash/Tris...sectshow1.html
You know what they say "a picture is worth a thousand words". I guess I could write this out. But when I started to write it out. I assessed the effort of writing it, and the amount of omissions that I would probably be guilty of if I wrote it. And decided that a movie was safer. Less confusion for all.
I use the computer to speed the little things you can do with a compass and ruler. There is nothing in that movie you cannot do with a compass and unmarked ruler.
Sincerely,
William McCormick
Having looked at your video, it is not at all clear to me what you are doing, although you do give the appearance at least of having trisected the angle.
There are several places in which it looks like you perform some sort of arbitrary subdivision. But that aside, it's worthwhile to point out that there is nothing stopping you from constructing angles that are arbitrarily close approximations to a true trisection of a given angle. My guess is that is what you are doing in this videothat is, that you have discovered a clever way to approximately trisect the given angle.
If that's the case, pat on the back. But, strictly speaking, that wouldn't be a solution to the "trisection problem" (which indeed has no solutions if you believe Galois theory is errorfree). To give an analogy, finding a rational number that is arbitrarily close to sqrt(2) is not the same as finding a rational number that is equal to sqrt(2).
Well, if you won't make a list of steps, can you at least make a stepbystep video. Draw a pair of intersecting line segments with an arbitrary angle between them, then start the video and show your construction.Originally Posted by William McCormick
Hope you are joking about sqrt(2). Quadring circle I will obtain a formula of what Pi is, related to sqrt(2). *Knowing* Pi as circumference divided by diameter of the circle,we can obtain a manner to admit existence of a rational value to define sqrt(2)...
Are you claiming that is a rational number? It appears that you are doing just that.Originally Posted by clearwar
You seem to be having a serious problem with understanding some basic facts about mathematics and mathematical proofs.
It has been PROVED that it is IMPOSSIBLE to trisect an arbitrary angle using only a compass and an unmarked straightedge. It is PROVED in every introductory class in abstract algegra that it is IMPOSSIBLE to represent as the ratio of two integers. These FACTS are not open to intelligent debate.
I suggest that you stop spouting nonsensee and learn a bit of elementary abstract mathematics. I suggest that you start with an introduction to abstract algebra. You will see there the proof that is not rational and you develop the basis for a study of Galois theory by which the impossibility of trisecting an arbitrary angle or solving the general quintic polynomial by radicals is proved.
If you progress a bit further you will also learn that is an algebraic number while is transcendental, and hence that any relationship between the two must also be transcendental in nature.
I trisect lines, in the movie, perfectly legit and feasible with total possible precision, with a compass and ruler.Originally Posted by salsaonline
Sincerely,
William McCormick
What I said.Originally Posted by MagiMaster
I have to wonder if this isn't an April Fools joke... It's hard to tell on forums.Originally Posted by clearwar
Come on guys this is grade school geometry. There is nothing you cannot draw with a compass and ruler.Originally Posted by MagiMaster
http://www.Rockwelder.com/Flash/Tris...sectshow2.html
Sincerely,
William McCormick
This is a simple question to determine. Are there proofs that it is impossible to trisect an arbitrary angle? YES. It is silly to say otherwise.
There are all sorts of mistaken ideas about math and here we see another. It might be instructive for those that haven't had the exposure to math to take some time and learn a little more before demonstrating things that do not work.
http://mathforum.org/dr.math/faq/faq...construct.html
This link is easy reading.
It would be instructive for those doing constructions to write up proofs that they are in fact trisecting angles. Forget giving the steps. That just dumps the proof on another person.
Time to do the proper thing. Do the proof. If your proof is correct then think of the glory. You'd demonstrate that some important and often studied proofs must be wrong. You could be the next Einstein or Copernicus and cause a revolution in math. Think of the money, the fame, the glory, and the math groupie hotties!
William: Once again you are muddling engineering with math, or, in the present case, technical drawing with math.Originally Posted by William McCormick
It has been explained to you that there are mathematical proofs that no angle be exactly trisected using compass and/or ruler  it is part of the socalled Galois Theory.
While, in general, I dislike appeals to authority, nonetheless, I go with DrRocket; I have open on my desk a proof  a mathematical proof  that no angle can be trisected in the way you claim. But since you are unlikely to understand it  or even give it houseroom  I shan't bother showing it. But ask away, if you want
Mod hat on Unless anyone other that William strongly objects, I propose locking this thread.
Dissenters?
W Obama
It is not quite true that NO angle can be trisected with a ruler and straightedge. It turns out that some angles can be trisected. What is true is that one cannot trisect an arbitrary angle in that manner. It is for instance rather simple to construct a 30 degree angle, and that permits trisecting a right angle.Originally Posted by Guitarist
Also, what I have quoted is not an appeal to authority. It is simply a reference to wellknown and rigorously proved theorems. No element of an opinion by any sort of authority is required. It is all rigorous mathematics.
Except for proving a lot of modern theories wrong, the rest is a good reason not to prove it. The hotties quickly become nagging bimbos. And Einstein was a fool in my opinion.Originally Posted by hokie
It is proven to six digits. If you trust math and computers. In that cadd program, those six digits are the worst it gets. The actual engine that the cadd program uses is much more exact then six digits. But six is all they guarantee.
Certainly no one I know can use a compass and a straight edge to draw better then six digits of accuracy. A pencil line is off by more then that. And that is what math is about.
Most of the super accurate machinery is in the ten thousands of an inch range. There are projectors and equipment that use projection to enlarge error so it can be seen. That equipment could see past six digits.
My inaccuracy if there is any is in the millionths. I could not imagine how else to prove it. Then how I just did.
Sincerely,
William McCormick
I as well vote to lock this thread. It is simply useless to discuss this matter with those that can't and more importantly REFUSE to understand this issue.
Agreed. As usual, William, if you want your ideas to be shown more respect, you need to make some effort beyond emotional arguments.
In this case, it's possible (in a sense) that you have some construction that trisects an angle. It's more likely that you have a construction that approximately trisects an angle. In either case, you would need to provide enough information for your construction to be recreated and you'd need to provide rigorous math to show exactly what your construction was capable of.
Since you refuse to do anything even remotely related to math or science, for whatever reason, you shouldn't be surprised when no one believes you've done what you say you've done. Your excuses about why you won't aren't helping.
Finally, the argument about six digits is a straw man. Sure, six decimal places may be pretty insignificant when we're talking about inches, but not so much when we're talking meters (less so when talking about miles). Besides, "good enough for government work" or "good enough for an engineer" isn't good enough in math.
As for the thread, yeah, lock it. It's degenerated too much. I might like to see something on Galois Theory, including how it can be used to prove the impossibility of this, but it'd probably be better to move that to another thread.
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