Trisection of angles-similarly

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Trisection of angles-similarly

Post by bob boben » Sun Sep 04, 2011 12:08 pm

The solution of mathematical tasks in the ancient Greek
Trisection of angles
angle=0° - no solution
180°>angle>0° - general solution (consists of 4 parts)

the first part

http://ultraphoto.org/images/c8ay0p1o0iufughx3o81.png" onclick="window.open(this.href);return false;

1.ruler AB
2.ruler AC
3.caliper A-AD
4.ruler DE

http://ultraphoto.org/images/djwbdd6r48czpuqorsxf.png" onclick="window.open(this.href);return false;

5.caliper D-DE
6.caliper E-DE
7.ruler FG intersects DE the point H ,DH=HE

http://ultraphoto.org/images/49vx9rqw92tr0mynuc2.png" onclick="window.open(this.href);return false;

8.caliper H-HE

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Re: Trisection of angles-similarly

Post by JO 753 » Sun Sep 04, 2011 10:50 pm

I dont see the trisection.

I dont see wut 8 iz illustrating.
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Re: Trisection of angles-similarly

Post by Tom Cara » Sun Oct 30, 2011 2:18 am

JO 753 wrote:I dont see the trisection.

I dont see wut 8 iz illustrating.

Hi - I am new here.

Is this of interest: http://users.tpg.com.au/musodata/trisec ... _angle.htm" onclick="window.open(this.href);return false; ?

Tom Cara.

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Re: Trisection of angles-similarly

Post by Gord » Sun Oct 30, 2011 1:19 pm

http://en.wikipedia.org/wiki/Angle_trisection" onclick="window.open(this.href);return false;
Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. The "solutions" often involve finding loopholes in the rules, or are just plain wrong.
Pseudomathematics. Hmmm, I wonder if it would be possible to design a Halloween costume and go to a party as "pseudomath?"
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Re: Trisection of angles-similarly

Post by JO 753 » Sun Oct 30, 2011 11:07 pm

Hi Tom. Welcome to the forum. :D

Thats pretty kool, except wut you appear to hav dun iz created a pretrisection for an angle insted uv trisecting a given angle. But!!! I only tried out diagram 1, since I'm a little short on time.

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Re: Trisection of angles-similarly

Post by Tom Cara » Mon Oct 31, 2011 1:08 am

JO 753 wrote:Hi Tom. Welcome to the forum. :D

Thats pretty kool, except wut you appear to hav dun iz created a pretrisection for an angle insted uv trisecting a given angle. But!!! I only tried out diagram 1, since I'm a little short on time.

Thanks JO -
You are right.
Diagram-1 will do that as you can construct the random angle on the arc from point C or H or I or J, as you choose, with the result being identical.
Diagrams 2 and 3 are for definite selected or given angles.
The point of Diagram-1 is that wherever the angle sits on the arc FG, it is automatically trisected within the construction.

Tom.

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Re: Trisection of angles-similarly

Post by JO 753 » Mon Oct 31, 2011 4:10 am

I think youv got it!

I just printed your diagrams and used a compass to test it. It wasn't perfect, but I suppoze its just sloppyness in the print & everything. I'll try it later on the computer.

In case youre interested, the same subject came up at Marilyn Vos Savant's website. I assume therez math websites also.

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Re: Trisection of angles-similarly

Post by xouper » Mon Oct 31, 2011 8:04 am


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Re: Trisection of angles-similarly

Post by JO 753 » Mon Oct 31, 2011 12:06 pm

DaMIT! Joe Keating beat me to it! (I wuznt paying attention to the name earlier. Did you just find this, Tom, or are you Joe Keating?)

The construction I did with Photoshop came out perfect*. I read & understand diagramz 2 & 3 and see that they also work. Its like he haz created a virtual device for tripling/trisecting an angle.

I messed around with a bunch uv ideaz for a month or 2 & didnt kum up with anything that workt more than once!

I kinda gave up anyway, so its not like I had a real chance.

So, he gets to vizit Wantzel'z grave and tell him to bite hiz shiny metal ass on hiz way to accepting the Nobel Prize. I'll just go back to machining parts and working on my rusty old beater.


*I make a big 1000 dpi picture with 3 pixel linez, using circles with a dot in the center az the compass. Way more precise than a stratej, pencil & compass ever coud be. A proper CAD program woud be even more precise.

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Re: Trisection of angles-similarly

Post by JO 753 » Sun Nov 06, 2011 4:27 am

xouper wrote:404 Not Found error
Looks like he realized hiz mistake and deleted it all.

Joez 2nd & 3rd diagramz also do not work az the ill mannered old fart engineerz at CR4 revealed. But I found a way to apply Joez invention to the trisection by the rulez uv the game. TRiSeK

Say 'Hello Mr. Keating. Welcome to the forum. I am honored to meet a future Nobel Laureate' everybody. Tom iz Joe keating.

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Re: Trisection of angles-similarly

Post by JO 753 » Fri Nov 11, 2011 7:53 am

I think maybe now I got it. Maybe? I'm not confident since I wuz so sure before and wuz rong.

So hav a look at TRiSeK 2 and tell me if I'm still a demented crackpot.

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Re: Trisection of angles-similarly

Post by Gord » Fri Nov 11, 2011 8:07 am

JO 753 wrote:...and tell me if I'm still a demented crackpot.
You're still a demented crackpot. :good:
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Re: Trisection of angles-similarly

Post by JO 753 » Fri Nov 11, 2011 9:53 am

DAMIT!

I prepared myself for the bad nyooz by watching Idiocracy.

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Re: Trisection of angles-similarly

Post by Gord » Sat Nov 12, 2011 2:10 am

Really. You pronounce it "nyooz." That's just so wrong....
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Re: Trisection of angles-similarly

Post by JO 753 » Sat Nov 12, 2011 4:16 am

My dialect iz kicking your dialects asss! :D

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Re: Trisection of angles-similarly

Post by Gord » Sat Nov 12, 2011 6:00 am

Yer prawnunseeayshun sucks bawlz!
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Re: Trisection of angles-similarly

Post by Austin Harper » Sun Nov 13, 2011 5:33 pm

If it makes JO feel better, he can just imagine that I pronounce the "s" in "island" and the "gh" in "through."
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Re: Trisection of angles-similarly

Post by JO 753 » Sun Nov 13, 2011 10:12 pm

That woud be scottish, rite? :lol: TIS o BRE NIGT TQNIGT!

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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 14, 2011 12:24 am

This wun iz REAL close: NIR

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Re: Trisection of angles-similarly

Post by Poodle » Wed Dec 14, 2011 12:44 am

Stick to your guns, Jo ...

From the OED.

news, n. (pl.)

(njuːz)

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Re: Trisection of angles-similarly

Post by JO 753 » Sun Dec 18, 2011 2:05 am

Herez a refinementuv the last wun thats at least 17 timez closer*, if not actually perfect: TRiSeKV2

*by a method uv measuring by pixelz.

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Re: Trisection of angles-similarly

Post by Harte » Tue Dec 20, 2011 5:58 am

Gord wrote:http://en.wikipedia.org/wiki/Angle_trisection
Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. The "solutions" often involve finding loopholes in the rules, or are just plain wrong.
I actually worked through that proof in class when I was a math major at Georgia back in the late 70's.
Unfortunately, these days, I've gotten rusty. I'm well on my way to pseudomath, I suppose.

The proof was righteous, so this is all just a load of old crap, to quote Ringo.

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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 21, 2011 1:58 am

A churlish lout in another forum suppozedly found this method to be shallow by a tiny smijen for a 60 degree angle.
The fellow had previously claimed to find it nearly 3/4 uv a degree off, going to 10 dijits past the decimal point. The number he quoted, didnt come anywhere near any point, so hiz credability wuz in the dumpster, so I woudnt expect him to make a mistake again.

Today I tried a variation wich puts the W point at 1/3 insted uv 1/2 way frum U to line ES on SRKL 5. I cranked the rez up to 2000 dpi, about az much az my old computer can handle since the file iz over 671 megabites.

It turned out to be way off. I drew another line going to the 1/2 point again and it appearz to rite on. So maybe the guy iz just pulling numberz out uv hiz ass.

Why dont you try to prove its rong, Harte? Not much advanced math involved, so your rust wont matter.

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Re: Trisection of angles-similarly

Post by xouper » Wed Dec 21, 2011 5:18 am

JO 753 wrote:Why dont you try to prove its rong, Harte?
I has already been proven that the trisection of an angle cannot be done with only compass and straightedge.

See for example:
http://en.wikipedia.org/wiki/Angle_tris ... ossibility

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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 21, 2011 7:06 am

Yes, except that I'm seeing evidence that this 'proof' MAY be false

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Re: Trisection of angles-similarly

Post by xouper » Wed Dec 21, 2011 1:05 pm

JO 753 wrote:Yes, except that I'm seeing evidence that this 'proof' MAY be false
I'm not.

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Re: Trisection of angles-similarly

Post by Harte » Mon Dec 26, 2011 4:37 am

JO 753 wrote:
Why dont you try to prove its rong, Harte? Not much advanced math involved, so your rust wont matter.
How many people here know what it takes for a set to be considered a field?

I was unaware that the synopsis of the proof was at wiki. Brings back old memories.

You might be thinking that the proof is geometric - it's not.

If it were, the Greeks would probably have proven it.

Like I said, I worked through it before. There's nothing wrong in the proof - it's righteous.

That's another way of saying rigorous, but way cooler.

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Re: Trisection of angles-similarly

Post by JO 753 » Mon Dec 26, 2011 8:15 am

Several uv the regularz here are well versed in math.

I'm not wun uv them. Thats likely the main reazon I can devote time & effort to this. If I knew beforhand that trisection iz impossible, I woudnt hav dun more than chuckle at Joe'z post and woud not hav been able to see the significance uv hiz trisector.

I hav found that ignorance often leadz to new ideaz. You hav to manage your mind carefully. In my line uv work, staying off the beaten path iz essential.

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Re: Trisection of angles-similarly

Post by JO 753 » Mon Dec 26, 2011 8:28 am

A fairly obvious way to refine the AG line in the TRiSeKsUN V2 iz to draw a circle with radius X centered on the intersection uv SRKL 1 and line AG and then draw line AG2 frum A to the intersection uv the new circle and line CD.

I've dun this with the C.a.R. app and the AG2 line iz about 10X more accurate. This refinement can be dun indefinitely. after 6 refinements the error didnt show up az a number anymore, but it coud still be seen by zooming way in.

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Re: Trisection of angles-similarly

Post by JO 753 » Mon Dec 26, 2011 1:23 pm

I've come up with a way to refine the AG angle shown in the ANGL TRiSeKsN V2 drawing that works faster than simply redrawing the radius X circle on the new line. (I'm calling that circle RX3 for now)

This method starts with that, but insted uv drawing the new line (AG2) to the intersection uv RX3 and line CD, you do this:

Draw a circle from the intersection uv RX3 & CD at the radius uv the length to the intersection uv linez AG and CD. Draw a second circle at that radius from the intersection uv AG and CD.

Now draw a line frum the intersection uv RX3 and the bottom intersection uv the 2 new circlez. Draw a line from the intersection uv this new line and CD to point A. This iz line AG2.

This process can be repeated indefinitely. However at an 84 degree angle, the RX3 circle intersected with CD at the 28 degree point on the AG2 line, at least az near az the C.a.R. program coud show! I zoomed in so far that the computer coudnt move the image anymore.

On a 27 degree angle I repeated it a 3rd time to AG3 and it was perfect to the limit uv the program.

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Re: Trisection of angles-similarly

Post by JO 753 » Tue Dec 27, 2011 8:06 am

xouper wrote:I has already been proven that the trisection of an angle cannot be done with only compass and straightedge.
I believe theory iz not quite matching up with reality here.

Since I have found a way to do it beyond a measurable error, the only objection remaining iz that a trisection may be off. Seemz to me the word 'impossible' iz wearing off uv this particular challenge.

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Re: Trisection of angles-similarly

Post by xouper » Tue Dec 27, 2011 12:37 pm

JO 753 wrote:. . . I have found a way to do it beyond a measurable error . . .
Sorry, but "beyond a measurable error" is not a valid form of mathematical proof.

Your trisection must have zero error, and you must prove that the error is zero. And prove it for ALL angles.

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Re: Trisection of angles-similarly

Post by JO 753 » Tue Dec 27, 2011 10:49 pm

Thats wut I'm saying.

Az a practical matter, the trisection method works. You can trisect an angl to the limit uv the precision uv wutever tool you are uzing.

If you are uzing a cheap pencil holder stamped steel compass and a plastic ruler, this method will trisect the angl so that the actual solution will be within the thickness uv your line.

Lets say you had to trisect an angle to some insanely hi level uv precision and you had the ultimate lazer micro line & circle scriber set. You woud be able to do it with this method. Even tho a naysayer coud still say 'theoretically, it may be off', the theoretically perfect trisection line will be within the molecule thickness line you drew.

Or if you are simply calculating everything. If you can still calculate an error at 20 googleplx decimal placez, just run the refinement process again.

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Re: Trisection of angles-similarly

Post by xouper » Wed Dec 28, 2011 12:28 am

JO 753 wrote:Thats wut I'm saying.

Az a practical matter, the trisection method works.

You can trisect an angl to the limit uv the precision uv wutever tool you are uzing.

If you are uzing a cheap pencil holder stamped steel compass and a plastic ruler, this method will trisect the angl so that the actual solution will be within the thickness uv your line.

Lets say you had to trisect an angle to some insanely hi level uv precision and you had the ultimate lazer micro line & circle scriber set. You woud be able to do it with this method. Even tho a naysayer coud still say 'theoretically, it may be off', the theoretically perfect trisection line will be within the molecule thickness line you drew.

Or if you are simply calculating everything. If you can still calculate an error at 20 googleplx decimal placez, just run the refinement process again.
As I already stated, all that is totally irrelevant to a mathematical proof.

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Re: Trisection of angles-similarly

Post by xouper » Wed Dec 28, 2011 12:33 am

JO 753 wrote:Az a practical matter, the trisection method works.
As a practical matter, there are easier ways to trisect the angle if you remove the restriction of using only compass and straightedge. If the intent is to be practical, then such a restriction is pointless.

In any case, the original problem was not about being practical. It is about mathematical proofs.

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Re: Trisection of angles-similarly

Post by Harte » Wed Dec 28, 2011 5:52 am

JO 753 wrote:
xouper wrote:I has already been proven that the trisection of an angle cannot be done with only compass and straightedge.
I believe theory iz not quite matching up with reality here.
What you refer to here as "theory," (no doubt your misuse of the term here is purposeful and entirely self-promoting) is in actuality a mathematical fact. and not a "theory" at all.

In fact, it is your claim that is the theory here, and not a very well thought out one at that, given that you're claiming to have established the obviously unestablishable.

The fact that you don't understand that the thing is mathematically impossible is on you and no one else.

It starts, as I said, with learning what it takes for a set to be considered a field. Once you get that, you can progress on to the definition of constructable.

The rest follows, as it says on the linked wiki page, from the concept of reducibility and from the rational root theorem, both of which are relatively simple concepts taught in many high school trigonometry or precalc classrooms.


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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 28, 2011 9:40 am

Maybe about time for me to read the article.

I just came up with a variation that starts off over 100 times closer. That will make it only 1 or 2 refinements away frum the limit uv my software to show a difference.

Been messing with this for 2 months already!

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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 28, 2011 10:41 am

Wutevr happend to Bob Boben?

I just came up with a way that makes the last thing I thought uv junk. It adz a few more initial linez, but it showed up perfect with no refining steps. Only tried it on a 45o so far, but I hav to get sum sleep.

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Re: Trisection of angles-similarly

Post by JO 753 » Wed Dec 28, 2011 12:54 pm

Coudnt sleep. Paranoia that sumwum else will discover this and post it first!

I think even you are going to hav trouble with this wun, xouper!

It starts the same az TRiSEK V2, but insted uv creating point W:

1.Draw a line frum the left intersection uv SRKL1 and line CD, intersecting the top uv SRKL4, to line ES. Call this line JK. (Line UV and line ES need to extend up past where they are in the drawing)

2. Draw a line frum the intersection uv SRKL3 and CD to the intersection uv JK and ES. Call this line LM
(SRKL3 iz going to be replaced by a line to mark the midpoint between B and S in the next picture)

3. Draw 2 lines from S to where JK and LM cross UV. Call theze NO and PQ

4. Draw 2 lines frum A thru NO and PQ where they intersect SRKL5 to CD. Call theze linez AG+ and AG-.
( minus meanz the angle iz less than wut it shoud be, plus iz more)

5. Draw SRKL6 radius X frum the intersectionz uv AG- and SRKL1 and SRKL7 radius X frum the intersection uv AG+ and SRKL 1.

6. Draw a line frum the intersection uv AG- and SRKL6 thru CD to the intersection uv AG+ and SRKL7. Call this line segment R.

Where it intersects CD iz point G. Draw a line frum A to G. Angle AGC iz, az far az my software can see, 1/3 uv angle ABC.

I've only tried it with a 45 so far. Since the real point W seemz to move around, its likely that this level uv closeness will not hold up for all anglez.

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Re: Trisection of angles-similarly

Post by xouper » Wed Dec 28, 2011 10:03 pm

JO 753 wrote:It starts the same az TRiSEK V2, but insted uv creating point W:

1.Draw a line frum the left intersection uv SRKL1 and line CD,
Is that intersection called point C? If so, why not simply say "draw a line from point C . . . "?
JO 753 wrote:intersecting the top uv SRKL4, to line ES.
What do you mean by "top" of circle4?