THERE IS A TIME FOR REFINEMENT. THEN THERES A TIME FOR REVOLUTION.
For those of you who aren't math geeks, there's a long running battle between educated mathematicians and inventive idiots like me over dividing an arbitrary angle into 3 equal angles using only a compass and straight edge.
Sounds as trivial as bisecting an angle, which you probably learned in grammar school, but apparently it is impossible. Greek mathematicians started trying thousands of years ago and Pierre Wantzel finally proved it can not be done in 1837.
Yet the battle continues!
A steady stream of trig neophites, unaware of the vast amount of high powered brain hours that have failed to produce a true trisection, get sucked in by the apparent simplicity and blinding promise of fame & fortune. I fell into the vortex myself in late 2010 at the Skeptic Society and Marilyn Vos Savant forums. In spite of embarrassing myself with early simple minded misconceptions, I forged onward to more complex embarrassing misconceptions. After several months and dozens of failed attempts, I basicly ran out of ideas and enthusiasm, but it was still simmering at the bottom of my brain like a ruined reactor core, occasionally belching up a glowing notion that would always turn out to have some fatal flaw.
Then on Oct 29 2011 Joe Keating posted a link to Approaching the Impossible in the Skeptic's Forum and I was back on the job with a vengence!
After a few quick tests I realized that Joe was on to something. If not the actual solution, something so close that I could not see any error. What eventually panned out from several weeks of trials and assaults on the CR4 engineering forum is that his trisector falls short only in that it is not chopping a given angle into 3, but rather creating 2 angles simultaniously at an exact 3 to 1 ratio. What the antagonists still fail to realize is that this is literally a half step away from the real thing.
To help clarify the dynamics, inspire and test ideas, I made 2 copies of a device for Joe and myself. Click on the pik below to see them in action (and also dispell any claim that its merely tripling an angle).
What the astute will realize is that it becomes a matter of determining that top line. And since the H point in Joe's diagram is on that line, any other point on the line finishes the job.
For quite a while I was using Photoshop and an actual compass and straight edge with pencil and paper to test ideas. Very tedius, and, even tho Photoshop can make far finer & precisely located lines & circles than the physical tools ever could and allows you to zoom in for microscopic views, it was actually slowing me down. FINALLY I got the C.a.R. program which enabled me to test and refine trisection ideas properly.
The result is that I have come up with a procedure that gets the angle within billionths of a degree to start with and has a way to refine this angle by leaps of 10 or even 100 times closer. A feature inherent in Joe's trisector is that the circle from point D will cross line E C above line B C if the angle is shallow and below if it is too steep. So instead of having to duplicate the angle you have come up with 3 times within the original angle in order to see the error, or continually bisecting your angle for only an infinite series of half as much error, this provides a way to see the error where it is and 2 ways to quickly draw a much closer angle.
Granted, this is really only like getting into a CaR and driving toward an unreachable destination instead of walking there, but I'm still experimenting with many other ideas presented by Joe's Trisector.
Here are some Photoshop drawings with the instructions written in Nooalf:
TRiSeKV3INSTR is just the instructions. It will be helpful in making sense of the CaR files and will download in a few seconds even if you're on dialup. If you don't read Nooalf yet, go here: Nooalf
TRiSeKV3LO is as low rez as I can make it without it becoming illegible.
TRiSeKV3FLaT is a crisp 1000 dpi picture if you want to see good detail. It's 8 meg, so go do something else for 5 or 10 minutes if you are on DSL.
Here are the latest versions of TRiSeK V3 in CaR:
The MQVUBL one can be adjusted just by using the move tool (the dot & curved arrow button at the top) to slide point A along SRKL 1. It should show up as a red X. If not, its the little circle in the middle of the red circle on the black diagonal line. Hold the left mouse button on it to move it around. It's virtually impossible to pick a specific angle like this, so I also made USiNDaNGL (assigned angle) so you can right click on the main AB angle and type in whatever angle in the edit box. I named everything the same as in the Photoshop drawing on the 'alias' line in the edit box. You will notice that the oRK V1 location is different from what is shown in the Photoshop drawing.
The CaR software is free and is very ergonomic. I was able to start using it in 5 minutes! There's way more to it than I've learned how to use so far. My compliments to Rene Grothmann & his team of programmers. (Or did he make the whole thing all by himself?!)
Mathematicians can still claim that the angle is not being perfectly trisected. But if it is not possible to measure the deviation, what does that mean? Better computers and more expensive software can probably find the error and refine the angle many more times than my setup can. I'm still experimenting here - coming up with refinements and variations on this proceedure and completely new ideas. Joe also occasionally sends me a new idea to test, and who knows how many young wippersnapper trigonometry savants are out there who could find an exact method. Because I'm not amazingly smart and yet have come so close, I suspect that it can be done.
Page originally posted on January 7th 2012. Edited and updated as warranted
Having fun yet? If you want to talk about this, I suggest joining Marilyn's or the Skeptics forum and jump into the topics already there.
It really pays to do the CaR tutorial. Today I discovered the move tool, thus eliminating a bunch of work building new test angles from scratch. I edited the above stuff from January 7th because the .zir Car files I had listed don't download and they were outdated versionz of the TRiSek anyway.
Wierdly, TRiSeK V3 works to about 104.4o. Substantially less accurate, but I would have expected it to completely collapse even slitely above 90.
I've been working on finding shorter processes and hopefully more accurate. I've come up with 3 or 4 substantially different ones that are reasonably accurate without refinement and I am of course still hoping for a revelation that leads to a perfect trisection.
This one SIMPL8 has only 8 circles and 8 lines to get within a thousandth of a degree. It relies on a variation of the refinement process. It gets more accurate as you approach zero, but is over .2o off at 90. Of course, you could divide angles in half or quarter or 1/8 etc., run this process and then multiply the rezult the same number of times to get extreem accuracy. Plus, the refinement process can be used at either or both stages.
A few gliches I've encountered with CaR: 1. Sometimes the angle measure will flip to the other side, so you see 145 instead of 35, for example. All you have to do is find the first angle point and slide it back over to the other side of the base point. 2. With a new version I'm working on the software will flip a circle over when the angle is over 76.5, making the trisek angle not be a trisek angle anymore. 3. The measure of the angle in the edit box got stuck at 9.something, so now I only get to see 10 decimal places.
At about 1:30 this morning I was pretty sure I had it. Every angle I tried was exact. No decimal places, just 20o, 15o, 24o, etc., even 30o for a 90, which is where TRiSeK V3 is least accurate! But now, (around 4 pm) the computer has changed its mind. It now shows errors after 13 decimal places.
Here's how it went:
For the last week I've been grooving to the sites & sounds of my YouTube favez on shuffle and working on what I call the 'crooked line perspective' in which the AZ line is considered as 2 or more segments in order to show direction and amount of error and hopefully lead to a way to precisely correct this crookedness. Then a few days ago Joe sent me this:
Completely different, yet looking as 'almost there' as the first one! I did a quick CaR construction to verify it: Joe's New Trisek. Move the red X up & down.
Amongst my first thoughts was that it could possibly be combined with the TRiSeK V3 to produce 100% unquestionable trisections. The key feature is the new connection to the top line by the OM vertical line. I was on my way to doing something more complicated wen I decided to try a line from this new O point to the CD oRKZ- intersection. Like majik it showed perfect numbers.
V4MQV .......... V4USiND
So, after watching Magnum Force, delivering a job and getting some sleep, here I am again looking at a reeeeeellly close almost insted of perfection. Some how during the intervening hours, 15o degenerated into 14.999999999999918o. 30o is now 30.00000000001o. In the daylite, my Nobel prize in the booming field of applied trigonometric twangularificationology is really a tarnished video arcade token. Perhaps the ghost of Wantzell was granted a few moments of influence to kibosh this threat to his accomplishment by the Adjustment Bureau, tweeking my computer's brain just enuff to thwart me again. OR, quite the opposite and far more likely, the little gremlin in my computer was fooling me with 'perfect' angles to get a few laffs.
Need to get Kolchak on the case....Waitaminit.
Just had another idea. And wuznt I working on a more complex arrangement before I tried that shortcut?
None of the more complicated ideas worked. Tried a very convoluted contraption that was perfect at 54o, but useless at any other angle. Spent the last 4 days making the Photoshop drawing for V4. I also tested it in a new way that goes far beyond what the software shows in numbers.
Gotta get some sleeeep. I've been in this chair so long I'm starting to waddle like Roger from American Dad.
First, download one of these so you can get an idea of what I'm yammering about:
V4 INSTR instructions only - no diagram.
TRiSeKV4 LO Good enuff if you aren't planning on framing it.
TRiSeKV4 PRINT pre-rotated for your convenience.
TRiSeKV4 HR 500 dpi Photoshop version.
This is actually less complicated than the V3, but I broke the steps down to their real individual steps instead of combining similar concurrent ones. You will notice there are several lines and arcs less than in V3. Simplification AND improved performance is always a good sign in R&D that you're getting close to a finished product!
Some of the errors indicated on the 24th were bogus - likely a rezult of having a bunch of erzatz features crowding up the construction during development. Errors shown now above 81o and below about 27o are likely real. But within the range of about 33o to 72o, the best test I have devised shows the construction to be perfect. (see the drawing) The trouble is that it relies on the accuracy of an angle drawn with the fixed angle feature of the software, which does all its calculations at only 16 decimal places maximum. The test is millions or billions of times more accurate than that, so the way the program calculates and draws things becomes the limiting factor. The only reliable confirmation I have is that it's not likely to be a coincidence that the angles match up perfectly. But what about the errors it does show below 27 and above 84? Is it real or the software miscalculating the V4 process? The test has produced inconsistent rezults, sometimes showing completely different circle sizes for the same angle. And something wierd happens around 76.5o - the RS line transitions from below the fixed test angle, which originates at point A, to above it. So somewhere around 76.5 it should be right on the line, yet the software will not do it. In one attempt this morning, it completely freaked out and erased everything! Luckily, restarting it brought back the last saved version.
As I have been informed, all CAD software runs at the same limits of precision, so I can't just buy something to take this to the next level. The goofball fraze 'you can't get there from here' actually makes sense in this case. I am ruluctantly forced to learn trigonometry now. Learning stuff! What a pain in the brain!
Joe sent me a boomarang.There was an incident.
I have a headache, but I'm still working on refinements. What I'm looking for now is an alternative or variation for the CJ JK lines; something that stays at a consistent relation to the true trisek line on SRKL4
Picked up some math books at from the library. Marilyn Vos Savant's book about Wile's proof of Fermat's last theorem. Also has some related stuff about cranks like me trying to do impossible stuff. Also got an actual math book to maybe learn something.
I succeeded in improving the CJ JK lines. Now it reads perfect over the whole range. Beyond actually, since it goes to 103 with 10 decimal places correct. At 104o it has a 5 at the end instead of 7.
I named it V4.1, like the software convention I've alwayz hated. Makes sense since I'll probably come up with a better refinement in a week and then some new revelation will lead to a trisector worthy of a V5 designation.
V4.1MQV .......... V4.1USiND
I'm not going to do the Photoshop thing for this. Instead, I'll email the CaR file to anybody who's interested. Breifly, the diference is the blue CJ JK lines are based on little circles now which keep the L point closer to where it needs to be.
Another thing I came up with this week is pretty obvious - put a test circle at a million kilometers! I tried a billion, but the software fails somewhere before that. It shows things there, but points jump around and things dont line up. A million is good enuff to show errors anyway, still assuming the fixed angles are precise.
I was testing ideas at 9o, 45o, 60o and 90o. V4.1 is 187 quadrillionthso under for a 9o (2.999,999,999,999,813). At 45 the test radius is .1586mm at point A and .0877mm at 1 million km. At 60o its .382mm and .189mm. At 90o its .5813mm and .5059mm. An interesting detail is that the line is below the A point till about 76.6o, then shifts above it. Same with V4, and I tried to zero in on an angle where its right on, but couldn't. That would be the angle at which the blue AN1 line would be a perfect trisection.
Seems I have already exceeded the computer & software's accuracy. I ran the V4.1 on my new Toshiba NB505 netbook and it produces different numbers. The 9 degree came up with 3.000 000 000 000 068o. Very significant that it's on the other side of 3o. Then I made a new one on the old Dell with 1,000,000 radius circles instead of only 1,000 and at 9o it says 3.000 000 000 000 1186o.
Want more accuracy? WeL, YU KANT GIT XAR FRUM HYAR!
On Feb 1, xouper from the Skeptic's Forum challenged Joe & I to construct triangle CID starting with only HIJ (see Joe's first trisek sketch). His claim is that Joe's thing is really only a version of a marked ruler trisection and that we would find it impossible.
Here you go, xoup: Xouper's Challenge
Originally posted in Marilyn Vos Savant's forum about 20 minutes ago in response to naysayers:
I'm reading Ms. Vos Savant's book about Wile's proof of Fermat's last theorem.
Very revealing insight into the world of math. Not the absolute prestine clarity of logic us laymen assume. Instead, everything is based on uncertain axioms, other proofs that aren't 100% accepted, and mind bogglingly complex labyrinths of thought that defy any sort of visualization.
If the simple, fundamental idea of parallelism is in doubt, why would anybody think that this trisection impossibilty proof is unquestionable?
Think of it this way: If little old me can get to quadrillionths of a degree in a matter of months with a 16 step construction, how far could a few thousand cooperating real brainiacs get in 10 years with no limit on how many steps their construction could have?
Or: I'm just churning thru ideas here one at a time. An existing program or a specially designed one could do the same thing a trillion times faster.
All the other trisection attempts I've seen are very simple constructions. The basic assumtion on the entire subject is apparently that there should be some simple trick of geometry that produces a perfect trisection.
Why should it be simple? Why should it even make sense? Especially, why should it make sense to our limited intelligence? It only has to work.
Why would someone who is well educated in math expect a 'proof' that it works when much more fundamental concepts have no proof and others have proofs that fill books and aren't universally accepted?
My contention today is that a bunch of perfect trisection methods can be created, but 'proving' they are perfect will be impossible for humans.
Turned the computer off and was going to leave it for a week, but I couldn't get the 'filter stage' idea out of my head.
Since my computer can't refine the final line and the L point can't be improved, what if some sort of pre-refinement could work? A filter stage to purify the brew before bottling. I seemed to get a good result once, and then it was coming up with seriously lousy numbers, like only 10,000ths of a degree. I'd become confused with the steps and the computer seemed to be bogging down, so I thought a week's vacaation would help.
I couldnt pay attention to anything, so gave up last night and fired up the Dell. Ran the Registry Booster and started work on the filter stage.
It seems to work. 'Work' meaning that it seems to be more accurate than V4.1. Not sure, because of the limited measurment accuracy of the computer & CaR, but it was consistently showing above perfect measuring from one side of the angle and below from the other. (14.999999999995 & 15.000000000001 for example). At 45, The gap between the final line and the software generated fixed test angle was the same at 1 million km as at point A (.00311), and at 90, everything lined up - the 30 degree fixed angle, the 'pre line' from the filter stage and the final trisecting line. Gap = zero at A and the 3 lines were so close together at a million km that the computer couldnt line up any new points with anything there.
So I made a fresh construction. It doesn't measure the same as the prototype, but it's the correct steps. I made the html with more functions this time. Click on the left & right trisecting angles to see the different decimal places. You can replay the whole construction, zoom with the mouse wheel, measure other angles, add new stuff etc. Notice the invisible circle.
I would like to see how it does in other programs - SolidWorks, Pro-E, Vellum, other geometry programs - anything thats made to do precision drafting work. I can email the CaR file or step by step instructions if this example isn't enuff. JO 753 at ZoL inc.
Started reading Dr. Underwood Dudley's The Trisectors last nite. His stated purpose is to dissuade people from trying.
"Well, it didn't work." begins the addendum to the introduction for the second edition. A good laff in the intro usually indicates a memorable read
Underwood, my friend, you have only been stoking the fire.
Exchanged some emails with Mark Stark - a man who really knows his trig and created a trisector that is superior to mine back around 2000, 2002. I'm sure he'd like me to make it clear that he wouldn't call it a 'trisector' but a 'trisection approximation', however precise it is.
And it is fantasticly precise.
Mark Stark's Trisek. Right click on the black angle line to change it.
MS MQVUBL. Left click on the red X. Click on the forward/reverse tool to bring up the construction replay. (I only did the second reiteration on this one, so the numbers may not be exactly 3 to 1)
Good thing I didn't stumble across this before! I would probably not have bothered to do any of this. / Too bad I didn't stumble across this before! No telling what I would have accomplished with all the time I've spent on this. There's an idea for you, Underwood!
You will see immediatly that it's way simpler than any version of my trisek. It's also good all the way up to 180o and accurate to 13 and sometimes 14 decimals for everything above about 12o. It gets down to only 8 when dividing a 1o. Also, CaR seems to like it. It measures consistently no matter where you put the points on the lines.
The consistency gave me an idea. Its always under by about 15 quadrillionthso, so what if you run the procedure from both legs, double one of them, then go way far out and divide a chord line between the 2? The error would then only be the difference between a precise line trisection and the arc that it should be, which at that distance over that narrow of an arc would be like 1/16th of a gorram planck length, gorramit!(planck length = 1.616 199(97) x 10-35; a unit so small, the late great wild eyed detail maniac, Theodore Wildi, didn't bother to include it in his book, Units & Conversion Charts!)
It took me a few hours of trying, but I did it. I had to put a perpendicular line 10 million kilometers out to see a gap wide enuff to work with, and the computer couldn't line the points up correctly, at least what it showed on the screen. It kept freezing up & I'd have to Ctrl/Alt/Del to shut the program down. Same thing several times trying to measure it. Finally I got it to at least try, but it came up with numbers that don't add up. For a 90 degree, it came up with 30.000 000 000 000 018o from the final line to one leg and 59.999 999 999 999 986o to the other. Adds up to 4 quadrillionthso over 90o.
Mark said that not even 1000 decimals is good enuff, but I think some serious mathematicians must be thinking 'but maybe if....' about Wantzell's proof.
I'm about half way thru the 'budget' (catalog) section of The Trisectors. Not the best writing you will ever read, but entertaining if you're a mathematician or have trisectionitis. The truly amazing thing is how wide the variety of attempts are. An idea anybody should get from that is 'what if I combine this one that is always 3% over with that one that's always 3% under and bisect the 2 approximate angles?'
I'm getting an inkling of why Professor Dudley doesn't do an analysis on the more complicated trisections, even if they seem to promise greater accuracy. A relatively simple one tranzlates into a long equation, so what would my TRiSeK V3 look like?
I'm starting to feel like I'm done, or maybe 'cured' is a better way to say it.
I did a little test earlier. I drew 2 lines on either side of The final line in V5 half a millimeter apart - the width of the led in a fine line mechanical pencil. A look at the 3 lines 5 million kilometers out and the final line is still well within this width; noticably off center, but how would CaR draw it on another computer? Or how would it turn out with a different program? To put it in perspective for the non math geeks who have somehow made it this far down the page, .5mm is the width of 5 human hairs and 5 million kilometers is about 12 times farther than the moon.
On top of that, the TRiSeK V5, and many of the trisections in Dudley's book, could be refined with the same steps I did with Mark Stark's trisector. Even if a new rule were imposed that limits the action to the local construction, we still have at least Mark's & my refinement steps to take the accuracy past the end of the universe.
So I no longer feel the urge to jump on the computer whenever I get a foggy notion about lines and circles intersecting. Not every little geometry factoid I learn gets thrown into the TRiSeK blender. The molten corium has spread out, seperated, disapated and diluted into relatively inactive blobs.
I'm still quite interested and will likely update this page occasionally, but unlike many of the poor slobs in Underwood's book, I can get on with my life...
...and back to my REAL obsession: Nooalf. The Future of Spelling! And this impossible dream actually has practical value!
Latest page update: Febuary 16 2012