What to Do When the Trisector Comes (1983) [pdf]
web.mst.edu
web.mst.edu
https://www.eff.org/awards/coop
However, I'm at a big disadvantage there because while general-form compass-and-straightedge angle trisections have been proven to be impossible, primality testing is known to be possible and we are actively soliciting solutions. (And even worse, we—like the Clay Mathematics Institute with its much more intellectually-important problems—actually do have prize money available, unlike the trisection problem.) It's just that several people per month refuse to believe that the problem is difficult and computationally intensive and was selected for precisely those reasons! Instead, they believe that they can solve it by pure reasoning, by finding a formula that generates prime numbers.
https://en.wikipedia.org/wiki/Formula_for_primes
It's even possible that this is the computer age's equivalent of the trisection and cube-duplication problems...
People I've corresponded with have exhibited all of the characteristics that Dudley describes here. :-(
The worst part is that (rather like the angle trisectors) a majority believe that mathematical results are tested experimentally and observationally, by trying them out and seeing if the results look right. For example, a lot of people have submitted formulas for primes which allowed them to find two or three or four primes (or simply two or three or four numbers that they didn't know how to factor, even though the factor program from coreutils can do it).
Also, like Dudley's correspondents who "think that the trisection is important" and even suspect it may lead to "practical alchemy" with transmutation of elements, we get claims from people who have found theories of everything, often including new physics theories and complete reinterpretations of mathematics and the foundations of mathematics, which they believe will be very important to society. One recent correspondent insisted that "10 is not a number, it is a concept".
Dudley described being sad that people have in some cases spent years of effort pursuing a mathematical impossibility. I often feel the same way because you can see people who are very, very passionate about mathematics and yet refuse to believe that mathematicians know anything about their subject or that one should study mathematics in order to make progress in it.
P=NP
https://www.amazon.com/Perfectly-Reasonable-Deviations-Lette...
We bisect to get the 1/2. Then we bisect the upper angle three times to get it down to 1/16 and we have 1/2 + 1/16. Then we bisect the upper 1/16 of the lower 1/16 that we just obtained, and bisect again. We now have the 1/64.
Now fraction 1/3 is 0.0101010101 ... in binary: 1/4 + 1/16 + 1/64 + 1/256 ...
Thus, though we can't trisect an angle, we have a process whose limit is a trisection.
All of the Immensely Important Engineering Problems which depend on trisection of an angle have a certain minimum precision that is required; if you meet that, you're good.
:)
I would love to see that collection. Anyone knows if anything like that is available online?
Edit: I think the second book there is actually just the first edition of the first book, and the title changed for the second edition.
"What is more infuriating than the condescension of the ignorant?"
This applies in so many domains. When you encounter these people, the best you can do is just to detach yourself from them (or them from yourself!) and move away as quickly as possible...
Why Trisecting the Angle is Impossible: http://www.uwgb.edu/dutchs/pseudosc/trisect.htm
(I've had that in mind ever since seeing (1), but haven't actually sat down with a straightedge and compass to try it; job and kid tend to get in the way :/)
(1) http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Lehman/emat6...
edit: someone commented, and then deleted their comment - but this explanation made it super clear as soon as you sketch this on a napkin:
> An easy way to see this is to try with an angle close to 180°
This makes it instantly and visibly clear why this doesn't work. Thanks!
Draw a quarter circle. Draw a line between any two points on the curve. Draw a ray from the center of the circle through the midpoint of the line. The ray also intersects the midpoint of the curve.
If you pick any other division of the line (say, 10%) instead of the midpoint, will the ray intersect the curve 10% along it's length? I don't know, but I don't think so.
It can't. The line and the curve are the same percentage of the way through at three points - the midpoint and both ends. Near the midpoint, the line is shorter than the curve per unit of arc, since it's basically parallel to the curve and closer to the center. Near the end, the line is longer per unit arc, since they're the same distance from the center but the line is slanted at an angle.
So if you plot out length travelled versus angle, you get two plots. The curve goes straight up-and-to-the-right, while the line will race above to start, peter out, cross over at the midpoint, and then catch back up at the end.
That quote from the article effectively summarizes the dynamic that powers present-day Internet comments.
I know there's a secret achievement for the 17-gon because I finished it once. I think there's a hidden achievement for the trisection too... see if you can do it! ;)
Also would have been nice to mention https://en.wikipedia.org/wiki/Neusis_construction though I appreciate that pre-Wikipedia, info was much more difficult to track down even if one was aware of it.
I prefer to see great patience and diligence in those cases where mathematicians do reply, and the facts I mentioned would be good to include (details of a proof, a slightly relaxed version of the problem (and other classic "impossible" problems) which does have a solution).
He can't be concerned with correcting their misapprehensions if he can't correct them.