
What to Do When the Trisector Comes (1983) [pdf] - dang
http://web.mst.edu/%7Elmhall/WhatToDoWhenTrisectorComes.pdf
======
schoen
It's amazing how similar this is to my experiences answering the
correspondence for

[https://www.eff.org/awards/coop](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](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.

~~~
empath75
> It's even possible that this is the computer age's equivalent of the
> trisection and cube-duplication problems...

P=NP

~~~
schoen
Yikes, I wonder what kind of e-mails Scott Aaronson gets.

~~~
cousin_it
He's also in the line of fire for quantum mechanics, so yikes indeed.

------
kazinator
We have a way to bisect angles with compass and straightedge. Repeated
bisection of angles yields any fraction of an angle that can be exactly
expressed in binary. For instance The 0.100101 fraction of an angle can be
found since it corresponds to 1/2 + 1/16 + 1/64, each of which are constructed
by individual repeated bisections.

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.

:)

~~~
GregBuchholz
Robert Yates wrote a book exploring approximations like this and other means
of trisection (graduated ruler, linkages, etc.).

[https://www.google.com/#q=the+trisection+problem+yates](https://www.google.com/#q=the+trisection+problem+yates)

------
tokai
"By the way, if you have any trisections or any works of mathematical cranks I
would be most grateful to have copies. I have sought such things out for
years, and my collection of analyzed trisections [...] is in it second
hundred."

I would love to see that collection. Anyone knows if anything like that is
available online?

~~~
schoen
Per someone else's mention elsewhere in the thread, it looks like Dudley
eventually published a book based on it: [https://www.amazon.com/Trisectors-
Spectrum-Underwood-Dudley/...](https://www.amazon.com/Trisectors-Spectrum-
Underwood-Dudley/dp/0883855143/) and also [https://www.amazon.com/Budget-
Trisections-Underwood-Dudley/d...](https://www.amazon.com/Budget-Trisections-
Underwood-Dudley/dp/1461264308/) (whose title was probably inspired by De
Morgan's "A Budget of Paradoxes").

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.

------
mindcrime
Money quote:

 _" 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...

------
empath75
For people that are curious:

Why Trisecting the Angle is Impossible:
[http://www.uwgb.edu/dutchs/pseudosc/trisect.htm](http://www.uwgb.edu/dutchs/pseudosc/trisect.htm)

~~~
RangerScience
Yes! I was wondering if it'd been proven impossible or not; I was (briefly) a
trisector in middle school when they told me no-one had done it, but all my
efforts amounted to were a deeper exploration of geometry. Which, at the time,
was a pretty good result.

------
pavel_lishin
If it's possible to trisect a line segment (1), why isn't it possible to
trisect an angle by drawing an arbitrary line that intersects it, trisecting
the segment thereby generated, and connecting the two points to the angle
origin?

(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...](http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Lehman/emat6690/trisecttri%27s/triseg.html)

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!

~~~
RangerScience
I suspect this is because the conversion from cartesian to polar coordinates
doesn't line up...? (it does line up for the midpoint?)

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.

~~~
ThrustVectoring
>will the ray intersect the curve 10% along it's length?

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.

------
pavlov
_" What is more infuriating than the condescension of the ignorant?"_

That quote from the article effectively summarizes the dynamic that powers
present-day Internet comments.

~~~
dang
I read this article years ago as a math undergrad and never forgot the title.
Re-reading it today I realized how well it applies to certain kinds of
internet commenters. But really the article is just a pleasure to read.

------
dahart
One of my favorite games of all time is the compass-and-straightedge simulator
"Ancient Greek Geometry":
[https://sciencevsmagic.net/geo/](https://sciencevsmagic.net/geo/)

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! ;)

~~~
pvg
An achievement originally unlocked by a 19 year old Gauss. The badge is a
17-pointed star on the base of his statue in Brunswick.

~~~
dahart
Indeed, and I cheated and used Gauss' construction; I could barely follow it,
I'd never come up with it on my own. :P

------
breadbox
In addition to publishing a book-sized collection of failed angle trisections,
Underwood Dudley is also the author of "Mathematical Cranks" (978-0883855072),
which gives a broad and fascinating descriptions of all sorts of examples of
the species. It remains one of my favorite books for the wide range of
examples and a few insights into how crankery develops in otherwise sensible
people.

------
theoh
This article would seem more generous if it sketched a proof of the
impossibility (using field extensions). Since it was worth proving for the
first time in the mid 19th C I don't think it's appropriate to make fun of mid
20th C folks who didn't have access to Internet or mathematical expertise. I
get that they're cranks but it's understandable that they hadn't got the memo.

Also would have been nice to mention
[https://en.wikipedia.org/wiki/Neusis_construction](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.

~~~
mikeash
The trouble is that they _did_ have access to mathematical expertise, in the
form of replies to the letters they sent to mathematicians, and pointing out
the proof that it was impossible did not deter them.

~~~
theoh
I think Dudley's attitude is regrettable. He uses the word "evil" in reference
to the incorrect ideas they have, and the time and money wasted by
mathematicians in replying to the cranks. He is more concerned with laughing
at them than correcting their misapprehensions.

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).

~~~
mikeash
It sounds to me like he displayed great patience and diligence for a long
time, and the new attitude of "evil" arose after observing that the former
approach essentially never worked.

He can't be concerned with correcting their misapprehensions if he _can 't_
correct them.

