
Proofs without Words and Beyond - adgasf
https://www.maa.org/press/periodicals/convergence/proofs-without-words-and-beyond-introduction
======
adenadel
If anyone is interested in proofs without words, my undergrad advisor has
published a bunch of them

[https://community.plu.edu/~edgartj/#pwws](https://community.plu.edu/~edgartj/#pwws)

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Dylan16807
What exactly is the first diagram supposed to be proving about angle
trisection?

If it's trying to show a way to solve the problem, it's kind of missing the
point of doing it with just a compass and straightedge. There are much simpler
tools you can add that let you trisect an angle. And _the device depicted
doesn 't actually work_, because when you try to align the ends the middle
points will poke through the paper.

If it's trying to argue that you need more tools than compass and straightedge
it's just nonsense because you could make an analogous picture with a bisector
yet angle bisection is trivial.

~~~
svat
All “proofs without words” require some imagination (and they can be extra
frustrating if one doesn't see the point quickly). In that sense, each one is
like a puzzle. I quite enjoyed this one.

All it's trying to prove is that the shown device works, and correctly
trisects any given angle. (The proof, if I had to put it into words, is
something like: let the outer bars be A and B, and let the middle bars be X
and Y. Because of the hinges, X is exactly halfway between A and Y, and Y is
exactly halfway between X and B, so X and Y trisect the angle.)

Here is a 33-second video:
[https://www.youtube.com/watch?v=sxwMGcshJI8](https://www.youtube.com/watch?v=sxwMGcshJI8)
— though it's an animation and the device is different, you can see there
won't be any problem with poking through the paper or having to bend it out of
shape. I find the depicted device elegantly simple and find it hard to imagine
something significantly simpler; what are some of the much simpler tools you
mention?

~~~
Dylan16807
The device in your video is designed differently and would work without any
fudging. It's key that the writing point is at a 90 degree angle to the device
spines, and each writing point has a different length.

> I find the depicted device elegantly simple and find it hard to imagine
> something significantly simpler; what are some of the much simpler tools you
> mention?

If you're starting off from the basis of "compass and straightedge" (and pen),
then all you need is to upgrade your straightedge with a fixed pair of marks.
If you're starting from nothing, all you need is a square of paper; origami
can trisect an angle.

~~~
svat
> It's key that the writing point is at a 90 degree angle to the device spines

But the device in the picture is exactly like that one! The three main spines
will always occupy exactly the same positions in either the device in the
picture or in the device in the video; only the interior hinges used to
enforce it are different. The device lies flat on the paper in both cases, and
the writing point (your pencil or whatever you stick in the holes) will of
course be at 90° angle to the device spines.

Actually a better search found names for these devices: see terms like
“Sylvester's Link Fan” and Laisant's Link #2 here:
[http://www.takayaiwamoto.com/Greek_Math/Trisect/Linkage/Link...](http://www.takayaiwamoto.com/Greek_Math/Trisect/Linkage/Link_Laisant.html)
— and a much better video showing almost exactly the device in the picture:
[https://www.youtube.com/watch?v=uTQVi7zskGU](https://www.youtube.com/watch?v=uTQVi7zskGU)
(except for quadrisection instead of trisection). (And the article does
mention the Laisant reference; hadn't noticed it earlier: see p.38 here:
[https://files.eric.ed.gov/fulltext/ED058058.pdf#page=49](https://files.eric.ed.gov/fulltext/ED058058.pdf#page=49)
)

> [...] straightedge with a fixed pair of marks [...] origami

Fair enough; these devices are indeed simpler to construct, but the actual
process of trisection of a given angle is not as simple as “plonk the device
on the paper, open it up to the angle, and there, it's already trisected”. Nor
is the proof that you've successfully trisected as straightforward.

